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Here are the summaries of the dedication sections:
Edward A. Silver dedicates this work to Maxine, Michelle, Norman, and Heidi
David F. Pyke dedicates this work to Susan, James, Daniel, and Cory
Ad majorem Dei gloriam Douglas J. Thomas dedicates this work to Traci, Alison, Kate, and Maya
This summary covers key aspects of lot sizing and inventory management for individual items when demand is uncertain.

It discusses different definitions of stock levels, whether backorders or lost sales are allowed, and key issues resolved by inventory control systems with probabilistic demand.

It introduces classifications of items (A, B, C) and the importance of that classification.

It compares continuous vs periodic review and describes four common types of inventory policies  (s,Q), (s,S), (R,S), and (R,s,S).

It discusses setting cost and service objectives and approaches to establishing reorder points (s) and orderupto levels (S).

It provides examples of finding the reorder point s in a continuous review (s,Q) system using different demand distributions.

It presents decision rules for establishing s and S in a (s,Q) system based on minimizing various costs or achieving service levels related to stockouts, shortages, and fill rates.

It discusses implied costs and performance measures that can be used to evaluate inventory policies.

Finally, it briefly introduces decision rules for periodic review (R,S) systems as well. So in summary, it covers the key concepts, approaches, and decision rules for inventory management of individual items under uncertain demand.
Here is a summary of the key points from Sections 6.9  6.12 of the chapter:

The Review Interval (R) refers to the time between reviews of the inventory level in an (R,S) system. It determines how often replenishment orders are placed.

The OrderUpTo Level (S) is the target level the inventory is replenished up to in an (R,S) system after a review. It is meant to provide a buffer of inventory.

Common assumptions include stationary stochastic demand, fixed lead times, no shortages allowed. Demand is often assumed to be normally distributed.

The common derivation used to determine R and S is to calculate the expected inventory and the expected stockout costs based on the demand distribution over the lead time and review interval. R and S are chosen to minimize total expected costs.

Variability in lead times can be handled by either looking at total demand over the full probability distribution of lead times, or by combining the demand rate distribution with the lead time distribution.

Exchange curves can show the tradeoff between inventory investment and customer service for different (s,S) policies on a single item or across a portfolio of items. They are useful for choosing optimal parameter values.

Modeling issues with nonnormal demand distributions are also briefly discussed.
Here is a summary of the key points from Section 10.2 Deterministic Case: Selection of Replenishment Quantities in a Family of Items of the textbook:

The section considers the deterministic case where demand for each item in a family is known with certainty.

The objective is to determine the replenishment quantities for each item to minimize the total inventory and ordering costs.

It presents a heuristic decision rule that determines the replenishment quantities based on the ratio of unit holding to unit ordering costs for each item.

Items with a higher unit holding to ordering cost ratio are assigned a larger replenishment quantity.

It proves that the total cost of the heuristic solution is within a bounded factor of the optimal solution.

Appendix 10A provides the derivation of the results presented in this section.
So in summary, it presents a heuristic for determining replenishment quantities in a deterministic multiitem setting and analyzes the properties of this heuristic approach.

This chapter discusses material requirements planning (MRP) and its extensions like capacity requirements planning (CRP) and distribution requirements planning (DRP).

MRP is a system used for planning and scheduling materials and production in a manufacturing environment, especially where items go through multiple stages of production.

It aims to overcome the weaknesses of traditional replenishment systems by taking a closedloop approach that considers end item demand as well as production and inventory constraints.

The basic MRP procedure involves collecting information on bills of materials, inventory records, planned orders and demand forecasts to generate a material requirements plan.

A numerical example is provided to illustrate how MRP works stepbystep to determine requirements for materials and components.

The material requirements plan output from MRP is used for production planning, inventory control and purchasing.

Extensions like CRP and DRP were developed to also consider capacity constraints and distribution requirements.

Weaknesses of MRP like inflexibility to changes are discussed, as well as enhancements through enterprise resource planning (ERP) systems.

The chapter concludes with a summary of the key concepts around MRP and its extensions.

The book covers inventory management and production planning decisions as important components of a firm’s total business strategy.

Sections address traditional inventory control systems for individual items, special classes of items like slowmoving items, and coordination of inventory across multiple locations or firms in a supply chain.

Production planning topics include aggregate production planning, material requirements planning, enterprise resource planning systems, justintime production, and shortrange scheduling.

The 11th edition expands coverage of lowvolume/erratic demand items using the Poisson distribution, updates supply chain planning software tools, and expands the use of composite exchange curves to manage multiple items.

Additional derivations are included in appendices to provide analytical depth without complicating the core text.

The book is intended for business, engineering, and operations research students and is also useful for practitioners and researchers.

The passage discusses the importance of inventory management and production planning and scheduling.

It notes that while the service sector has grown, manufacturing is still considered vital by nations for several key reasons.

Food and machine tool manufacturing are important for national security, as countries need domestic capacity for food production and manufacturing tools.

The textile/apparel industry employs many lowskill workers, so it helps reduce unemployment.

Other industries like microprocessors, computers and automobiles are also vital as they drive innovation, productivity gains, and highskilled jobs.

Overall, while services have grown, manufacturing remains strategically important for economic and employment reasons. Effective inventory management and production planning are thus still critical functions.
This passage discusses the importance of inventory management and its impact on the economy. Some key points:

Inventories play a fundamental role in business cycles and economic fluctuations. During expansions, optimism can lead to excess inventory buildup which triggers recessions as inventory is unwound.

Inventory carrying costs in the U.S. exceeded $2 trillion in recent years, comprising a significant portion (810%) of GDP. Effective inventory management is therefore economically important.

Excess inventory levels have been linked to negative stock market reactions and poor firm performance. Inventory management affects firm value.

While manufacturer inventory levels have declined slightly over time, inventory has shifted more downstream to wholesalers and retailers due to factors like increased global sourcing and product variety.

Most firms do not fully understand inventory management complexities throughout the supply chain. Improved practices could potentially lead to 20%+ cost savings for many firms without hurting service.

The passage advocates approaches in the book to strengthen economies by helping firms implement better manufacturing, logistics and inventory management methods. This can add value and benefit the overall economy.

Inventory management and production planning impact corporate finance in several ways. Inventories are classified as current assets on the balance sheet. Reducing inventory levels lowers assets and can improve liquidity ratios like the current ratio.

Changes in inventory levels can affect the income statement. Increased or better allocated inventory can boost sales revenue. Lower inventory carrying costs reduce operating expenses. More efficient production scheduling also lowers labor expenses.

Inventory turnover is a key performance measure, calculated as annual sales divided by average inventory. Higher turnover means inventory is generating more sales. But it’s a simplified measure and too high a focus on it can be problematic if it leads firms to take nonoptimal actions just to increase turnover numbers.

In general, the finance function is concerned with how inventory and production strategies impact key financial metrics like assets, liquidity, revenue, expenses and overall corporate performance and profitability. Effective coordination is needed between operations/production and finance.

A company wanted to reduce inventory levels by ordering inventory in smaller lots and producing less. This led to smaller inventory levels initially but then caused service issues and stockouts as inventory depleted.

Smaller lot sizes increased costs and profits declined. Strategically, just focusing on inventory turnover numbers without considering other factors like customer service can negatively impact the business.

Inventory is a component of working capital. Interest paid to finance inventory increases operating expenses. Improving inventory management can positively impact profitability and shareholder value by reducing expenses and increasing sales.

Droms (1979) provides a nontechnical overview of key elements in financial reports like revenue, expenses, assets, liabilities, equity/shareholder value. Inventory ties into working capital, cost of goods sold, and operating expenses on the financial statements.
So in summary, narrowly focusing on inventory turnover without considering other factors like production lot sizes, service levels and costs can undermine profits and shareholder value over time according to this description. Inventory management links to the broader financial picture of the company.

In the 1950s1970s, low cost was the primary objective for most U.S. manufacturing firms. Quality rose in importance in the mid1970s1980s due to Japanese competition.

Quality, delivery, flexibility, and cost are the four main operations objectives. Quality has multiple dimensions like performance, reliability, etc. Delivery considers speed and reliability. Flexibility includes volume, new products, and product mix.

Ten management levers help achieve the objectives: facilities, capacity, vertical integration, quality management, supply chain management, new products, process/technology, human resources, inventory management, and production planning/scheduling.

The objectives remain constant over time but the emphasis shifts. Early in a product’s life, flexibility and delivery are crucial. Later, quality and cost take priority as design stabilizes.

Competition increasingly focuses on “timebased competition” through rapid new products/delivery due to commoditization of cost and quality among competitors. Levers like supply chain management become more important.
So in summary, it outlines the evolution of operations objectives, defines the objectives and levers, and notes how the emphasis shifts over a product’s life and with changing competitive landscape.
This passage discusses several key points about inventory management and production planning and scheduling:

It introduces the concept of “levers” or strategic management levers that relate to supply chain management. Key levers mentioned include inventory management, production planning and scheduling, vertical integration, and new products.

It notes that while new supply chain initiatives emerge, inventory management and production planning/scheduling continue to be challenging areas for many managers. Even with realtime data availability, many firms don’t utilize the data effectively.

Managers must understand the value of information and choose appropriate inventory and production planning policies for their situation. Goals of the book are to communicate principles to help managers provide insight in this area.

Manufacturing strategies should ensure consistency among objectives, policies for each lever, and among the levers themselves. Incentives/rewards need to match quality goals, for example.

Measures of effectiveness should relate to overall objectives and consider tangible and intangible factors. Cost is difficult to estimate and management cares about aggregate levels like inventory/service, not individual item optimization.

Decisions in inventory and production planning should not be made in isolation and interact with other areas like distribution, transportation, quality, and pricing.

The chapter provides an overview of strategic issues while later chapters will focus on designing specific inventory and production planning systems for different contexts.
Here are the key points about frameworks for inventory management and production planning/scheduling:

These problems are extremely complex due to the large number and diversity of stockkeeping units, demand patterns, supply modes, costs, etc.

Companies face tens of thousands or even hundreds of thousands of unique inventory items to manage.

Items differ in cost, physical attributes, storage needs, demand patterns, substitution/complementary relationships, etc.

Demand can occur continuously, intermittently, in large or small batches. It varies by item.

The basic decisions for each item are: inventory review frequency, reorder points/times, and order quantities.

These decisions must be made for thousands of items in a coordinated, systemwide way to meet company objectives.

Humans have limited cognitive abilities to comprehend and solve these complex, multivariable problems optimally.

Frameworks, aids, and models are needed to help organize information, identify relevant factors, and recommend solutions within the bounds of rational human decisionmaking.

Sections 2.32.5 discuss classification schemes, operational tools, and foundational inventory models that serve as frameworks to help address this complexity.
Inventory management and production planning decisions are complex problems that involve many interconnected factors. No single decision maker can rationally conceive of all the relevant details. Therefore, frameworks and conceptual aids are needed to help simplify and systematize these decisions.
Specifically, inventories can be functionally classified into categories like cycle stock, safety stock, anticipation stock, etc. to help managers better understand and control aggregate inventory levels. Computer systems and exceptionbased reporting can also help expand a manager’s span of control over vast amounts of inventory data.
Meanwhile, individual inventory items can be grouped into more manageable organizational categories. And theories should focus on identifying the most important variables to consider rather than attempting to account for all minor factors. Overall, the goal is to design decision systems that help complement, not replace, managers’ existing capabilities and approaches for dealing with these multifaceted production and inventory problems.

Land’s End, Dell, and other direct mail companies have negligible pipeline inventory because they deliver directly to consumers. Pipeline inventory will be discussed further in Chapters 11, 15, and 16.

Decoupling stock is used in multiechelon systems to allow decentralized decision making without every decision at lower levels immediately impacting higher levels. Inventory acts as a “boundary” between organizations/divisions.

Chapters 11 and 12 will discuss multiechelon inventory management and supply chain management, which aim to capture systemwide effects and manage them effectively.

Inventory can be classified into six functional categories based on organizational purpose: cycle stock, congestion, safety stock, anticipation, pipeline, and decoupling stocks.

Items are further classified using an ABC analysis based on their value of annual usage (Dv). Class A items account for the top 520% of items but 50%+ of total usage value. Class B accounts for most remaining items and usage. Class C receives the least attention.

The ABC classification forms the basis for designing individual item decision models, with Class A receiving the most attention and resources.

The passage discusses frameworks for inventory management and production planning, including ABC classification of inventory items and a productprocess matrix for classifying production systems.

For ABC classification, items are divided into A, B, and C categories based mainly on dollar value of the item. Class A items account for the majority of investment and are controlled closely. Class B items are also important but can be controlled more by computer. Class C items are numerous but account for a small part of investment; control systems for them must be kept simple.

The productprocess matrix classifies production systems into four types  job shop, batch, assembly line, and continuous process  based on differences in their product/market characteristics and layout. These differences impact the appropriate production planning and scheduling systems to use.

Production processes are discussed in the context of the product lifecycle framework, with an emphasis on adapting the functional areas and control aspects as a product moves through the stages of introduction, growth, maturity, and decline.

Grouping inventory items and using simplified decision rules can help manage the large number of class B and C items more efficiently.
That covers the main points regarding frameworks for inventory management and production planning discussed in the passage.

Process industries tend to have welldefined capacity that is limited by the bottleneck or pacing operation. Their output rate is relatively inflexible.

Job shops, batch production, and assembly lines have more flexibility in output rates and what can be produced. Their bottlenecks can shift depending on the workloads.

Process industries aim to run at full capacity continuously to maximize efficiency and output. This requires highly reliable equipment and scheduled preventative maintenance during shutdowns.

Changing capacity takes much longer for process industries due to equipment needs and regulations. Other industries can adjust capacity more incrementally.

Process industries tend to use fewer raw materials that are more variable. Assembly uses more materials that must be wellcoordinated.

Output is more standardized and products are grouped into families for process industries. Other industries offer more customized products.

The nature of operations, challenges, definitions of capacity, and maintenance approaches vary depending on the industry type and characteristics of production.
In summary, the key factor is that process industries have defined capacity limits centered around dedicated equipment and standardized product output, whereas other industries have more flexibility in capacity and customization.
Here are the key points about the frameworks presented:

The productprocess matrix maps products based on their lifecycle stage (variety to commodity) against production processes (job shop to continuous flow). It shows that products and processes should be aligned strategically.

Different production planning and scheduling systems are suited to different areas of the matrix. Job shops use rulesbased systems, batch processes use optimized production technology (OPT), repetitive manufacturing uses materials requirement planning (MRP) and justintime (JIT), and continuous processes use periodic review/cyclic scheduling.

The ease of associating raw materials and parts with production schedules depends on the matrix position, going from easiest in continuous processes to most difficult in batch/assembly to easier again in job shops.

Important cost factors for inventory and production decisions include unit variable/value cost (v), inventory carrying costs, setup/changeover costs, production run costs, shortage/stockout costs, and costs of adjusting production capacity/workforce.

The productprocess matrix, alignment of products and processes, and appropriate planning systems provide a strategic framework. Understanding relevant costs informs tactical inventory and scheduling decisions.

Inventory holding costs (GE) include the opportunity cost of capital tied up in inventory and the opportunity cost of warehouse space used for inventory. These costs are difficult to measure with traditional accounting systems.

The opportunity cost of capital is the return that could be earned by investing funds elsewhere. It is difficult to define in practice as alternative investment opportunities change daily. Companies set a fixed cost of capital as a policy.

Inventory often has mixed financing sources, each with different interest rates, so the true cost of capital should be a weighted average. Risk also affects the cost of capital.

The ordering/setup cost (A) includes administrative and production changeover costs like ordering, receiving, processing invoices, wages for machine setups, quality issues during production changes, and lost opportunity costs during setup/machine adjustment periods.

Shortterm insufficiency costs include expediting emergency orders, rescheduling, split lots, emergency shipments, substitution of less profitable items. Longterm costs are more difficult to measure and include potential lost customers, goodwill, and future business from unhappy customers.

The passage discusses different types of costs involved in inventory management models, including stockout costs, carrying costs, ordering costs, and system control costs.

Important variables that affect inventory decisions are also identified, such as replenishment lead time, whether the item is produced versus nonproduced, and demand patterns.

There are three main types of modeling strategies: 1) Detailed modeling and optimization of a limited set of decision variables (e.g. economic order quantity model). 2) Broader modeling with less optimization (e.g. MRP). 3) Minimization of inventories with little formal modeling (e.g. JIT/OPT philosophies).

Which strategy to use depends on the situation and whether a clearly defined optimization is possible given the complexity. Simpler models are generally preferable when possible.

It is important to incorporate important factors into models while keeping them as simple as possible to avoid costly erroneous decisions from incorrect modeling.

As inventory and production models become more complex, they require more managerial time, undergo longer design and maintenance periods, and have higher system control costs due to issues like personnel turnover. Considerable judgment is needed in developing appropriate models.

Little recommends models should be understandable, complete, evolutionary, easy to control and communicate, robust, and adaptive.

When modeling inventory and production situations, analysts should: use objectives consistent with organizational goals rather than pure optimization; develop heuristic decision rules rather than mathematically optimal ones; present results suitably for management review; start simply and add complexity gradually; analyze special cases first like two items before generalizing; introduce probabilities cautiously; and model to analytic decision rules if possible.

Cost measurement can be imperfect in practice. Cost accounting systems allocate fixed and overhead costs across inventory in various arbitrary ways. Job order costing tracks individual order costs more precisely but is more expensive than process costing. Valuation methods like actual, predetermined, and standard costing also involve allocation assumptions.

Variances between standard and actual costs arise because standards may not accurately reflect ideal or normal rates, or they have not been adjusted for process improvements. Positive variances can artificially inflate costs.

Predetermined cost allocations in cost accounting systems are usually arbitrary and may not accurately reflect the actual costs incurred by products. But they ensure all costs get allocated.

Absorption costing charges all overhead costs according to a formula like direct labor hours. Activitybased costing attempts to allocate actual overhead costs incurred by each product.

Direct costing classifies costs as fixed or variable. Fixed costs are not allocated to products. Relevant costs for decision making include future and opportunity costs.

Allocated overhead from cost accounting may not reflect relevant costs for inventory decisions. Top management may view carrying costs as policy variables rather than attempt explicit measurement.

Exchange curves can design inventory systems based on cost policy variables specified by management to achieve desired aggregate operating characteristics like inventory levels and order frequency, rather than relying entirely on explicit cost measurements.
Here is a summary of the key points about a production planning and scheduling system from the passage:

A major production planning study involves 6 phases: consideration, analysis, synthesis, choosing among alternatives, control, and evaluation.

The consideration phase focuses on conceptualizing the problem and studying organizational objectives, structure, inventory levels, current planning procedures, available resources, and appropriate modeling strategy.

The analysis phase involves detailed data collection on uncontrollable variables like item characteristics, demand patterns, lead times, and costs. It also looks at controllable variables like storage locations, reorder policies, production capacity, and prices.

The synthesis phase brings together the information from analysis to establish relationships between objectives, uncontrollable factors, and means of control, often resulting in a mathematical objective function.

The choosing alternatives phase manipulates the model/objective function to determine reasonable settings for controllable variables like order quantities, production levels, and decision rules for repetitive decisions.

The control and evaluation phases focus on implementing the planned system, maintaining it over time, and ongoing assessment and improvement.
In summary, it outlines a structured process for production planning system development including problem scoping, data analysis, mathematical modeling, alternative evaluation, and ongoing system management.

Inventory and production management systems should allow for sensitivity analysis to see how changes in input data might affect results. This gives managers a sense of how robust the decision is.

For a “oneoff” decision that is not repetitive, there is no need to develop a detailed framework for repeated use in the future. A single, adequate decision is sufficient.

During implementation of a new inventory or production system, it is important to monitor any transient effects as parameters are adjusted. This could lead to temporary increases in inventory levels, ordering rates, or stockouts that need to be balanced.

A phased, gradual implementation approach is usually better than changing all parameters at once to avoid overload during the transition period.

Physical stock counts are important for accuracy but can be disruptive and costly. Cycle counting, where items are counted during each replenishment cycle, is a better approach that rations resources more effectively.

Cycle counting inventory levels regularly is important for maintaining accurate records but often gets deprioritized, especially when business is busy. Having a dedicated team for cycle counting can help ensure it continues.

Estimating inventory amounts through weighing can be an effective alternative to literal counting for some product types.

Providing incentives for those responsible for counting can help reduce inaccuracies in inventory records from inaccurate physical counts.

Cycle counting effort should be allocated in proportion to annual dollar usage of items, but other factors like accessibility, size, and potential for resale on black markets should also be considered.

Stratified sampling techniques can help reduce required sample sizes for tasks like auditing inventories.

The passage discusses challenges with cycle counting getting relegated due to competing priorities when business picks up. It suggests designating a permanent team to focus on cycle counting yearround to help address this issue. It also notes that weighing can serve as an effective estimating technique for certain product types as an alternative to literal counting. Finally, it discusses how incentives and consideration of additional factors beyond just dollar usage can help improve cycle count accuracy.

The data provided annual usage (demand) in units and unit variable cost for various SKUs/items (labeled 0Z and 1A1T).

Total annual usage across all items was 1,570 units.

The top 20% of items in terms of dollar usage (calculated as annual usage x unit variable cost) accounted for approximately 51% of the total dollar usage across all items.

Carrying costs factors that may change with production lot size include qualityrelated costs. Larger lot sizes could lead to higher scrap/rework costs if quality issues aren’t detected until the full lot is produced.

For one example item, the fixed cost of ordering/setup (A) included items like purchasing order processing fees. This was included because it is incurred with each production run/order placed. Carrying charge (r) included storage costs like warehouse rent  this was included because inventory has ongoing carrying costs over time in storage. Unit variable cost (v) included direct material and labor costs.

Components ignored included overhead allocations and opportunity costs, due to difficulty separating for this single item.
Here are summaries of the key papers:

erger, R. J. 1983 discusses applications of singlecard and dualcard kanban systems for pullbased replenishment in justintime production systems.

Shank, J. K. and V. Govindarajan 1988 critiques cost allocation methods based solely on production volumes, arguing they fail to account for business risk.

Silver, E. A. 1981 reviews and critiques operations research models in inventory management, focusing on assumptions, formulations, and implications.

Silver, E. A. 1996 questions a revision to the productprocess matrix framework for classifying industries.

Simon, H. A. 1957 proposes models of rational and social decision making.

Singhal, V. R. 1988 and Singhal and Raturi 1990 and 1994 incorporate business risk into inventory models.

Spencer and Cox 1995 analyze the productprocess matrix and its application to repetitive manufacturing.

Swann 1989 discusses the importance of basics versus novelty in operations research.

Taylor, Seward, and Bolander 1981 classify process industries as different from discrete part manufacturing.

White 1969 considers infrequent but impactful contingencies in operational research models.

Zanakis et al. 1989 categorize and survey heuristic solution techniques.

Ziegler 1996 discusses software for automated cycle counting in inventory management.

Zipkin 1995 models inventory networks with planned inventory levels and feedback.
Here is a summary of the key parts of Section 3.63.9 from the passage:
Section 3.6 addresses how to deal with unusual kinds of demand, such as nonrecurring demands or spikes in demand.
Section 3.7 discusses encouraging and using human judgment in forecasting. It raises issues around incorporating subjective input from experts.
Section 3.8 talks about developing forecasts for special classes of items, such as new or substitute products.
Section 3.9 returns to tactical and strategic issues for forecasting, specifically around choosing between different forecasting methods or procedures.
In summary, these later sections discuss:
 Handling unusual demand patterns
 Incorporating human judgment
 Forecasting for specific item types
 Choosing forecasting methods
They address practical issues in applying different forecasting techniques for different situations, as well as leveraging subjective human input.

Egression refers to the time period t in a multiple regression model. Multiple regression includes two or more exogenous (external/independent) variables to predict the dependent variable.

Autoregression models the demand in period t as a function of past periods’ demands. For example, xt = a1 + a2xt1 + εt models demand in period t (xt) as a linear function of the previous period’s demand (xt1) plus an error term.

A leading indicator is a variable that changes before another variable it is correlated with. Multiple regression forecasts the variable of interest indirectly based on its association with leading indicators.

The BoxJenkins approach uses autoregressive moving average (ARMA) models to represent demand based on past demands and error terms. It involves selecting the appropriate ARMA model and estimating its parameters based on historical demand data.

Individualitem shortterm forecasting often uses simple models like simple/exponential smoothing that are inexpensive and practical for highvolume forecasts compared to more complex techniques. Common underlying models assumed include level, trend, and multiplicative seasonal models.

Exponential smoothing is a technique for forecasting time series data that changes gradually over time. It places more emphasis on recent data points compared to older ones.

Simple exponential smoothing assumes the underlying demand pattern has no trend or seasonality, just random variation. It uses one smoothing parameter (alpha) to update the forecast.

Holt’s exponential smoothing accounts for a linear trend in the data. It uses two smoothing parameters (alpha_HW and beta_HW) to update both the level and trend forecasts over time.

Brown’s exponential smoothing is a simplified version of Holt’s method that only uses one smoothing parameter, making it easier to apply when there is a trend but no seasonality in the demand pattern.

Exponential smoothing techniques are useful when the underlying demand consists of a basic level or trend, plus random variation over time. They provide a systematic way to use new data to update the forecasts while still relying more on recent observations.
The passage summarizes the HoltWinters exponential smoothing procedure for dealing with time series data that exhibit trends and seasonality. Some key points:

HoltWinters is an extension of simple exponential smoothing and Holt’s linear trend method to address seasonal patterns in the data.

It uses three smoothing constants: α for the level/mean, β for the trend, and γ for the seasonal components.

The updating equations compute new estimates for the level, trend, and seasonal factors each period based on the prior values and the smoothing constants.

The forecasting equation is the sum of the level, trend projected to the target period, and the seasonal factor for that period.

Brown’s method is a special case of HoltWinters where β is a function of α, simplifying it to use a single smoothing constant.

Initialization typically uses ordinary least squares regression on historical data to estimate the initial level and trend values.

The procedure minimizes the sum of geometrically weighted forecast errors, allowing more recent errors to have greater influence than older ones.
So in summary, HoltWinters extends Holt’s linear trend method to explicitly account for seasonality in the time series using three exponential smoothing parameters. The procedure iteratively updates the level, trend and seasonal factors each period to minimize a weighted forecast error metric.
Here is a summary of the information provided:

The data shows quarterly sales figures from Q1 2011 to Q4 2014 for product EDM617.

An initialization procedure using ratiotomoving average is performed to separate the trend and seasonal components in the historical data.

Moving averages are calculated and centered to estimate the level/trend for each historical period.

Seasonal factors are estimated by dividing demand by the centered moving average. These are averaged across similar periods in different years.

The seasonal factors are normalized to sum to 4 (the number of quarters in a season). This gives the initial seasonal indices estimates.

The data is deseasonalized using the seasonal indices to obtain estimates of the trended values.

A regression line is fit to the deseasonalized data to estimate the initial level (a0) and trend (b0) parameters for the exponential smoothing model.
So in summary, the initialization procedure uses moving averages and regression to separate trend and seasonality, in order to estimate the initial parameters for the HoltWinters exponential smoothing forecasting model.
Here is a summary of the key steps in the procedure described in Sections 3.4.3 and 3.4.4 of the passage:

The historical demand data for item EDM617 is used to initialize the estimates of the trend (a0 and b0) and seasonal (F) factors at the end of period 0 (end of 2014).

Linear regression is performed on the historical demand data, after adjusting for the estimated seasonal factors, to obtain the initial trend line parameters a0 and b0.

The initial seasonal factors F are estimated by calculating the average normalized demand for each period over the historical data.

Forecasts for future periods are made using the HoltWinters exponential smoothing model, which combines the estimated trend, level and seasonal factors. The forecast for a future period t is given by (at + bt)(Ftp).

When actual demand data becomes available for a period, the forecast error is used to update the level, trend and seasonal factors using the HoltWinters smoothing equations.

The updated parameters are then used to generate a new set of forecasts for future periods.

The process is repeated as each new actual demand data point becomes available.
The example uses data for item EDM617 to illustrate the initialization and forecasting process. The initialized parameters are estimated as a0 = 76.866 and b0 = 1.683. Forecasts are generated and updated as actual demand is observed.

The passage discusses conducting a search experiment to establish reasonable values for the smoothing constants in exponential smoothing models.

For representative items, the demand history is divided into two sections  the first is used to initialize the model parameters, and the second is used to evaluate forecast accuracy for different combinations of smoothing constant values.

The combination of values that minimizes a measure of forecast error across similar items is selected. Exact minimization is not required as the results tend to be insensitive to deviations from optimal values.

Large smoothing constant values (over 0.3) should be viewed with caution as they may indicate a trend model is more appropriate than a simple exponential smoothing model.

Reasonable ranges of values for the smoothing constants in simple exponential smoothing, Holt’s linear trend method, and Winters’ seasonal method are provided based on the literature and experience.

Measures of bias (average error), dispersion (standard deviation of errors), and accuracy (MSE, MAD, MAPE) are discussed for evaluating forecast performance and identifying opportunities for improvement.

The estimate of the mean absolute deviation (MAD) for oneperiod ahead forecasts is calculated as the average of the absolute errors over n periods.

The estimate of the standard deviation of forecast errors over a lead time (σL) can be obtained by converting the estimate of the standard deviation of oneperiod ahead forecast errors (σ1).

The relationship between σL and σ1 is assumed to be σL = Lcσ1, where L is the lead time in number of periods and c is an empirical coefficient.

To estimate c, forecasts are simulated for various lead times using a forecasting model like Winter’s method. The standard deviation of forecast errors (σL) is calculated for each lead time and divided by the oneperiod ahead standard deviation (σ1) to obtain estimates of c.

The mean squared error (MSE) can also be used instead of MAD to initialize σ1 and update it over time using exponential smoothing. The square root of MSE provides a reasonable estimate of σ1.
So in summary, it presents a method to estimate σL from σ1 through an empirical coefficient c, to account for the difference between the forecast update period and lead time when calculating safety stock.

The standard deviation of Lperiodahead forecasts (σL) relative to the 1period forecast standard deviation (σ1) is computed for lead times L = 2, 3, 4, 5, 6.

These values are plotted on a loglog scale, with log(σL/σ1) against log(L).

If this follows a straight line relationship, it implies σL/σ1 = Lc, or log(σL/σ1) = clog(L).

For the data shown, the line with slope c=0.6 provided a reasonable fit, implying σL = L0.6σ1.

This relationship could also arise if forecast errors at different periods are independent with a standard deviation of σ1.

However, the assumptions of independence and constant σ1 are only approximations. An analysis for each individual system is recommended to estimate the true slope c.
So in summary, the relationship between forecast error standard deviations at different lead times was examined to test assumptions and estimate parameters of the forecast error distribution. A loglinear relationship provided a good fit for this dataset.
This passage discusses two types of corrective actions that can be taken in response to inaccurate demand forecasts:

Changing the smoothing constants used in adaptive forecasting models. This involves making the parameter estimates adjust or “smooth” more quickly toward their true but unknown values. Various adaptive forecasting techniques are described that dynamically adjust the smoothing constants over time based on forecast errors. However, research also indicates these methods don’t always perform better than regular nonadaptive smoothing.

Human intervention to make significant changes. This occurs when biases are sufficient or smoothing remains high. It may involve manually adjusting parameter values or changing the demand model itself. Intervention is guided by analyzing recent forecast errors and demand trends to identify factors like temporary promotions or permanent market changes that impacted demand. Intervention can override forecasts temporarily or make longerterm adjustments to the model or parameters. Guidance is provided on when and how to incorporate human judgment into forecasting.

Judgmental input from subject matter experts is often needed to supplement statistical forecasting models, which do not account for certain important factors.

Factors external to the organization like economic conditions, regulations, competitor actions, and consumer preferences require judgment. So do internal factors like price changes, promotions, new products, and distribution changes.

The Delphi method, where experts privately submit forecasts and receive anonymous feedback, can help structure the judgmental input process.

Shortterm forecasting is the focus, but many ideas also apply to mediumrange forecasting.

Guidelines are proposed for structuring a committee to provide judgmental input, such as including different functional experts, developing individual forecasts privately first, and providing feedback on forecast accuracy.

Special classes of items like new products with limited history and intermittent demand items require customized approaches within statistical forecasting models. Estimation techniques are described for initializing new items.

Standard deviation of demand forecasts for new items cannot be statistically estimated since there is limited or no historical demand data. Models other than the one given in Equation 3.72 have been proposed in the literature to estimate forecast uncertainty for new items.

Demand that is intermittent (occurring infrequently) or erratic (high variability relative to the mean) is difficult to forecast using standard exponential smoothing methods. The Croston method forecasts the time between demand occurrences and the size of individual occurrences separately to improve forecasts for intermittent demand.

Replacement/service part demand depends on failure rates of equipment, which in turn depend on timelines of new/repaired equipment introductions and distributions of equipment lifespans. While the relationships can be stated, operationalizing them mathematically is complex.

Terminal demand tends to decrease geometrically over time. Plotting demand vs. time on a semilog graph produces a straight line that can be used to forecast future demand levels during the termination phase.

Statistical accuracy of different forecasting methods can be compared using measures like mean squared error, mean absolute deviation, etc. However, correlated forecast errors limit the ability to make strong statistical claims about relative performance. Combining forecasts can potentially outperform individual methods.

Research has shown that simple methods like equalweighted forecast combinations perform as well or better than more complex optimal combination methods.

Exante (outofsample) testing is important to evaluate forecast accuracy, not expost testing on the same data used to develop the model.

A landmark study by Makridakis et al. found that simple exponential smoothing and Winter’s method performed well for shortterm forecasts, while a more complex method by Lewandowski was best for longterm horizons. Ratiotomovingaverage deseasonalization worked well. Adaptive methods did not outperform nonadaptive ones.

Statistical accuracy is not the only consideration  forecasting methods should also be costeffective and support decision making. Simulation testing on system costs is recommended when choosing methods.

Robustness to data quality issues is also important.

While objective models often outperform subjective judgments on average, informed human input is still important.

Biases, lack of credibility/communication, and low organizational support can hinder forecasting. Contingency planning for errors is also important.

The company was trying to measure and forecast timeseries data (trends in demand, sales, etc.).

Six highpowered timeseries analysts who worked on forecasting were laid off.

This resulted in substantially lower overall costs associated with forecasting.

It also led to a better forecasting strategy, implying the new approach was more efficient and effective than relying so heavily on expensive analysts.

The goal of laying off the analysts was to reduce costs while still having an effective forecasting process. It appears the company was able to improve its forecasting strategy and lower costs by reducing the number of analysts.

Exponential smoothing and HoltWinters methods are commonly used techniques for forecasting time series data. Exponential smoothing can handle trend and seasonality with extensions like HoltWinters.

Judgemental adjustments can be combined with statistical forecasts to improve accuracy by incorporating domain expertise. Various techniques exist for combining multiple forecasts like averages or weighted averages.

Accuracy metrics like mean absolute deviation and mean squared error are used to evaluate forecasting models. Deseasonalized errors should be used to evaluate seasonal models.

Initialization and parameter selection are important considerations. Parameters like alpha control the influence of past observations and can be optimized. Crossvalidation is commonly used.

Forecasting needs to account for factors like trends, seasonality, level shifts, outliers. Models need to be selected and tuned appropriately for the characteristics of the time series data.

Forecasting involves balancing accuracy and complexity. Simpler models are generally preferred if their accuracy is sufficient for the intended use of forecasts.
Here is a summary of the key papers referenced:

Dickinson (1975) discusses the combination of forecasts from different methods.

Eilon and Elmaleh (1970) examine adaptation limits in inventory control models.

Ekern (1981) revisits adaptive exponential smoothing models.

Eppen and Iyer (1997) apply Bayesian updates to improve fashion buying decisions.

Fildes (1979, 1983, 1989) evaluate various forecasting models and techniques including extrapolative models, Bayesian forecasting, and model selection rules.

Fildes et al (2009) empirically evaluate the effectiveness of forecasting adjustments by experts.

Fisher and Ittner (1999) study the impact of product variety on assembly operations.

Gardner (1983, 1990, 2006) examines approximate inventory control rules and evaluates forecast performance in inventory systems.

Granger and Newbold (2014) cover forecasting economic time series.

Hyndman et al (2008) discuss exponential smoothing approaches using state space models.

Makridakis et al (1982, 1990) analyze forecasting accuracy competitions and propose a sliding simulation technique.

Maddala and Lahiri (1992) introduce econometrics.
Many of the other papers discuss forecasting models, techniques, theory and applications in areas like operations management, production, inventory control and demand forecasting.
This summary analyzes key papers on extrapolation (time series) forecasting methods from major forecasting competitions and journals. It discusses the results and implications of the M2, M3 and other forecasting competitions run by Spyros Makridakis that compared various extrapolation methods. It also summarizes several of Makridakis’ publications on forecasting frameworks, methods and applications. Overall, it provides an overview of the literature on evaluating and comparing extrapolation forecasting techniques based on time series data.

Section 4.1 introduces the assumptions of the basic EOQ model, such as constant deterministic demand, no quantity restrictions, no cost changes over time, etc. These assumptions are later relaxed in the chapter.

Section 4.2 derives the EOQ formula by minimizing the total relevant costs, which are the ordering/setup costs and inventory carrying costs given the parameters.

The chapter goes on to introduce realworld complications like quantity discounts (Section 4.5), inflation (Section 4.6), quantity limitations (Section 4.7), and noninstantaneous replenishments (Section 4.8).

Section 4.9 discusses other factors in choosing reorder quantities.

Section 4.10 shows how to develop exchange curves, an important approach when the cost parameters like carrying costs are implicitly specified rather than explicitly evaluated.

In summary, the chapter introduces the basic EOQ model and its assumptions, then gradually relaxes the assumptions to incorporate more realworld considerations and complexity. It emphasizes both optimizing given current parameters and also “changing the givens” through process improvement to further reduce costs.

The EOQ (economic order quantity) model considers determining the optimal order quantity when demand is deterministic, replenishment lead time is zero, and backorders are not allowed.

The costs involved are ordering costs (A+Qv) and carrying costs (Ivr). The total relevant costs per unit time (TRC) is the sum of these costs.

TRC follows a Ushape, with ordering costs decreasing and carrying costs increasing as Q increases. The EOQ minimizes TRC.

The EOQ formula is derived by setting the derivatives of ordering and carrying costs equal: EOQ = √(2AD/vr)

TRC at EOQ is simply TRC(EOQ) = √(2ADvr)

Deviations from EOQ result in small cost penalties according to the sensitivity analysis formula PCP = p2/(2(1+p)) where p is percent deviation.

Numerical examples are provided to illustrate calculating EOQ, TRC, reorder time, and inventory turnover for a sample item.

Quantity discounts are common in practice and ignore them can lead to missed savings opportunities.

The most common type of quantity discount is an “allunits” discount, where orders at or above a breakpoint quantity receive a discounted unit price for all units.

To determine the optimal order quantity when an allunits discount is present, there are two relevant cost expressions  one for quantities below the breakpoint and one for quantities at or above the breakpoint.

The expression incorporating the discount will always yield lower total costs for the same quantity. Therefore, if the minimum of that expression is above the breakpoint, it is the optimal quantity.

There are three possible optimal solutions  order at the breakpoint to take advantage of savings if acquisition cost reduction outweighs higher carrying costs; order at the unadjusted EOQ if carrying costs dominate; or order at the adjusted EOQ with discount if it is below the breakpoint.

An efficient algorithm evaluates these cases to determine the true optimal order quantity when an allunits discount is present.
So in summary, quantity discounts need to be accounted for and can impact the optimal order quantity compared to the standard EOQ model without discounts.
Here is a summary of the key points regarding accounting for inflation when determining optimal order quantities:

Inflation impacts both fixed replenishment costs (A) and unit variable costs (v) over time, causing costs to increase at the inflation rate.

Two approaches are considered: 1) Price is established independent of ordering decisions 2) Price is adjusted each replenishment lot.

Under the first approach, total discounted present value of future costs is minimized, since costs change over time but price does not. Costs at time t are Aeit and veit, where i is the inflation rate and t is time.

Under the second approach, both costs and revenues change with each replenishment lot since price is adjusted. This impacts the optimal order quantity decision.

Analytically modeling inflation is complex as order quantities may change over time. As an approximation, it is assumed quantities remain the same for future replenishments.

The present value of costs is used, discounting by the discount rate r. Optimal order quantity is determined by minimizing the total present value of costs over the planning horizon.

Quantity discounts are not considered as the optimal quantity may be insensitive to inflation near discount breakpoints.
So in summary, accounting for inflation involves discounting future costs to determine the optimal replenishment order quantity that minimizes total present value of costs over time.

When inflation is high, all cash is quickly used to buy inventory since the increase in price exceeds the interest earned on invested capital. This section assumes more reasonable inflation levels.

The optimal order quantity that minimizes total costs under inflation is approximately equal to the EOQ multiplied by a correction factor of (1  i/r), where i is the inflation rate and r is the discount rate.

As inflation approaches the discount rate, the cost penalty of using the standard EOQ formula increases but remains quite small in percentage terms of total costs.

When price is set as a fixed fractional markup on the variable unit cost, both costs and revenues are impacted by the choice of order quantity. The optimal order quantity is approximated as the EOQ multiplied by a correction factor of (1 + fi/r), where f is the fixed fractional markup and i is the inflation rate.

There are physical constraints like maximum time supply, storage capacity limits, minimum order quantities from suppliers, and discrete unit sizes that may prevent using the EOQ result and require adjusting the order quantity.

If replenishment occurs at a finite rate rather than all at once, the economic production quantity (EPQ) model is more appropriate than the EOQ model.

The derivation from Figure 4.1 is modified to account for finite replenishment rates, as shown in Figure 4.7. The average inventory level is now Q(1  D/m)/2, where D is demand rate and m is production/replenishment rate.

The total relevant costs are given by TRC(Q) = AD + Q(1  D/m)vr^2. The optimal order quantity that minimizes costs is the finite replenishment economic order quantity (FREOQ), also known as the economic production quantity (EPQ).

The EOQ formula is modified by a correction factor of 1/√(1  D/m). As D/m increases, indicating demand is closer to the replenishment rate, the correction factor decreases toward 1.

Figures 4.8 shows the correction factor values for different D/m ratios and the cost penalties from ignoring the correction factor. The cost penalty is small until D/m is large.

As D/m approaches 0 or 1, the FREOQ reduces to the standard EOQ, as the assumptions of the basic model are met. For D/m > 1 the model breaks down as demand exceeds production capacity.

The section then briefly discusses how the EOQ model can be modified to incorporate other realworld factors like nonzero lead times, payment periods, different types of carrying costs, and setup/freight costs. More complex formulas result but maintain the basic EOQ approach.
Here is a summary of the key points about the choice of replenishment lot size:

The Economic Order Quantity (EOQ) model aims to minimize total relevant costs by balancing setup costs and holding costs. It determines the optimal order quantity.

When freight/transportation costs are included explicitly in the model, they can significantly impact the optimal order quantity due to economies of scale in shipping larger quantities. It may be cheaper to order more than the EOQ to take advantage of lower perunit freight rates.

Onetime opportunities to procure items at reduced unit costs require a different approach than the standard EOQ model. The current order quantity will differ from future orders since the unit cost changes.

A heuristic approach is used to determine the optimal order quantity in these situations by maximizing the improvement in total costs out to a certain time horizon from ordering more at the old price.

The optimal order quantity formula accounts for factors like demand rate, future EOQ, current and future unit costs, holding costs. It determines how much to order above the future normal replenishment quantities.

There may be practical constraints like inventory obsolescence that limit how much excess inventory is carried due to a special procurement opportunity.

The chapter discusses determining optimal order quantities (Q) when demand is approximately level (static).

Under a set of assumptions, the economic order quantity (EOQ) formula is derived as Q* = √(2AD/r), where D is demand rate, A is fixed order cost, and r is carrying cost rate.

It may be difficult to explicitly determine values of A or r. An alternative approach is to consider aggregate inventory costs and constraints for a group of items.

If EOQ is used for each item, the total average cycle stock (TACS) and total number of replenishments (N) per period depend on the ratio A/r through equations.

A hyperbolic “exchange curve” relates TACS and N, and any point on the curve implies a value of A/r.

Management can select a desired operating point on the curve based on aggregate goals, thereby implicitly determining an appropriate value of A/r, A, or r.

The procedure provides a way to improve current practices and determine better parameter values even if individual cost factors are difficult to estimate.
So in summary, it presents a methodology to determine replenishment quantities and corresponding cost/frequency tradeoffs when considering the aggregate inventory situation rather than individual items.
Here is a summary of the provided text:

The economic order quantity (EOQ) model is commonly used in practice to determine the optimal order quantity that minimizes total relevant costs. The EOQ is derived by selecting an operating point on the aggregate exchange curve of total average stock versus total number of replenishments per year.

The EOQ is fairly robust to deviations  small inaccuracies in the order quantity don’t produce large cost errors. This justifies its widespread use in practice over more complex models. Conditions change over time though, requiring periodic recomputation of the optimal quantity.

Several simple modifications have been made to the basic EOQ model to expand its applicability. In the next chapter, variations for fluctuating demand rates will be discussed.

Three important extensions will be covered in later chapters: stochastic/uncertain demand (Chapter 6), coordinated multiitem ordering (Chapter 10), and multiechelon inventory systems (Chapter 11). In each case, the basic EOQ will play an important role.

Several problems are then posed related to applying the EOQ model and analyzing the impacts of changes in parameters like the ordering cost (A) and annual carrying cost rate (r). Extensions considering backorders, quantity discounts, and alternative inventory policies are also proposed.
In summary, the text provides an overview of the EOQ model and its common usage, before outlining some extensions and modifications that will be covered in subsequent chapters. It establishes the EOQ as a foundational concept that remains applicable even as assumptions are relaxed. A series of problems are then given to apply and analyze the basic EOQ formulation.

The EOQ formula aims to balance the fixed ordering costs and variable carrying costs to minimize the total annual cost. It determines the order quantity that results in equal marginal costs for ordering and carrying inventory.

Applying the EOQ formula to the example given:
 A = $10 (ordering cost)
 v = $0.24/unit/year (carrying cost)
 D = 900 units/year
 L = 10 weeks
 r = 0.24 $/$/year

The Dartmouth Bookstore case study involves calculating the EOQ, total costs at EOQ vs other order quantities, and analyzing the impact of quantity discounts and inflation on the optimal order quantity.

Monopack is advised to incorporate inflation into its EOQ calculation since inflation will impact carrying and ordering costs over time, thus potentially changing the optimal order quantity.

The key consideration is how to balance ordering fixed costs with inventory carrying variable costs given demands and costs to minimize total relevant annual costs. Quantity discounts, inflation, storage fees, and other considerations may further impact the optimal order size.
The passage discusses solving inventory optimization problems when demand is approximately level. It presents a case study where a bookkeeper compiled time and cost data for various activities involved in processing domestic and imported orders.
It then outlines a multipart optimization problem to analyze the current ordering process and determine if switching to an economic order quantity (EOQ) policy could reduce costs. The problem involves:
a) Identifying any missing costs from the bookkeeper’s data.
b) Calculating the company’s carrying cost.
c) Computing the cost of a single domestic vs imported order.
d) Determining the EOQ for each of the six sample items.
e) Calculating total cycle stock under the current ordering rules.
f) Calculating total relevant costs (ordering and holding) using EOQs.
g) Comparing total costs from parts e) and f).
h) Computing order replenishments under each system.
i) Graphing an exchange curve and analyzing cost impacts of changing order quantities.
j) Comparing total costs and estimated savings of switching to an EOQ policy for all items.
The passage also includes derivations in an appendix for some equations used in the analysis, such as the percentage cost penalty of using an incorrect replenishment quantity.
Here is a summary of key points about temporary price discounts:

Temporary price discounts are offered for a limited time period to incentivize customers to purchase sooner rather than later. They aim to increase sales volume during the discount period.

Models have been developed to incorporate temporary discounts into inventory control and economic order quantity (EOQ) calculations. This allows determining optimal order quantities and reorder points when discounts are temporarily available.

Factors that impact optimal order quantities include the size and duration of the discount, expected demand during the discount period, holding and setup costs, etc. Larger orders may be justified to take advantage of discounts.

However, large orders run the risk of excess inventory if demand does not increase as expected. Models try to balance these factors to maximize savings from discounts while minimizing costs of excess inventory.

Empirical research has examined the realworld impact of temporary discounts on customer purchase behavior and sales patterns. Understanding customer response helps improve inventory and pricing models that incorporate temporary promotions.

Managing inventories and order quantities optimally during discount periods requires coordination between production, purchasing, pricing and inventory control functions.
Here is a summary of the key points about lot sizing for individual items with timevarying demand:

When demand rate varies over time, it is no longer optimal to use a constant replenishment quantity as with the EOQ model. The analysis becomes much more complex.

Three main approaches are introduced: 1) Straightforward use of EOQ (not truly optimal but simple), 2) Exact optimal procedure, 3) Approximate heuristic method.

Common assumptions for all three include known demand over a finite planning horizon and infinite supply.

A numerical example is used throughout the later sections to illustrate the approaches.

The exact optimal approach analyzes the total relevant costs over the planning horizon and determines the optimal replenishment quantities. But it is computationally intensive.

The heuristic approach uses simple rules to determine replenishment quantities in each period, aiming to approximate the optimal solution. It is less accurate but more efficient than the exact method.

Quantity discounts can be handled by adapting the procedures to the timevarying demand case.

Aggregate considerations are still important, as illustrated through the exchange curve concept applied to this setting.
So in summary, it introduces more complex methods to deal with timevarying demand compared to the EOQ model, focusing on optimal versus heuristic solutions and examples to illustrate the approaches.

The passage discusses three approaches to dealing with deterministic, timevarying demand: 1) using the basic EOQ formula with average demand, 2) using an exact mathematical model (WagnerWhitin algorithm), and 3) using an approximate heuristic method.

It presents the assumptions used in the models, including known timevarying demand, replenishments only at period starts, no shortages allowed, etc.

It provides a numerical example of monthly demand for a product to illustrate applying the company’s “three month rule” of producing enough to satisfy 3 months of future demand.

Applying this rule to the example data, it calculates the resulting inventory levels, replenishments, carrying costs, and total costs.

The key point made is that as demand information becomes more uncertain over a longer time horizon, the less accurate any replenishment decisions based on that information are likely to be. Rolling schedules that maintain a constant horizon are preferable in practice.

The company uses a fixed EOQ approach for production planning when demand is approximately constant. This means ignoring time variability and using the same EOQ quantity for all replenishments.

For the example given, the average demand is 100 boxes/month over 12 months. So the EOQ is calculated as 164 boxes.

To apply this approach, cumulative demand is tracked until it reaches the closest total to the EOQ. This determines the replenishment quantity and period covered.

For the example, the first replenishment of 214 boxes is determined to cover demand through April.

Using the fixed EOQ approach slightly reduces total costs versus the company’s 3month rule, but increases inventory levels.

The WagnerWhitin method can optimally determine replenishment quantities and timing when demand is timevarying. It uses a dynamic programming algorithm to evaluate all options at each period.

The key assumptions are deterministic demand, replenishments only when inventory hits zero, and either finite horizon or predefined ending inventory.
So in summary, it outlines different approaches for production planning when demand varies over time, from a simple fixed EOQ to the optimal WagnerWhitin method.
The passage describes the WagnerWhitin algorithm for determining optimal lot sizes for inventory with timevarying demand. While it provides the lowest total costs, it has some drawbacks that limit its practical application.
Specifically, the algorithm requires defining an end point for the demand pattern, which may not always be feasible. It also assumes replenishments can only occur at discrete intervals.
As a result, simpler heuristic approaches are often preferable, as they can capture much of the potential cost savings with lower implementation and control costs. The passage then introduces several heuristic methods that have been proposed as practical alternatives to the WagnerWhitin algorithm for inventory lot sizing with variable demand.

The passage discusses various commonly used heuristics for lot sizing with timevarying demand, since fixed EOQ may not perform well in that setting.

It begins by introducing the SilverMeal (or least period cost) heuristic, which selects the replenishment quantity to minimize the total relevant costs per unit time over the period covered by that replenishment.

Other heuristics mentioned include expressing EOQ as a time supply or periodic order quantity (POQ), and a lotforlot (L4L) approach.

An example numerical problem is used to illustrate the application of the SilverMeal heuristic. For this example, it gives the same solution as the more complex WagnerWhitin algorithm.

The heuristics aim to provide reasonable suboptimal solutions when demand patterns are variable, as they only consider a limited future time horizon and are easier to apply than exact algorithms. Fixed EOQ may not perform well in variable demand settings.
Here is a summary of the key points about the Least Unit Cost (LUC) heuristic from the passage:

The LUC heuristic is similar to the SilverMeal heuristic, except that it accumulates requirements until the cost per unit (rather than the cost per period) increases.

Calculations are shown for determining the first replenishment quantity using LUC for a numerical example.

Applying LUC to the full 12month period results in a total cost of $558.80, which is 11.5% higher than the optimal solution found by the WagnerWhitin algorithm.

Research has shown this level of performance compared to the optimal is typical for the LUC heuristic. In other words, it usually performs reasonably well but not perfectly.

The LUC heuristic focuses on minimizing the cost per unit rather than the cost per period like SilverMeal. But in practice it does not significantly outperform SilverMeal and is more complex to implement.
So in summary, the LUC heuristic is an alternative to SilverMeal that focuses on unit cost rather than period cost, but research shows it does not typically improve substantially on SilverMeal’s performance while being more complex to implement.
Here are the key points summarized from the passage:

The SilverMeal heuristic is a wellknown algorithm for solving the lot sizing problem with timevarying demand. It determines the optimal replenishment batch sizes and timings over a planning horizon.

The heuristic was tested by deliberately changing the input parameter A/vr from its correct value. The costs of the resulting replenishment pattern were then computed using the true A/vr value and compared to using the correct value from the start. This was done for various percentage errors in A/vr. The results showed cost penalties remained small even for reasonably large A/vr errors.

Methods have been proposed to reduce “nervousness” or frequent changes to the replenishment schedule, such as incorporating a penalty cost for changing previously established plans. This allows accommodating timevarying setup costs as well.

The heuristic was extended to handle quantity discounts by checking if meeting a discount threshold is costeffective versus the regular replenishment timing.

Developing an “exchange curve” to choose an optimal A/vr value for multiple items is more complex than for basic EOQ due to the discrete decision variable in SilverMeal. It requires running the heuristic for a sample of items and scaling the results.

In summary, the SilverMeal heuristic provides an effective approach for the lot sizing problem with timevarying demand and has been enhanced to address various practical extensions and applications.
Here is a summary of the key points in Problem 5.15:

The problem considers a transportation/distribution system where inventory accumulates according to supply processes and is depleted by dispatches (rather than stocking problems where inventory levels are depleted by demand).

Periodic review is assumed, with opportunities to dispatch (reduce inventory to zero) at the end of each period.

The goal is to minimize the tradeoff between inventory holding costs incurred each period and the fixed cost per dispatch.

The system starts empty at the beginning of period 1.

Wt denotes the set of items supplied in period t whose first opportunity for dispatch is at the end of period t.

HC(Wt) is the cost of holding items Wt in inventory for one period. If held for multiple periods before dispatch, the holding cost is multiplied by the number of periods.

A is the fixed cost per dispatching event.

The problem is to determine the optimal dispatching policy  i.e. which sets of items Wt to dispatch at the end of each period t in order to minimize total costs over the planning horizon.
So in summary, it poses a lot sizing problem for a transportation system where inventory accumulation follows supply processes rather than demand depletion, with the goal of minimizing holding and dispatching costs.
A dispatching policy is needed to determine in each time period whether to dispatch or not dispatch in that period. The goal is to develop a decision procedure, such as the SilverMeal heuristic, that minimizes the relevant costs per unit time between dispatch moments.
The procedure should be illustrated using the example given, which involves dispatching freight trains from a switchyard terminal. Cars waiting to be dispatched to a particular destination W accumulate costs per hour that vary by car. The example gives the hourly holding costs for cars destined for W that were switched each hour for the first 7 hours.
The fixed cost of dispatching a train to destination W is $800. The procedure should determine when to dispatch trains to minimize total costs based on the hourly holding costs accumulated for cars waiting to be dispatched to W. The illustration should show the decision for the first two dispatch moments based on this example.
Here is a summary of the paper:

The paper discusses lot sizing for individual items when demand is timevarying or probabilistic rather than deterministic.

Previous chapters assumed deterministic demand patterns, but in reality demand is often uncertain or probabilistic.

When demand is probabilistic rather than deterministic, the relevant costs for selecting order quantities become more complex and sensitive to demand uncertainty.

This chapter models demand as a probabilistic process and formulates the lot sizing problem as a stochastic dynamic program.

It reviews methods for solving stochastic lot sizing problems exactly or approximately through approximations, simulations, or heuristic algorithms.

Some key models discussed include solutions for items with random yield, lost sales, multiechelon supply chains, compound Poisson demand processes, and spare parts inventory.

The chapter provides an overview of approaches for dealing with probabilistic demand in lot sizing decisions, which is more realistic than deterministic models but also more complex from a modeling and solution methodology perspective.
This summary covers the key points from the provided text:

When demand is probabilistic/uncertain rather than deterministic, inventory modeling becomes more complex conceptually and computationally.

There is uncertainty regarding how often to monitor inventory levels, when to place replenishment orders, and how large orders should be.

Different types of control systems are needed to resolve these issues under uncertainty.

Whether backorders are allowed or sales are lost impacts inventory metrics like safety stock and performance.

The chapter will cover important terminology, review A/B/C classification, continuous vs periodic review systems, common control systems, shortage cost/service metrics, and methods for determining reorder points and order quantities under probabilistic demand.

Sections will present models for continuous review systems, discuss implied performance vs objectives, address periodic review systems and lead time variability, and show how total buffer stock relates to service for different policies.

The focus is singlestage systems; multiechelon systems require different control approaches covered later.
So in summary, it introduces the key challenge of probabilistic demand and previews the chapter’s coverage of approaches and models for inventory control under uncertainty.

Inventory management policies need to address four key questions: how important is the item, how often should stock status be reviewed, what form should the inventory policy take, and what cost/service objectives should be set.

Items are typically classified as A, B, or C based on their importance to the business. A items are most critical.

Stock status can be reviewed continuously or periodically. Continuous review provides better customer service but higher costs, while periodic review has predictable workload but requires more safety stock.

The main forms of inventory policies are:

Order point, order quantity (s,Q) system  Order a fixed quantity Q when inventory position hits reorder point s. Simple but not ideal for large variable demands.

Order point, order upto level (s,S) system  Order enough to raise inventory to order up level S when hitting reorder point s. More flexible for variable demands.
Managers need to select classification, review method, and policy form to optimize costs and service for each item based on its characteristics and business needs.
This passage summarizes different inventory control systems and methods for setting safety stock levels:

It describes four common inventory control systems: (s,S) continuous review; (R,S) periodic review; (s,Q) continuous review with fixed order quantity; and (R,s,S) which combines elements of the other systems.

When demand is probabilistic, there is a chance of stockouts. Different approaches can be used to balance stockout risks vs inventory holding costs.

Approaches include: using simple metrics like safety factors; minimizing costs by considering shortage costs; targeting service levels to meet customer needs; and setting levels to optimize aggregate performance across all items.

There is no single best approach  it depends on factors like the competitive environment and customer priorities. Equivalencies exist between some service measures and shortage costing methods.

Different approaches may be suitable for different item classes within the same company. The goal is to present the key options and considerations rather than an exhaustive treatment.
Here is a summary of the more common measures used in establishing safety stock (SS) levels:

Equal time supplies  Set SS to the same number of time periods of forecasted demand for all items (e.g. 2 months). Does not account for different demand uncertainties between items.

Fixed safety factor  Set SS as the product of a safety factor (k) and the standard deviation of forecast errors over the lead time (σL). Uses a common k for all items.

Cost per stockout occasion (B1)  Sets SS to minimize the total cost of stockouts, assumed to be a fixed cost B1 per occasion.

Fractional charge per unit short (B2)  Sets SS to minimize costs where a fraction B2 of unit value is charged per unit shortage.

Fractional charge per unit short per time (B3)  Similar to B2 but the charge is per unit shortage per time period.

Cost per customer line item short (B4)  Sets SS to minimize costs where there is a cost B4 for each incomplete customer order line.

Probability of no stockout per cycle (P1)  Sets SS to achieve a target probability P1 that demand is fully met within each replenishment cycle.

Fraction of demand met from stock (P2 fill rate)  Sets SS to achieve a target fraction P2 of demand that is met from inventory without backorders.

Ready rate (P3)  Sets SS to achieve a target fraction of time that inventory levels are positive.

Average time between stockouts (TBS)  Sets SS to achieve a target average time between stockout occurrences.

Minimize expected stockouts (ETSOPY)  Allocates a fixed total SS among items to minimize the expected total stockouts per year.

Minimize expected shortage costs (ETVSPY)  Allocates SS to minimize expected total shortage costs per year.

The allocation interpretation focuses on allocating a fixed total safety stock (TSS) among items to minimize the expected total variable stockout profit loss per year (ETVSPY).

There are two decision rules that lead to the same allocation as minimizing ETVSPY:

Assuming the same value for the factor B1 for all items, and selecting the safety stock to minimize total carrying and shortage costs.

Specifying the same average time between stockouts (TBS) for each item.

This aggregate view of allocating a limited safety stock resource may be more appealing to management than trying to explicitly determine B1 values or setting arbitrary TBS values for each item.

The goal is to find the optimal safety stock allocation among items to minimize total costs, given a fixed total amount of safety stock available. This provides a practical approach that management may find easier to implement than more complex analytical models.

The passage describes a process for determining the optimal reorder point (s) in a continuousreview inventory system with a fixed order quantity (Q) and probabilistic demand.

It assumes demand follows a normal distribution and discusses using the normal distribution to model forecast errors and lead time demand.

It defines the common notation used, including parameters like demand rate, lead time, reorder point, order quantity, safety factor, etc.

It notes that while Q is assumed fixed, in reality s and Q are interdependent and their optimal values depend on each other. However, assuming a fixed Q makes practical sense, especially for B items.

It discusses how the equations and decision rules for determining s depend on the costing method or service level criterion used. Different rules will be provided based on the inventory costing approach.

It emphasizes that once optimal parameters are determined, efforts should also focus on “changing the givens” like reducing lead times, demand variability, and required service levels to further lower inventory costs and improve service over time.
So in summary, it lays out the common assumptions and framework for determining the reorder point s in a continuousreview (s,Q) system using a normal demand assumption and different costbased decision rules.
Here is a summary of the key points about establishing the reorder point value s:

The reorder point s is determined indirectly by first calculating the safety stock, and then setting s equal to the forecast demand plus the safety stock.

Safety stock is typically expressed as a multiple of the forecast error standard deviation, according to the formula Safety Stock = k * σL, where k is the safety factor.

The goal is to determine the appropriate value of k based on the costs and service level considerations. Different decision rules are used depending on whether a target service level, shortage cost, or safety factor is specified.

If a safety factor k is specified directly, then the reorder point is simply calculated as s = forecast demand + k * σL.

If a perstockout cost B1 is specified, then the total relevant costs are modeled and k is determined to minimize total costs, which balances ordering, carrying and stockout costs.

Under the assumption of normal demand forecast errors, the stockout probability can be expressed as a function of the standard normal distribution, Pu≥(k). Other parameters like expected stockouts per cycle can also be determined based on the normal assumption.

The general approach is to determine the safety factor k first, then calculate the reorder point s using the relationships s = forecast demand + safety stock and safety stock = k * σL.

The total expected annual cost when using a B1 cost model (fixed dollar amount per unit shortage) has three components: carrying cost, ordering cost, and shortage cost.

A decision rule is provided to determine the safety factor k. It involves checking if DB1/√(2πQvσLr) < 1. If so, use a formula to calculate k. If not, use the minimum allowed k.

The reorder point s is then calculated as the forecasted mean demand + k times the demand standard deviation.

A similar procedure is described for a B2 cost model (fractional charge per unit short) and a B3 cost model (fractional charge per unit short per unit time), with the appropriate formulas to calculate k.

Numerical examples are provided to illustrate the decision rules. Graphs also show how the total cost varies with different values of k.

The decision rules result in higher safety factors (larger k values) being assigned to slower moving and more expensive items, as expected intuitively.

The minimum allowed k value should be used if the formulas cannot determine a k that satisfies the cost model constraints.

This decision rule is for determining reorder points when a specified average time between stockout occasions (TBS) is given.

Step 1 checks if the order quantity Q is greater than average demand during the TBS period.

If so, proceed to Step 2 and set the safety factor k to its minimum allowed value.

If not, Step 1 continues by selecting k to satisfy a probability function equal to Q divided by average demand during TBS.

Step 3 calculates the reorder point as the forecast mean demand plus k times the demand standard deviation, rounded up to the next integer if needed.

An example is provided to illustrate the calculation steps.
So in summary, this rule determines the safety factor and reorder point needed to achieve a given average time between stockouts, by either setting k to its minimum or choosing it to satisfy a probability function, depending on the order quantity level.

The inventory problem specifies a demand rate D = 200 units/year, order quantity Q = 30 units, and desired target batch size (TBS) value of 2 years.

Using Equation 6.31, this gives a value for the Zscore (k) of 1.44.

From tables, k=1.44 corresponds to a safety stock (SS) of 18.9 units.

The base stock (s) is calculated as SS + mean demand during lead time, which is 58.3 + 18.9 = 77.2 units, rounded to 78 units.

Equations 6.22 (B2 costing) and 6.31 (TBS measure) are equivalent, with TBS equivalent to B2/r.

Increasing the TBS value lowers the value of pu≥(k) from Equation 6.31, increasing the safety factor k and thus the SS.

Allocating a total safety stock (TSS) to minimize the expected total stockout per year (ETSOY) or expected total value of shortages per year (ETVSPY) both follow the same decision rule as using B1 or B2 costing, respectively.

Nonnormal demand distributions can also be used, such as Gamma, Laplace or Poisson, depending on the characteristics of the demand.

A fill rate of 98% can yield a low cycle service level of 59.87% because with a large order quantity, many customer demands are met from stock on the shelf before the lead time even begins. So frequent stockouts at the end of the replenishment cycle do not significantly lower the fill rate.

The inventory policy allows for a low cycle service level, which may be acceptable to management or potentially problematic. The policy should be adjusted if the low cycle service level is an issue.

Similar logic can be used to determine implied values for cost parameters, like a unit shortage cost of 5.6% of the unit value based on the cycle service level. This value may be consistent with management objectives or seem incorrect, in which case a different inventory policy should be considered.

Any input performance objective (like shortage cost, service level, etc.) can be used to determine an implied value for k, which is then used to find the reorder point/orderupto level. The values of the other input objectives can also be implied.

In summary, the fill rate may mask low cycle service levels when order quantities are large. Management should evaluate if low cycle service levels are acceptable or if the inventory policy needs adjustment. Implied values from input objectives can help evaluate policy fit with objectives.
Here is a summary of the key points about lead time from the passage:

Lead time refers to the time between placing an order and receiving the goods. It’s a key variable in setting reorder points and orderupto levels.

If lead times are uncertain or variable, it increases the safety stock (SS) needed to protect against this additional uncertainty.

Seasonal variations in lead time can be accounted for if the pattern is known. Increasing lead times due to supply issues can also be accounted for.

Efforts should be made to reduce lead time variability through things like transportation mode choices, ordering process improvements, and closer supplier cooperation/relationships.

If some lead time variability remains, two approaches can be used: 1) Measure total demand over full lead time or 2) Measure demand rate/period and lead time separately and combine distributions.

The first approach uses measurements of actual demand over lead times directly to set reorder points. The second assumes demand and lead time are independent and combines their distributions.

Nonnormal distributions like Gamma can also be used if demand over lead time is skewed or lead time variability is high. Close supplier partnerships are important for lead time improvements.
Here is a summary of the key points about dsheet:

dsheet describes research on combining demand distributions and lead time distributions for inventory modeling purposes. Lead times that take on just a few values (like 2 or 4 weeks) cannot be accurately modeled using a normal distribution, according to research by Eppen and Martin.

There is extensive research on accounting for variable lead times in inventory models, dating back to the 1970s. Many studies assume fixed demand and lead time distributions. Some work looks at “changing the givens” by investigating how to reduce average or variability of lead times.

Exchange curves can show the tradeoff between total safety stock (TSS) and aggregate service level when using a certain decision rule for setting reorder points across multiple items. They allow finding the preferred service level for a given inventory investment constraint.

A simple example compares the current approach of setting reorder points as equal time supplies versus allocating the TSS to equalize stockout probabilities per cycle across items. Reallocating TSS in this way can improve aggregate service for the same investment.

The chapter outlines the general methodology for developing exchange curves based on specifying an input objective like fill rate and computing implied performance measures like stockout probabilities.

The table shows numerical results of allocating safety stock (SS) using different methods: equal time supply, reorder points based on equal probabilities of stockout, reorder points based on a B1 rule, and reorder points based on a B2 rule.

Allocating SS using a B1 or B2 rule further improves the aggregate measures of expected stockout occasions and expected value short compared to equal probabilities of stockout.

A B1 rule minimizes expected total stockouts, while a B2 rule minimizes expected total value short for a given inventory budget.

Exchange curves can compare different SS allocation methods across a range of aggregate service levels like expected stockouts or expected value short. Curves based on B1 and B2 rules typically perform better than equal time supply.

Composite exchange curves combine SS exchange curves with EOQ cycle stock curves to show the threeway tradeoff of total stock, replenishment frequency, and aggregate service level.

The optimal SS allocation depends on the specific service objective being optimized. B1 and B2 rules are more sophisticated than equal time supply since they consider itemlevel demand variability and uncertainty.
Here is a summary of the key points from Equation 4.30 in Chapter 4 and the discussion of individual items with probabilistic demand:

Equation 4.30 provides the formula for computing the reorder point (R) in an (R,S) inventory control system given a desired service level (P1). It is based on the normal distribution and involves the mean demand during lead time (xL), standard deviation of demand during lead time (σL), and the zscore associated with P1.

(s,Q) and (R,S) systems are two common approaches to inventory control when demand is probabilistic. The (s,Q) system uses a reorder point (s) and fixed order quantity (Q). The (R,S) system uses a reorder point (R) and orderupto level (S).

For both systems, the reorder point is determined using a service level parameter (e.g. P1, TBS) and information about the demand distribution during lead time such as mean (xL) and standard deviation (σL).

Examples are provided to demonstrate how to compute reorder points and other quantities for each system given relevant demand and cost parameters.

Aggregate exchange curves are an important concept for analyzing the average and expected costs associated with different reorder point/service level combinations.

Realworld probabilistic inventory situations tend to be more complex, with extensions involving multiitem, multiechelon, and coordinated control problems.
Here is a summary of the key details from the subject:

An (s,Q) inventory policy is used, where inventory is reviewed continuously.

There is a fee of $0.325 per gallon per day that inventory is short.

Inventory carrying cost is $0.20 per dollar of inventory per year.

The lead time (L) for replenishment is 30 days on average, but has a standard deviation of 6 days.

The customer is proposing alternative shortage cost structures:
 A flat $1,000 charge per stockout occasion
 A $65 charge per gallon short per cycle

Under each alternative, what would the optimal policy be? Are the proposed changes acceptable?
So in summary, it involves determining the optimal (s,Q) policy under the original shortage cost structure, and then evaluating how the policy and costs would change under the two proposed alternative shortage cost structures.

The manager considers a fractional shortage charge of 0.50 for each unit short and an inventory carrying charge of 0.30 $/$ per year.

For each transportation alternative, the optimal inventory policy depends on minimizing total relevant costs, which include replenishment costs, carrying costs, and shortage costs.

Alternative A has lower replenishment costs but higher uncertainty (larger standard deviation), while Alternative B has higher replenishment costs but lower uncertainty.

In general, Alternative B would be preferred since it has lower uncertainty, even though replenishment costs are higher. The lower uncertainty reduces expected stockout costs and carrying costs needed to achieve a given service level.

To determine the specifically optimal alternative, a quantitative cost analysis considering all relevant cost parameters under each alternative would need to be conducted to identify which minimizes total expected costs.
So in summary, Alternative B is likely preferable due to its lower uncertainty, but a quantitative cost analysis is needed to say definitively which alternative minimizes total costs for this specific case. The inventory policy under each should aim to achieve the optimal ordering quantity and reorder point that balances all relevant costs.
The passage summarizes a technical explanation regarding how to calculate inventory levels and costs using probabilistic demand modeling. Specifically:

Inventory levels (SS) are expressed as the sum of forecast demand and a safety stock factor.

Safety stock is expressed as the product of two factors: a safety factor (k) and the standard deviation of forecast errors over the lead time (σL).

This allows determining k to achieve a desired service level using the normal distribution function.

Two common service level measures are discussed  P1 considers stockouts within the lead time, while P2 considers the fraction of demand satisfied from inventory.

Formulas are derived for setting k based on each service level measure, under assumptions of complete backordering or lost sales when stockouts occur.

A Lagrange multiplier approach is presented for allocating a total safety stock budget (TSS) to minimize expected stockouts per year across multiple inventory items.

Formulas are provided for calculating the standard deviation of demand during the lead time, assuming probabilistic/variable lead times and demand that follows certain distributions (e.g. normal, gamma).

Two general approaches for determining expected costs per unit time are contrasted  either directly calculating total relevant costs, or using a results from renewal theory about the expected cost per replenishment cycle.

Finally, a formula is presented for calculating expected stockouts per replenishment cycle if demand follows a gamma distribution.
Here is a summary of the papers:

de Kok (1990) discusses hierarchical production planning for consumer goods.

De Treville et al. (2014) values lead time in inventory management.

Disney et al. (2015) examines fill rate in a periodic review model with autocorrelated demand.

Ehrhardt (1979) develops a power approximation for computing (s,S) inventory policies.

Eppen and Martin (1988) determines safety stock with stochastic lead time and demand.

Ernst and Pyke (1992) discusses component part stocking policies.

Federgruen and Zheng (1992) presents an efficient algorithm for computing optimal (r,Q) policies.

Fortuin (1980) analyzes spare part requirements for aftersales service.

Gallego (1992) develops a distributionfree procedure for the (Q,R) inventory model.

Gallego and Moon (1993) reviews and extends the distributionfree newsboy problem.

The papers examine various issues in inventory management models, including production planning, lead time valuation, fill rates, computing optimal policies, safety stock, component parts, spare parts, and distributionfree procedures. A variety of inventory models are analyzed, including (s,S), (r,Q), (Q,R), and newsboy models. Both theoretical analyses and algorithms are presented.
Here are some key points about managing important or Acategory inventory items:

A items typically have higher total costs associated with replenishment, carrying inventory, and shortages compared to other items. They are often highvolume items.

Factors beyond just annual dollar usage (Dv) can also dictate an item be in category A, like being essential to a product line even with low sales.

More sophisticated control systems than for B items are warranted due to higher associated costs.

Manual managerial oversight is important in addition to decision rules/models, which cannot incorporate all factors. Managers should use subjective judgement.

Perpetual inventory records should be maintained, particularly for expensive items.

Minimum order quantities should balance costs vs. shortages. Higher service levels may be needed.

Safety stocks and reorder points may need manual adjustment based on demand trends, supplier performance, etc.

Critical item review meetings with suppliers can improve coordination and response times.

Flexible replenishment methods like kanban may be suitable for highvolume, lowcost items.
So in summary, A items require more customized, managerial oversight compared to basic decision rules for B items, to minimize total costs given their higher replenishment, inventory carrying and shortage costs.
Here is a summary of the key points regarding simultaneous determination of safety stock (s) and order quantity (Q) for fastmoving items:

For important Alevel items, it is better to simultaneously determine s and Q rather than the sequential approach covered in Chapter 6.

Equations 7.3 and 7.4 can be used to solve for s and Q simultaneously, assuming normal demand and a B1 shortage cost structure.

An iterative process is used, initially setting Q = EOQ and solving for k, then using the k value to find a new Q, and repeating until convergence.

The simultaneous Q value will always be larger than the EOQ, while the simultaneous k value will be smaller than or equal to the sequential k value. This makes intuitive sense given the relationships between Q, s, and shortage costs.

A numerical example is provided to demonstrate the iterative process of converging on the simultaneous s and Q values.
So in summary, simultaneous determination provides a more optimal solution for s and Q compared to the sequential approach, with the iterative process converging the values.

The passage describes sequential and simultaneous approaches for determining the reorder point (s) and orderupto level (S) for an (s,S) inventory control system.

In the sequential approach, the order quantity Q is set equal to the economic order quantity (EOQ). Given this Q, the reorder point s is determined. Then the orderupto level S is set as S = s + Q.

In the simultaneous approach, s and S are selected together to minimize total relevant costs while taking into account the “undershoots” (how far below s inventory falls when an order is placed).

The passage provides an equation that can be used to approximate the distribution of undershoots when the orderupto level S is much larger than the average transaction size.

This distribution of undershoots is combined with the distribution of lead time demand to determine s and S simultaneously such that a target stockout probability or fill rate is achieved.

A numerical example is provided to illustrate applying the sequential approach. The simultaneous approach is described but no example is worked through.

The revised power approximation is a heuristic method for determining values of the (R, s, S) parameters for inventory control systems when there is a fractional charge (B3) per unit short at the end of each review period.

It uses equations derived from regressing optimal parameter values found through laborious analysis against representative problem characteristics. The equations give approximate values for the order quantity Q = S  s and reorder point s.

A numerical example illustrated applying the method to find s and S values for a given item with specified characteristics like demand rate, lead time, costs, etc.

The method was derived by assuming general forms for Q and s equations and fitting parameters to optimize against a wide range of problems. It has been shown to perform well in most circumstances compared to exact but impractical analyses.

Overall, the revised power approximation provides a practical heuristic approach for determining (R, s, S) parameter values when backorders are penalized on a perunit basis each period (B3 costing).

Ehrhardt et al. (1981) and Ehrhardt (1984) present approximations for inventory models with autocorrelated demand and random lead times respectively.

Naddor (1978) is also referenced for related research.

The decision rule presented determines the reorder point s such that a specified fraction P2 of demand is satisfied directly from stock.

This is a heuristic procedure originally developed by Tijms and Groenevelt (1984) that can be adapted for the (s,S) policy as well as other service measures like P1 and TBS.

The rule is given as Equation 7.25, which typically requires a trialanderror solution but can be simplified under certain conditions.

A numerical example is provided to illustrate the application of the simpler form of the rule given by Equation 7.28.

The rule aims to set s such that the expected shortage per replenishment cycle equals the specified fraction 1P2 of the expected replenishment size, as per Equation 7.29.

Some discussion is provided on deriving and interpreting the decision rule. Further references on related research are also given.

The heuristic described involves precomputing (s, S) values for various mean demand levels under stationary assumptions. It then approximates the nonstationary problem by averaging demand over an estimate of the optimal reorder period from the stationary problem.

Tests show the error from using this heuristic averages only 1.7%, so it is very fast and accurate.

However, the heuristic should be avoided if demand is expected to decline rapidly as a product becomes obsolete, as the stationary approximation will not be valid in that case.

Chapters 8 and 9 address inventory problems where demand declines rapidly over time.

Managing multiple suppliers reduces inventory costs but increases relationship and coordination costs. The inventory implications of sourcing decisions are often overlooked.

Research looks at models for dual sourcing that balance use of fast and slow suppliers to minimize costs while maintaining acceptable safety stock levels. Good dual sourcing policies are more complicated to implement than single supplier policies.

Some key works on dual sourcing, emergency ordering, expediting, and other extensions to the basic inventory models are highlighted.
So in summary, the heuristic is easy to implement but provides an accurate approximation only under certain stationary demand assumptions. The passage recommends alternative approaches when demand behavior deviates from those assumptions.
Here are the summaries:
i. Simultaneous best (Q, k) pair
To find the optimal order quantity Q and safety factor k, we need to solve two equations simultaneously:

Equation 7.3: The derivative of the total cost function with respect to Q is set to 0.

Equation 7.4: The fraction of demand satisfied from stock is equal to the target service level.
Solving these two equations simultaneously will give us the optimal Q and k values that minimize total costs.
ii. Item characteristics:
The item has:
 Annual demand (D) = 1,000 units/year
 Lead time demand standard deviation (σL) = 80 units
 Ordering cost (A) = $5
 Unit holding cost (r) = 0.16 $/unit/year
 Unit price (v) = $2/unit
 Target fraction of demand satisfied from stock (P2) = 0.5/year
 Shortage cost parameter (B1) = Not specified
Given these item characteristics, we can calculate the EOQ and solve the two equations to find the simultaneous best Q and k values.

The expressions for the optimal order quantity Q and reorder point k are derived for an inventory system with continuous review (s,S) policy and normally distributed demand.

Taking the derivative of the expected total net relevant cost function NTRC(k,Q) with respect to Q and setting it equal to 0 leads to an expression for the economic order quantity EOQ.

Taking the derivative with respect to k and setting it equal to 0 leads to an expression relating the optimal reorder point k to the safety factor B1, demand parameters, and EOQ.

The optimal reorder point k is the value such that the probability of stockout during the lead time is equal to the target service level 1/B1.

Specifically, the expression equates the standard normal cumulative distribution function evaluated at k/√A/Q to the target stockout probability 2/B1σLQ.

This derivation utilizes the fact that the derivative of the standard normal c.d.f. is the standard normal p.d.f.
So in summary, the optimal policy parameters Q and k are derived by taking derivatives of the expected cost function and setting them equal to 0, resulting in expressions relating the optimal values to system demand and cost parameters.

The article discusses inventory control methods for slowmoving items classified as category A or B. These are important items that are slowmoving.

For highervalue A items where benefit of accuracy is higher, it recommends using a normal distribution to model forecast error if average demand over lead time is at least 10 units. Otherwise, a discrete distribution like Poisson is more appropriate.

For slowmoving A items, it recommends using (s,Q) ordering policies and approach of explicit costing of shortages discussed in Chapter 6.

Poisson distribution has a single parameter of average demand over lead time. Relationship between average and standard deviation is specified.

Discrete nature of Poisson distribution is both blessing and curse  it allows modeling in discrete units but discrete math creates implementation problems.

Traditional approach of determining reorder point s after reorder quantity Q is discussed, and challenges with discrete distributions are noted. Alternative approaches may be needed.

Section discusses control of important slowmoving items, recommending adapted (s,Q) policies and consideration of discrete demand distributions like Poisson for low demand levels.

The derivation of parameters like reorder point (s) and order quantity (Q) initially ignored the impact of these parameters on shortage costs. This led to a sequential approach of first optimizing Q and then s.

However, s and Q are actually interdependent as Q influences shortage costs. A larger Q means smaller shortage costs.

An exact approach would simultaneously optimize s and Q to minimize total relevant costs per year (ETRC), as expressed in equation 8.2. This is more theoretically sound but also more computationally complex.

For a B1 (fixed shortage cost per stockout) cost structure, indifference curves can be used to simultaneously select s and Q by evaluating equation 8.6 on a graph of Qvr/DB1 vs average demand.

For a B2 (fraction of unit value charged per unit shortage) cost structure with Q=1, equation 8.4 provides an easy decision rule for s. With Q>1, the more complex equation 8.5 is used but is harder to implement.

Overall, the sequential approach tends to result in only small cost penalties compared to the exact simultaneous approach, so it remains a justified simplification for most inventory situations.
This summary outlines key points about managing Class C (slowmoving, lowvalue) inventory items with steady demand:

For C items, the goal is simple procedures that minimize control costs per item, not maximizing savings.

Inventory records may not be needed for some C items, relying instead on reordering when stock runs out. If used, records should have a long review interval like 6 months.

More complex inventory optimization techniques like EOQ are unwarranted. Assign items to a small set of time supplies (like 6, 12, 18 months) based on factors like shelf life and obsolescence, not classical EOQ parameters.

Estimate demand rate simply by calculating usage between reorder periods, rather than sophisticated forecasting. For example, in a twobin system calculate as inventory before order + order quantity  safety stock, divided by time between orders.

Select reorder point (s) conservatively to avoid stockouts, given low service level requirements for C items. Adjust upward if shortages have high implicit costs.
So in summary, it recommends very simple inventory management procedures for C items that minimize control costs and effort, rather than complex optimization techniques applied to higher value/importance items.
Here is a summary of the key points about managing inventory with orderupto levels:

An orderupto (R,S) system specifies a review period R and an orderupto level S. Inventory is reviewed every R periods and replenished up to level S if needed.

The orderupto level S effectively acts as the reorder point, as inventory is replenished up to this level at each review.

Setting the orderupto level S involves selecting a safety factor k based on the expected time between stockouts (TBS), using formulas provided. This allows specifying service level through TBS.

A simple implementation of the (R,S) system is the “semiautomatic stock control system” where S is set periodically but orders equal demand over each review period.

Grouping similar slowmoving items and coordinated replenishment based on an item in the group needing replenishment can reduce setup costs compared to individual replenishment.

Items with declining demand patterns, like those nearing product lifecycle end, require special consideration of timing and sizing of replenishments to optimize inventory and meet remaining demand. Formulas are provided for deterministic linearly declining demand.
So in summary, the orderupto level approach provides a practical way to set and implement replenishment policies for slowmoving items while achieving a target service level.

The strategy discusses reducing excess inventories of slowmoving and lowvalue (Class C) items. This is an important problem area as product life cycles shorten.

Excess inventory can be caused by replenishment errors or overestimating demand. It’s important to identify excess items and decide on remedial actions.

A distribution by value (DBV) list ranks items in descending order of dollar usage. Items with zero usage in the past year appear at the bottom.

Calculating coverage (expected months of inventory) for each item and ranking by coverage provides additional insights.

The example table shows 4.2% of total inventory value is tied up in zeromover items, and 6.9% has coverage over 5 years. This informs management on the overstock situation severity.

Where usage data is unavailable, a “dust test” of storage facilities can help identify slowmoving excess stock by appearance. Managing Class C inventory is an ongoing process involving periodic reviews.
The section discusses rules and factors to consider for disposing of slowmoving or excess inventory. Any stock that is overly dusty, deteriorated, or occupying valuable storage space is a candidate for disposal. Two approaches to the disposal decision are presented:

Deterministic demand  Dispose of the amount of inventory that exceeds the EOQ plus projected demand over the disposal period.

Probabilistic demand  Use formulas to determine the optimal number of units to save based on variables like salvage value, carrying costs, and expected demand over time.
Options for disposing of excess stock include using it for other products, transferring to other locations, promotions, markdowns/sales, returns to suppliers, auctions, or scrap value. The effects on financial statements like income and taxes should be considered.
Factors for deciding whether to stock an item or produce/purchase to order for each demand include system costs to stock, variable costs under each approach, backorder costs, setup/ordering costs, demand patterns, and supplier relationships. Both cost and noncost aspects need to be evaluated.
Here are the key points from the passage:

When deciding whether to stock an item or not stock it, the following factors need to be considered: demand for the item when not stocked, fixed setup costs, carrying costs including inventory holding costs and obsolescence, demand pattern in terms of size and frequency, and replenishment lead time.

It may be better to not stock an item locally if an internal supplier like a central warehouse can stock it for multiple locations, considering replenishment lead times and customer wait tolerance.

A simple decision rule is presented to determine whether to stock an item based on comparing costs of stocking vs not stocking under specific assumptions. The assumptions and decision rule are explained.

Some extensions to the basic decision rule are discussed to make it more generalized, such as allowing different costs for stocked vs unstocked items, and periodic review models.

Estimating parameters like average time between demands and average demand size can be difficult for slow moving items with sparse data, and care needs to be taken to prevent frequent changes in stocking decisions.

Managing inventories of large numbers of slow moving or low value (Class C) items requires simplicity due to low annual costs involved. Input from supervisors familiar with items is also important.
So in summary, the passage outlines the key factors to consider in “stock versus no stock” decisions for slow/low value items, presents a basic decision rule, and discusses some extensions and challenges in accurately applying the decision rule. Simplicity is important given the low annual costs associated with Class C inventory items.
Here is a summary of the key points in the prompt:

A C item is often purchased from the same supplier as an A or B item. An order is placed for the B/A item every 2 months.

Demand for the C item is constant at 18 units/year. Unit value is $3. Additional fixed cost to include C item in a B/A order is $1.20. Carrying charge is 0.24 $/unit/year.

Consider ordering the C item in quantities that will last 2 months, 4 months, 6 months, etc. (integer multiples of the B/A item time). Determine which time supply is preferred.
To summarize, the prompt provides demand and cost data for a C item that is often ordered with an A or B item. It asks which integer multiple of the 2month B/A order cycle (2 months, 4 months, 6 months, etc.) would be preferred for ordering the C item, given its constant demand rate and the carrying and fixed order costs. The optimal time between C item orders should be determined based on a comparison of relevant inventory carrying and ordering costs.

The company stocks components from inventory that are used to assemble larger products. The lead time to acquire components from inventory varies between 13 days.

Inventory records are not updated on a perpetual basis. Instead, physical inventory counts are taken periodically, with the review period ranging from 25 weeks depending on the item.

The company uses an orderupto inventory control system for each component, setting a reorder point.
Some suggestions for improving inventory management:

Implement a perpetual inventory system to more accurately track inventory levels in real time rather than relying on periodic physical counts. This would allow reordering to be triggered based on actual usage rather than estimates.

Standardize review periods across similar items for better consistency. Consider taking counts more frequently for fastmoving items.

Evaluate reorder points and order quantities using a datadriven approach factoring in demand trends and variability. The current approach may not be optimized.

Consider implementing safety stock to better ensure availability and reduce stockouts during lead times.

Automate the replenishment process to speed up ordering and reduce manual errors/delays. Integrate with supply chain system for smoother flows.

Establish annual usage and demand forecasts by item to aid in planning and inventory positioning.
Here are the key points about an inventory replenishment policy for the case of a linear decreasing trend in demand:

The demand is assumed to follow a linear decreasing trend over time. This captures situations where demand declines gradually over the replenishment cycle or planning horizon.

A common policy is to set a periodic review cycle and replenish inventory up to a target level each period to meet expected demand.

The target inventory level each period accounts for the expected decline in demand and ensures sufficient stock is on hand. It decreases each period in line with the declining demand trend.

The order quantity each period is the difference between the target inventory level and the onhand inventory level. It decreases over time as both demand and the target inventory level decrease.

Shortages may be allowed and backorders considered in the formulation. Alternately, lost sales can be assumed if shortages are not permitted.

The objective is to choose replenishment periods and target inventory levels each period to minimize total inventory holding and ordering costs over the planning horizon.

Mathematical programming and simulation models can be developed and solved to determine the optimal policy parameters under different cost and demand assumptions.
So in summary, it involves periodic replenishment up to a decreasing target inventory level that reflects the linear downward trend in demand over time.

Northern Canada supply run  The question asks how many supplies should be brought in by boat prior to the long winter freezeup. This is estimating inventory needs for an isolated community until boats can no longer access it in winter. Factors to consider would include population size, expected duration of freeze, typical consumption rates, and safety stocks.

Farmer crop planting  The question is about how much of a particular crop to plant in a specific season. Factors to consider would include expected yields, market demand forecasts, storage/processing capacity, and selling price. The goal is to match production to demand.

Toy manufacturer product run  A “fad” toy shows strong potential sales. The question is how many units to produce in the initial run. This is a newsvendor problem with uncertainty around demand. Overproducing risks leftover inventory that may not sell later. Underproducing risks lost sales. The optimal order balances the costs of overage and underage.

The newsvendor model aims to determine the optimal order quantity (Q*) that maximizes expected profit for goods facing uncertain demand.

One approach is to plot the cumulative demand distribution and locate the value where cu/(cu+co) lies on the vertical axis. Then move horizontally to the curve and down to read the optimal Q*.

An equivalent approach is to formalize the profit maximization problem. This gives the same solution as the marginal cost analysis approach but also allows deriving an expression for expected costs/profits for any Q value.

If demand is normally distributed, the problem can be solved by finding the value of k where the normal cumulative distribution equals cu/(cu+co). This gives a simple formula for Q* and expected profit/costs.

Sensitivity of expected profit to the Q value chosen can be analyzed, showing the impact of selecting quantities above or below the optimal Q*.

Modifications to the model address practical issues like fixed ordering costs, discrete demand values, and starting with inventory on hand.

The problem involves determining the optimal stocking levels (Qi) for multiple items (n items), given a common constrained resource (budget W).

The solution procedure uses a Lagrange multiplier (M) to account for the cost of exceeding the budget constraint.

Starting with an initial M value, Equations 9.12 and 9.13 are used to determine the Qi values that satisfy the constraint for each item.

The total cost of the Qi’s is compared to the budget W. M is adjusted up or down accordingly and the process repeated until the total cost is very close to W.

This traces out an exchange curve of total expected profit vs. total budget. The optimal solution occurs where the curve is tangent to the budget constraint.

The multiplier M represents the marginal value (in profit terms) of adding $1 to the budget. It allows incorporating the budget constraint into the individual stocking decisions.

The approach can be adapted to other types of constrained resources like space or capacity, by adjusting the units in the constraint comparison.
So in summary, it provides a methodical way to determine the optimal allocation of a constrained resource like budget among multiple stockkeeping items.
If the budget is changed somewhat from the original proposed budget of approximately $70,000, then the expected total profit and optimal order quantities may change as well. The summary provided:

Calculated the multiitem newsvendor model for different budget constraints (values of M, the budget multiplier).

Showed that at the original proposed budget of $70,000 (M = 0.5093), the expected total profit is $28,755 and the optimal order quantities are provided.

Noted that increasing the budget could lead to a substantial improvement in the total expected profit. The profit would start decreasing if the budget is too high and requires investing the entire amount.

Three values of M were identified that correspond to points where the critical fractile for one product equals zero, making it unattractive to produce.

Changing the budget somewhat would result in a different optimal solution with different expected total profit and order quantities for the items. Trial and error would need to be done to find the new value of M that uses the changed budget amount.
So in summary, if the budget is changed, the expected total profit and optimal order quantities output from the multiitem newsvendor model may update to a new optimal solution.
Here is a summary of the calculations:
500  100  25 + 200 = 575
Since demand is normally distributed, we can use a spreadsheet formula or Appendix II to find k = 0.512.
Using k = 0.512, the optimal stocking quantity (Q) is 110 units for the postponed system.
The expected profit for the original system is $15,637 The expected profit for the postponed system is $15,976
Adopting the postponed differentiation results in a slightly higher stocking quantity (110 vs 109 units) and slightly higher profit ($15,976 vs $15,637).
The cost of overstocking is lower in the postponed system since the $25 transformation cost is not incurred for overstocked items. This allows the firm to stock more units and increase profits slightly over 2%.
The passage is discussing numerical analysis of a singleitem, singleperiod inventory problem with and without postponed product differentiation.
Some key details:

It presents a table comparing the original system (with separate glossy and matte products), a postponed system (where the product type is delayed), and key metrics like expected profits.

The postponed system has higher total expected profits ($32,312 vs $29,442) due to reducing the mismatch between supply and demand, lowering expected units short and units left over.

It notes that if demands had to be ordered in advance, the profit could be increased slightly more by using the lower cost of understocking for matte boards when determining the order quantity.
So in summary, it is analyzing the inventory costs and expected profits of a single product using a standard newsvendor model, and comparing it to a “postponed product differentiation” approach where the specific product type is determined later, showing the flexibility benefit of postponing differentiation.

The parameters of the demand probability distribution are initially unknown. Prior knowledge is encoded using probability distributions over possible parameter values.

As demand is observed early in the season, the probability distributions are updated to incorporate the additional information using techniques like Bayesian updating.

Forecast accuracy can be improved by only using sales data from the first 20% of the season.

Demand forecasts can leverage patterns of past demand for similar products as well as estimates of total life cycle sales and seasonal variations.

Inventory policies are developed using multiperiod newsvendor models to control inventory levels over the season in response to demand.

Reorders and markdowns are determined based on the fraction of initial inventory sold over time to identify fast vs slow moving styles.

Capacity can be reserved ahead of time with suppliers by paying a premium to ensure availability for peak demand.

For components shared across products, it may be better to hold safety stock at the component level rather than the finished product level to reduce inventory needs.

The impact of demand variability on optimal order quantities depends on relationships between selling price, production cost and unit cost.
Here is a summary of the key points from section 9.8 Inventory Control of Perishable Items:

Perishability refers to the physical deterioration of inventory units over time. It causes decreased demand for older units.

Perishable items can have either fixed or random lifetimes. Fixed means units last a set period then lose value. Random means lifetime varies randomly.

For fixed lifetime items, extensive research looks at policies like (S1,S) with Poisson demand and positive lead times. Approximations have been developed for models with lead times and (s,Q) policies.

For random lifetime items assumed to deteriorate at a constant rate, research examines models where demand decreases over time. Heuristics have been developed for settings with timevarying demand as well.

Most research assumes a firstinfirstout (FIFO) issuance policy, but in retail a lastinfirstout (LIFO) policy may be observed when customers select items.

One paper examines a model where shelf life is fixed but useful life is random, exploring the interaction of orderupto level and shelf life date.

Analysis of probabilistic demand and lifetimes is very complicated. Research looks at optimal disposal quantity in a Poisson demand model.
Here are my summaries:
a. What kinds of data (objective or subjective) should he collect?
The man should collect both objective and subjective data. Objectively, he should collect data on historical sales amounts and patterns for the particular items he plans to restock. Subjectively, he should gather input from his wife on her preferences and opinions regarding what leftovers can be safely frozen versus what should be eaten fresh. Gathering both types of data will help him make the most informed decision.
b. His wife says that certain types of leftovers can be put in the freezer. Briefly indicate what impact this has on the analysis.
If certain leftovers can be frozen, this increases the man’s options for what to purchase and how much to purchase. Items that can be frozen have less risk of going to waste if not eaten fresh, so he could potentially buy more of those types of items knowing they can be stored longer term in the freezer. This means his analysis of how much to purchase should account for the freezer storage option as a way to reduce potential waste. He would be able to buy more of Freezable items versus nonfreezable items.
The newsvendor formulation is relevant to this situation because it deals with deciding how much inventory of a perishable good to purchase/produce in situations where demand is uncertain. The newsvendor approach helps optimize this stocking decision by balancing the costs of overstock versus understock.

v1 = unit acquisition cost for copies ordered prior to the day in question (i.e. the standard/usual ordering method), expressed in $/copy

v2 = unit acquisition cost for an emergency additional order placed on the day in question, expressed in $/copy

It is assumed that v1 < v2, since the emergency order has a higher cost.

p = selling price per copy

g = unit salvage value per copy if copies are left over/unsold

It is assumed g < v1 < v2 < p, in terms of the relative sizes of these unit costs and prices.

Q1 = order quantity placed prior to the demand being realized (i.e. the standard order)

Q2 = additional/emergency order quantity placed after demand x is known, to meet any excess demand above Q1
So in summary, this sets up the basic parameters of acquisition costs, selling price, and salvage value for the newsvendor model with two ordering opportunities  a standard order Q1 before demand x is known, and a potentially highercost emergency/additional order Q2 placed after observing demand x.

The expected profit is calculated as the profit under perfect information (unit margin p  v times expected demand) minus the per unit overage cost (unit cost  salvage value) times expected leftover units, and the underage cost (unit margin p  v plus penalty cost B) times expected shortage units.

For normally distributed demand, the expected profit formula involves finding the optimal order quantity Q by setting the derivative of expected profit with respect to Q equal to 0. This leads to an equation involving the standard normal CDF.

For discrete demand, the expected total cost formula is developed and the optimal Q is found by selecting the smallest Q where the difference in expected total costs between Q and Q+1 is greater than 0.

For the constrained multiitem case, a Lagrangian approach is used to maximize total expected profit subject to a budget constraint. The partial derivatives with respect to the Lagrangian multiplier and order quantities lead to similar expressions as the single item case, except the unit cost is replaced by the Lagrangian multiplier times the unit cost.
So in summary, it outlines the basic expected profit/cost models for inventory optimization under different demand distributions (normal, discrete) and extends it to a constrained multiitem setting using a Lagrangian approach.
Here is a summary of research papers 87, 93–108:

Several papers study inventory models for items that deteriorate or become obsolete over time, like perishable goods or fashion items. They develop models to determine optimal replenishment and production schedules considering factors like timevarying demand, uncertain lifetimes, and different types of demand distributions.

Other papers examine solutions like component commonality where multiple products share common parts to reduce costs. They analyze the tradeoffs between reduced inventory carrying costs and increased production/ordering costs.

A few papers focus specifically on seasonal or fashion goods, developing techniques for forecasting demand, coordinating production and inventory, and determining optimal sale and markdown strategies.

Several analyze various aspects of the classic “newsboy problem” involving determining optimal ordering quantities for a single product with uncertain demand. Papers generalize the problem in different ways, like incorporating multiple products, suppliers, or time periods.

Optimization methods studied include both exact formulations and heuristic solutions. Papers also compare the effectiveness of myopic versus coordination production/inventory policies.
This chapter discusses coordinated replenishments of multiple items at a single stocking location. Some key points:

Items may be coordinated if they share a common supplier, transportation mode, or production facility. Coordinating replenishments can reduce costs.

With deterministic demand, the economic order quantity (EOQ) model is extended to account for a major fixed setup cost for replenishing the entire group of items, plus a minor fixed cost per item. Optimal order quantities and a replenishment schedule are determined.

With probabilistic/uncertain demand, a joint replenishment policy is developed that coordinates orderupto levels and order points across items to minimize total expected costs. Heuristics are presented for multiproduct systems.

Production environments add complexity as productive capacity must be shared each period across items. Approximation methods are discussed to determine optimal production and inventory policies.
So in summary, this chapter develops models and heuristics for optimally coordinating replenishments or production of multiple items stocked at a single location to reduce costs when items share resources or facilities. Both deterministic and probabilistic demand settings are considered.
Here is a summary of key points from Section 10.3:

The section extends the coordinated replenishment model to allow for quantity discounts, where the unit purchase cost may decrease if the total order quantity reaches a certain breakpoint.

It considers a situation where grouping items can achieve a freight rate reduction if a carload size replenishment is met.

The goal is to select replenishment quantities for each item that maximize savings from quantity discounts or freight reductions while minimizing ordering/setup costs.

The model chooses integer values for “mi”, which represents the number of time periods between replenishments for each item i. Items with higher setup costs or lower demand rates would have higher mi values, meaning less frequent replenishments.

The optimal values of mi are chosen to minimize the total relevant costs per time period, as defined in the objective function presented.

Once the best mi values are determined, they are used to calculate the optimal time period T between replenishments for the whole group.
So in summary, the section extends the basic coordinated replenishment model to incorporate quantity discount or freight rate incentives for larger total order quantities.

This section considers coordinated inventory replenishments for multiple items when group discounts are offered based on the total quantity or value of items ordered.

Taking advantage of discounts reduces replenishment costs but increases inventory carrying costs, making the optimal solution more complex than when no discounts are involved.

A heuristic approach is proposed that considers three possible solutions: 1) where discounts are always achieved, 2) where replenishments are just at the discount breakpoint, and 3) with no discounts.

The method first calculates replenishment quantities assuming discounts are achieved. It then checks if the smallest replenishment meets the discount breakpoint quantity. If so, that solution is used. If not, it compares the cost of the nodiscount solution vs. a solution at the breakpoint. The lower cost solution is chosen.

This allows a reasonable compromise without explicitly modeling the increased complexity of scenarios where replenishment cycles differ depending on whether discounts are achieved.

The paper discusses coordinating order quantities when it is possible to take a discount on some replenishments but not others. This additional flexibility can yield significant savings.

Previous related research includes work by Russell and Krajewski (1992) on coordinating order quantities across multiple locations. Sadrian and Yoon (1992, 1994) looked at consolidated versus direct shipments. Katz et al. (1994) considered the tradeoff between quantity discounts and inventory holding costs. Xu et al. (2000) developed a model for coordinated replenishments across multiple suppliers.

The key idea is that by allowing discounts to be taken on some replenishments but not others, it provides more flexibility that can lead to overall cost savings compared to always taking discounts or never taking discounts. The paper outlines an procedure to determine the optimal strategy that balances these tradeoffs.

The algorithm allocates a major setup cost A across multiple products in a family to determine economic order quantities (EOQs) and time supplies.

It starts by calculating EOQs and time supplies based just on minor costs for each product. The product with the smallest time supply is labeled product 1.

A small portion α1 of A is initially allocated to product 1, increasing its EOQ and time supply.

α1 is increased until product 1’s time supply matches that of the product with the next smallest time supply (product 2).

Then α1 and α2 are increased together to keep their time supplies balanced, while more of A is allocated.

This continues until the time supplies equal that of the next product (product 3), at which point α3 is also allocated.

The process repeats until the entire setup cost A has been allocated (Σαi = 1).

This ensures each family order accounts for the full setup cost and products are replenished in balanced cycles.

Miltenburg models demand for inventory items as independent diffusion processes, which implies the total demand over an interval has a normal distribution.

Items are often reordered when above the reorder point, even if not actually needed, because another item is triggering the replenishment. This residual stock provides additional safety stock.

The distributions of residual stock after each allocation must be evaluated to set reorder points accordingly and maintain the desired service level.

For periodic review, residual stock is reasonably approximated as normal. For continuous review, a spike at 0 plus truncated normal above fits better.

An example applies Miltenburg’s approach to coordinate replenishments of beams needed in full truckloads. Reorder points are set based on acceptable vs expected shortages. Products are allocated truck capacity to maximize time until next order.

Production environments also involve coordination but are more complex due to capacity constraints. Methods can still apply in certain settings like continuous flow processes. The economic lot scheduling problem aims to optimally schedule production runs considering changeover times.

The economic lot scheduling problem (ELSP) aims to find an optimal production schedule that minimizes total inventory and setup costs while meeting demand and production capacity constraints.

A pure rotation schedule, where each product is produced once per cycle, is a common heuristic approach. It ensures products are not produced simultaneously.

The paper presents a procedure to find the optimal cycle time T for a pure rotation schedule. T is set to maximize total relevant costs or satisfy the capacity constraint, whichever is larger.

A numerical example applies the procedure to three products produced on one bottleneck machine. The optimal cycle is found to be 5 weeks, with each product produced in sequence within that cycle.

Research has looked at more complex ELSP variants that allow varying batch sizes and cycle times over time, aiming to find truly optimal schedules rather than heuristic solutions. Overall, the ELSP aims to coordinate production to minimize costs while meeting demand.

The capacitated lot sizing problem (CLSP) involves determining the production schedule for multiple items over multiple time periods, taking into account timevarying demand and production capacity constraints.

Dixon (1979) developed an early heuristic approach for the CLSP that uses a forwardlooking method to determine production quantities for the current period based on demand forecasts for a limited number of future periods.

The heuristic aims to minimize total relevant costs per unit time by selecting lot sizes that balance setup and holding costs. However, it must also consider capacity constraints.

It works by first setting each item’s lot size to the minimum amount required. It then increases the lot size of the item with the largest marginal cost decrease per unit of capacity used, until capacity is fully utilized or lot sizes are no longer decreasing costs.

It also performs a lookahead to ensure sufficient production to meet future demand within capacity. Some later adjustments can improve initial solutions.

Many other heuristics have been developed since, as the CLSP is an important and difficult production planning problem with capacity constraints.

lmeier and Derstroff (1996) present an approach to solving multiitem inventory problems.

Van Nunen and Wessels (1978) present a different conceptual approach than the Dixon and Silver heuristic. Their approach first ignores capacity constraints and solves each item problem independently. It then adjusts the solution to make it feasible with minimal cost increases.

Variations of this problem consider backordering of demand (Eppen and Martin 1987; Pochet and Wolsey 1991; Millar and Yang 1994).

Some authors examine problems with startup costs and reservation costs for having a facility available (Karmarkar and Schrage 1985; Karmarkar et al. 1987; Sandbothe 1991; Hindi 1995; Coleman and McKnew 1995).

When demand is probabilistic, two basic approaches are used  developing a cyclic schedule with control rules, or heuristics that directly decide production sequencing and quantities (Bourland and Yano 1994; Gallego 1990, 1994; Federgruen and Katalan 1996c).

Issues with shipment consolidation are also discussed (Higginson and Bookbinder 1994, 1995; Tyworth et al. 1991; Van Eijs 1994; Bausch et al. 1995; Hall and Racer 1995).
The key procedure highlighted is Van Nunen and Wessels (1978), which takes a different conceptual approach than the Dixon and Silver heuristic by first ignoring constraints and solving individual problems, then adjusting the solution to be feasible.

A supplier is offering Ptomaine a 3% discount on any order totaling at least $1,350 in value (before the discount).

Fishman’s partner, L. Talks, likes the idea of saving on the purchase price by taking advantage of the quantity discount offer.

However, Fishman is skeptical and worries that tying up so much money in inventory may not be advisable.

The question asks whether Ptomaine should take advantage of the supplier’s quantity discount offer, given the potential downside of increased inventory investment highlighted by Fishman.
Overall, this summarizes a scenario where a supplier is offering a quantity discount to Ptomaine, but Fishman has reservations about the increased inventory investment required to meet the discount threshold. The question posed is whether Ptomaine should take the supplier up on the discount offer.
Here are the key steps to find the best order quantities:

The problem involves finding coordinated replenishment quantities for 10 items purchased from the same supplier.

The replenishment cost A is a function of the total order value V, as defined in the equation provided.

Two approaches are considered:
a) Order a fixed time supply (e.g. 1, 2, or 3 months) of all items. Calculate the order quantities, costs, and select the best time period.
b) Use a modified SilverMeal heuristic that iteratively calculates the order quantities based on an estimated setup cost A. Start with an initial estimate of A based on historical average order value. Compute order quantities, recalculate actual total value V, update A, and repeat until convergence.

For both approaches, the calculations would need to be done for each replenishment period over the 12 month horizon while ensuring at least 97.25% fill rate for each item.

The total replenishment and holding costs can then be computed and compared to select the best approach.
So in summary, the key steps are to 1) Formulate the problem, 2) Develop the algorithms for the two approaches, 3) Iteratively calculate the order quantities and costs for each approach over the time horizon, and 4) Compare the total costs to select the best approach.

It presents a multiitem inventory model with coordinated replenishments at a single stocking point. The objective is to minimize total replenishment and holding costs.

It develops a mathematical formulation of the problem with decision variables denoting replenishment quantities. The formulation includes constraints like inventory balance equations and nonnegativity conditions.

It derives an expression for the optimal replenishment quantities in terms of model parameters like demand rates, ordering costs, holding costs etc. This gives the optimal coordinated replenishment policy.

It proves that this policy provides a lower bound on the best possible total cost by choosing parameter values that satisfy constraint conditions.

Extensive references are provided on related research in multiitem inventory planning problems, coordinated replenishment models, lot sizing, production scheduling etc. This models builds on prior work in these areas to analyze a single stocking point replenishment problem.
In summary, it presents a mathematical formulation and optimal solution approach for a multiitem replenishment problem with coordination at a single stocking point, along with background context on related inventory optimization research.
Here is a summary of the papers referenced in the request:

Federgruen, A. and Z. Katalan (1994) develops approximations for queue size and waiting time distributions in general polling systems.

Federgruen, A. and Z. Katalan (1996a) analyzes customer waiting time distributions under basestock policies in single facility multiitem production systems.

Federgruen, A. and Z. Katalan (1996b) examines the impact of setup times on the performance of multiclass service and production systems.

Federgruen, A. and Z. Katalan (1996c) studies the stochastic economic lot scheduling problem using cyclical basestock policies with idle times.

Federgruen, A. and Z. Katalan (1998) determines production schedules under basestock policies in single facility multiitem production systems.
The papers by Federgruen and Katalan analyze various models related to multiitem inventory systems, production scheduling, queueing and waiting times under different policies. They develop approximations, analyze performance impacts and determine optimal schedules and policies.
This summary covers operational research papers related to production planning and scheduling, inventory management, joint replenishment problems, and lot sizing problems. Some of the key topics covered in the papers summarized include:
 Production planning with timedependent capacity bounds
 Heuristics for loading problems and scheduling problems
 Dynamic scheduling for multiitem single machine production systems
 Improvements to cycle lengths heuristics
 Inventory models with storage space and investment limitations
 Planning and scheduling for cyclically scheduled manufacturing systems
 Algorithms for singleitem capacitated lot sizing problems
 Determining joint replenishment policies for discountconnected stock lines
 Integrated inventory models for multiple vendors and buyers
 Complexity and heuristics for multilevel capacitated lot sizing problems
 Reviews of multiitem single level capacitated dynamic lot sizing heuristics
 Stochastic economic lot scheduling problems
 Modeling for stochastic demands in multiple item procurement
 Heuristics for multiitem procurement under stochastic nonstationary demands
 Production scheduling at a fastener manufacturer
 Models incorporating discounts, stochastic demands, and constraints
 Heuristics for dynamic lot sizing problems with coordination
 Lagrangian heuristics for capacitated multiitem lot sizing with backordering
 Production/inventory control under random demands for multiple products
The papers proposed and evaluated methods addressing different variations of production planning, scheduling, inventory management, and lot sizing problems. Many involved stochastic elements, constraints, or coordination across multiple items or time periods. Heuristics and optimization methods were commonly developed and assessed.
Here is a summary of the key points about multiechelon inventory management:

Multiechelon inventory systems involve managing inventory levels across multiple locations in a supply chain, from suppliers to production to warehouses to retailers.

Supply chain management aims to optimize coordination and inventory deployment across an entire supply chain rather than optimizing individual firms.

Models of multiechelon systems can help optimize inventory levels across locations and evaluate changes to the supply chain structure.

Deterministic demand models were initially analyzed but probabilistic demand adds complexity.

Base stock policies are commonly used where inventory is reviewed periodically and replenished up to a target level.

Serial, tree, and push systems are examples of multiechelon structures that have been analyzed.

Product returns and repairs introduce another level of complexity as recovery inventories must be managed in addition to new inventory.

Coordination across organizational boundaries is an important challenge addressed in later chapters on supply chain coordination.
The summary highlights the key considerations around modeling and analyzing inventory management across multiple locations in a supply chain as presented in the introduction to the chapter.

This section discusses deterministic demand in a multiechelon inventory system, where demand rates are known with certainty. This is an idealized case but useful for establishing basics.

It first considers a simple serial system with one central warehouse, retailer warehouse, and retail outlet, where demand is steady/level.

The decision variables are the replenishment quantities QW at the warehouse and QR at the retailer. QW must be an integer multiple of QR.

With deterministic demand, the warehouse inventory does not follow the usual sawtooth pattern even though end demand is constant.

Instead of physical inventory, it uses the concept of “echelon stock” which is the inventory at or passing through an echelon but not yet committed to customers. Echelon stock has a simple sawtooth pattern.

It can then compute average echelon stock levels simply and calculate total inventory costs by multiplying each by holding costs, rather than using physical inventory which is more complex with withdrawals between echelons.
Here is a summary of the key points about y carrying costs in multiechelon inventory management:

In a multiechelon system, the same physical units of inventory may be counted in multiple echelon inventories, leading to potential doublecounting of carrying costs.

To avoid this, each echelon inventory should only be valued based on the value added at that echelon. So intermediate components are only valued at downstream stages based on the further processing they enable.

For decisions about where to stock inventory (upstream vs downstream location), the relevant holding cost is the incremental cost of moving the product to the next downstream location. This incremental cost is known as the echelon holding cost.

Optimization models for determining optimal reorder quantities at each stage use echelon holding costs and valuations to minimize total relevant costs (setup costs plus carrying costs).

The models determine reorder quantities that balance setup frequency against carrying costs in a way that considers the downstream impacts of inventory decisions at each stage.

For a twostage serial system with deterministic demand, closedform expressions can be derived for the optimal reorder quantities and frequencies at each stage. More complex multiechelon networks require iterative or heuristic solution approaches.
This passage discusses inventory management in a multiechelon supply chain system. Some key points:

It introduces a threestage supply chain system as an example to illustrate issues in multiechelon inventory management.

Direct replenishment lead times are given between each stage.

Naively applying singlestage inventory models at each stage ignores cost interdependencies between stages and can result in excess/duplicated safety stock.

Demand variability increases as one moves up the supply chain due to batching from downstream stages.

Defining service level gets more complex in a multiechelon system. Stockouts at one stage impact downstream stages.

Partial shipment policies need to be considered if a downstream order exceeds upstream inventory.

Several approaches have been developed to model multiechelon systems, including approximating as simpler models, restricting to nested policies, and tracking each demand unit through the system.

Local vs global information and centralized vs decentralized control are important dimensions to consider in multiechelon system design. Global/centralized approaches aim to optimize the whole system but require more coordination.
So in summary, it discusses the key challenges in modeling inventory management across multiple stages of a supply chain due to cost and service interactions between stages.

Decentralized control systems involve independent decision making at separate locations, often with a “pull” approach where locations pull stock from suppliers based on their own needs.

Centralized control with global information is typically best as it allows for coordination across the entire system. But this requires cooperation across multiple parties.

If decentralized control must be used, each location can apply the methods from chapters 47 with probabilistic lead time formulas to gain some benefits.

One approach that assumes decentralized control but shares global demand information is the base stock control system. Each location establishes reorder points and orderupto levels based on end customer demand rather than orders from downstream locations.

For serial inventory systems specifically, De Bodt and Graves propose a procedure to set reorder points and order quantities that accounts for stock levels across echelons. Reorder points are set based on expected demand plus a safety factor, where the factors satisfy equations involving costs, demand parameters, and other values.

Numerical examples are provided to illustrate applying the procedures to a twoechelon example of warehouse and retailer inventory.
I apologize, but without more context it is difficult to provide a meaningful summary of over 500 numbers listed sequentially. Can you provide some additional details on what these numbers represent or relate to? A effective summary would need to group or categorize the information in some way.
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