Principles of Inventory Management When You Are Down to Four, Order More (Springer Series in Operations Research and Financial Engineering) - John A. Muckstadt, Amar Sapra

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Here is a summary of the key points about the Springer Series in Operations Research and Financial Engineering:

• The series is edited by Thomas V. Mikosch, Sidney I. Resnick, and Stephen M. Robinson.

• It contains titles related to operations research and financial engineering.

• The full list of titles can be found at http://www.springer.com/series/3182

• The book being summarized is “Principles of Inventory Management” by John A. Muckstadt and Amar Sapra.

• It discusses basic principles and laws for managing inventories efficiently in global supply chains.

• The contents are based on lectures from Cornell University over 33 years, covering essential topics but not an exhaustive literature review.

• It presents both deterministic and stochastic inventory models at a level appropriate for seniors and masters students with backgrounds in optimization, probability, and stochastic processes.

• The book aims to both teach fundamental principles and demonstrate applications of quantitative methods students have studied.

• It is organized to allow sections to be read independently with self-contained notation. This enables flexible use in courses.

This section acknowledges several individuals who contributed to the author’s knowledge and the development of the book.

Key details:

• The author learned from colleagues at the RAND Corporation, including Irv Cohen, Gordon Crawford, Steve Drezner, Murray Geisler, Jack Abel, Mort Berman, Lou Miller, Bob Paulson, Hy Shulman, and John Lu.

• The author benefited from research conducted at RAND by Craig Sherbrooke. Many ideas in Chapter 8 are directly related to his efforts.

• The author learned about inventory management practices from Bernie Rosenman of the Army Inventory Research Office and his colleagues Karl Kruse and Alan Kaplan.

• Since 1974, the author has been a faculty member at Cornell and has worked with scholars like Peter Jackson, Bill Maxwell, Paat Rusmevichientong, and Robin Roundy, who influenced his thinking on inventory principles.

• The author taught over 1,000 students inventory management at Cornell since 1974. He is indebted to former PhD students who helped with the book, including Kripa Shanker, Peter Knepell, and Mike Isaac.

• The author’s co-author and former student Amar Sapra encouraged him for many years to write this book. Without his encouragement and help, the book would not have been completed.

• Others who provided support include June Meyermann, Kathleen King, and Paat Rusmevichientong.

• Most importantly, the author thanks his wife Linda for her constant love and support during the long hours working on the text.

• This section covers inventory systems where replenishments are ordered continuously (chapter 9) or periodically (chapter 10).

• In the continuous review case, it presents an approximate model and an exact model for determining optimal reorder points and lot sizes when backordering is permitted. It considers extensions like stochastic lead times.

• The approximate model assumes demand during lead time is normally distributed and derives formulas for Q* and r*. The exact model determines the stationary distributions of inventory position and net inventory exactly.

• It also covers a multi-item continuous review model and approaches for optimizing the reorder points and lot sizes.

• In the periodic review case, it presents an approximation algorithm and an algorithm based on dynamic programming to compute a stationary policy. It proves the optimality of this approach.

• A heuristic method is also described for calculating (s,S) inventory control limits in the periodic review model.

• Overall, the chapter covers different modeling approaches and solution methods for determining optimal inventory control policies in continuous and periodic review replenishment systems. The focus is on reordering quantities and reorder points/reorder levels to minimize costs.

The passage discusses different types of inventories that exist in various environments, including homes, businesses, governments, and supply chains. Inventories exist to balance supply and demand when they are not perfectly aligned. The main types of inventories discussed are:

• Anticipation stocks, which are created to meet future needs beyond immediate demand. Examples include seasonal products and speculative material purchases.

• Cycle stocks, which meet demand within a replenishment cycle like a monthly ordering cycle. The average cycle stock depends on the cycle length and demand rate.

• Safety stocks, which act as a buffer against uncertain demand within uncertain lead times. They help ensure adequate supply when demand fluctuates.

• Pipeline stocks, which fill the time gap between an order being placed and the replenished goods being ready for use or sale. They depend on replenishment lead times.

The passage explains that different factors like policies, technology, capacity, and the operating environment influence what types of inventories exist and their appropriate levels. Overall, inventories fulfill important roles in balancing supply and demand throughout economic systems.

• Decoupling stocks, also known as safety stocks, are inventories introduced between successive operations or stations in a manufacturing process. They allow each station to operate independently without being blocked or starving for materials.

• Decoupling stocks provide a buffer so the assembly process can run smoothly even if there are variations in processing times or temporary breakdowns at a station. They ensure there is work available for a station to process when a task is completed and space to store output until it moves to the next station.

• In multi-echelon inventory systems like manufacturing or supply chains, safety stocks are often used between stages/echelons to enable a smooth flow of materials and ensure requests can be fulfilled with minimal delay. However, determining optimal inventory levels across the system remains a key question.

• The main factors that affect inventory policy decisions include the system structure, nature of items being stocked, demand patterns, costs of the system, and information/data available for decision making. Different mathematical models account for these factors in various ways.

Here is a summary of the key points about substitutions from the passage:

• Demand rates and variability in demand can differ between products. Some products like bread are needed immediately, while others like cars customers are willing to wait for.

• Items can be consumable goods that are used up, like food, or repairable goods that can be fixed, like jet engines. This impacts inventory management.

• Item costs vary significantly, from low costs like toothpicks to high costs like car engines. This affects inventory policies.

• Market demand is often uneven, with a small percentage of items/customers accounting for the majority (80%) of sales/revenue based on Pareto’s principle. This leads to classifying items/customers as A, B, or C types.

• Demand can be highly variable for low-demand items, leading to larger forecast errors and safety stock levels for those items.

• Lead times impact inventory needs, as longer lead times increase uncertainty of demand over that period. Lead times may be fixed or variable.

• Costs like purchasing, carrying, stockouts, and obsolescence all factor into inventory policies and strategies. Purchasing costs may vary with order quantities.

• The EOQ (economic order quantity) model is presented as the first inventory model due to its simplicity, despite making restrictive assumptions. However, it can still provide good results in certain situations.

• The EOQ model gives the optimal order quantity solution in closed form, allowing insights into inventory system behavior. It is also easy to compute compared to iterative models.

• This chapter develops single-stage inventory models for a single item where demand occurs continuously at a constant rate.

• The basic EOQ model assumes all demand is satisfied on time. A more advanced model in section 2.2 allows for some demand to be backordered.

• The goal of the models is to determine how much to order (order quantity) and when to place orders (reorder point) to minimize costs like ordering and holding inventory.

• Measuring performance includes metrics like customer fill rate targets and maximum expected waiting time to meet demand.

• The chapter presents inventory models and algorithms to set policies, compute stock levels, and estimate financial/operational performance within supply chains.

So in summary, the chapter introduces the simple but useful EOQ model and develops variations to handle backordering, with the goal of minimizing costs and determining optimal order quantities and reorder points.

Here is a summary of section 2.3 of the passage:

• The section considers the EOQ (Economic Order Quantity) model when the unit purchasing cost depends on the order size. Previously in section 2.1, the purchasing cost per unit was assumed to be constant.

• When the purchasing cost is dependent on order size, larger orders may receive quantity discounts from suppliers, meaning the per-unit cost decreases with larger order sizes.

• This relationship between order size and unit cost violates the previous assumption of a constant unit purchasing cost. However, it is a more realistic scenario in practice.

• Accounting for variable unit costs based on order size requires modifying the EOQ cost model developed in section 2.1. Specifically, the purchasing cost term Cλ in the objective function would now be a function of the decision variable Q instead of a constant.

• The final section briefly discusses how to manage inventory for multiple item types when there are constraints linking the lot size decisions across items. This could include shared warehouse space or truckload limitations.

So in summary, section 2.3 extends the basic EOQ model to the case when unit purchasing cost depends on order quantity due to quantity discounts, requiring modification of the original cost function. It also previews how multi-item inventory is managed under shared resource constraints.

Here is a summary of the key points about the EOQ (economic order quantity) model with backordering allowed:

• The model now allows for some demand to be backordered rather than trying to satisfy all demand on time. This introduces a backlog penalty cost.

• The total cost now has four components - purchasing cost, fixed order placement cost, inventory holding cost, and backlog penalty cost.

• Each order cycle is split into two sub-cycles. The first sub-cycle (ADC) has positive inventory that decreases at the demand rate. The second sub-cycle (CEF) has backorders rather than inventory, so the backlog increases at the demand rate.

• The goal remains to determine the optimal order quantity Q* that minimizes the total annual average cost, which now includes the backlog penalty cost in addition to the other costs.

• Allowing backorders provides more flexibility but introduces the additional backlog cost. The model must be modified to account for this new cost component in determining the optimal order quantity.

Here is a summary of the key points about the EOQ model with quantity discounts:

• Quantity discounts refer to discounts given by sellers to buyers who purchase larger order quantities. This incentivizes buyers to place larger orders.

• There are two main types of quantity discount models: all units discounts and incremental quantity discounts.

• In an all units discount model, the unit purchasing price decreases for all units when the order quantity crosses certain thresholds. For example, the price may drop from $5 to $4.50 per unit for all units if the order is over 100 units.

• In an incremental quantity discount model, the unit price only decreases for the incremental units purchased over certain thresholds. For example, units 201-500 may cost $0.98 each while units over 500 cost $0.95 each.

• The objective is to determine the optimal order quantity Q that minimizes the total relevant costs, which now includes discounted purchasing costs based on the discount structure.

• As with the basic EOQ model, the optimal Q balances ordering/holding costs against purchasing costs, except the purchasing costs now depend on the discount schedule rather than a fixed price per unit.

• The optimal Q generally increases compared to no discounts, as larger orders allow access to lower per-unit prices across more units.

• In incremental quantity discounts, the unit purchasing price decreases incrementally for quantities beyond certain thresholds, rather than all units getting the discount.

• Suppliers offer these discounts to encourage customers to purchase larger quantities per order. Large orders help subsidize the buyer’s inventory holding costs.

• The total annual cost function is segmented, with different cost levels for different quantity ranges based on the pricing tiers.

• To find the optimal order quantity, an algorithm starts with the lowest cost tier and checks if the EOQ is feasible there. If not, it moves to the next higher tier. It stops when it finds a feasible solution with minimum cost.

• For incremental discounts, the purchasing cost and average cost expressions are written considering the incremental nature of the discounts applied beyond each threshold quantity.

• The optimal order quantity is the one that minimizes the total annual cost function, which balances purchasing, holding and ordering costs across the different discounted pricing tiers.

• The algorithm for determining the optimal order quantity in the incremental quantity discount case involves three steps:

1. Compute potential optimal order quantities Q*j for each discount level j by setting the derivative of the cost function Zj(Q) equal to 0.

2. Check which potential Qj values are feasible, meaning they satisfy qj ≤ Q < qj+1.

3. Calculate the costs Zj(Q*j) for the feasible solutions and select the minimum cost as the true optimal order quantity.

• The algorithm is then illustrated with a numeric example involving three discount levels.

• The optimal solution is found to be Q* = 294.39 units, with a corresponding average annual cost of $2611.45.

• This emphasizes that the optimal solution cannot be equal to the breakpoint quantities qj, and must be determined using the three-step algorithm.

So in summary, the passage describes an algorithm to solve for the optimal order quantity in the incremental quantity discount case, and illustrates it with an example problem. The key aspect is that the optimal Q* cannot be a breakpoint quantity qj.

Here are the key points about power-of-two (PO2) policies:

• PO2 policies consider reorder intervals (instead of order quantities) as the decision variable, which is more practical for production planning.

• The reorder interval must be an integer multiple of a base planning period (TL), so possible values are T = nTL where n is a positive integer.

• Rather than find the optimal solution, PO2 policies restrict the reorder interval to be a power of two times the base period (T = 2kTL). This makes the interval very easy to implement in practice.

• It is proven that the worst-case cost under a PO2 policy is no more than 6% higher than the optimal cost. So PO2 policies provide a good compromise between optimality and practicality.

• PO2 policies can be applied to both single-stage and multi-stage inventory/production systems. The base period and constraints may differ depending on the characteristics of each stage.

• By formulating the problem with reorder interval as the decision variable, it is easier to ensure availability of all components needed for production at each stage in a multi-stage system.

• The passage discusses inventory management policies where the reorder interval (T) is constrained to be a power of two multiple of the base planning period. This is known as a power-of-two (PO2) policy.

• There are advantages to using PO2 policies over other polynomials for scheduling production. The worst-case costs are lower for PO2 policies compared to other polynomials like power of three.

• An example is given of an automotive manufacturer that scheduled production on a weekly, bi-weekly, or monthly basis, following a PO2 structure.

• Balancing machine loads is easier with PO2 policies if multiple products share a machine, by ensuring their production cycles do not fully overlap.

• For a single-stage system, the optimal PO2 reorder interval T is the smallest power of two that satisfies certain cost inequalities, compared to higher power of two intervals.

• The cost of the optimal PO2 policy is proved to be at most 6% higher than the true optimal cost in some cases.

• Serial inventory systems with multiple stages are introduced, along with definitions of demand rates and lead times at each stage.

• The model considers a serial system with n stages where components are produced sequentially from stage 1 to stage n.

• The reorder interval Ti for each stage i must be a power-of-two multiple of a base planning period TL, so Ti = 2liTL where li is a nonnegative integer.

• A nested policy is assumed, meaning if stage i orders, stages 1 through i-1 also order simultaneously. The policy is also stationary, with fixed reorder intervals Ti over time.

• The objective is to minimize the total average annual cost, which includes the fixed ordering cost Ki/Ti for each stage i plus the holding cost giTi at each stage. The holding cost rate gi is proportional to the echelon holding cost h’i.

• An initial relaxation RP is solved that doesn’t impose the power-of-two constraint on Ti, just the nesting constraint Ti ≥ Ti-1. The solution to RP is then used to solve the full problem P with the power-of-two constraints.

• Definitions are provided for subgraphs and components that will be relevant to modeling subsystems within the overall serial system.

The passage provides explanations and examples regarding solving an optimization problem (RP) on a serial system using a “string algorithm”.

Some key points:

• (RP) seeks to minimize the total cost of holding inventory across stages in a serial system, given holding costs and reorder costs for each stage.

• The problem can be solved optimally by partitioning the serial system into ordered subgraphs and assigning a common reorder interval to each subgraph.

• A string algorithm is presented to compute this optimal partition in a step-by-step manner. It works by sequentially examining each node and either adding it to the current subgraph or starting a new subgraph, based on comparingreorder interval ratios.

• Thealgorithm ensures the reorder intervals are non-decreasing between subgraphs, satisfying the feasibility conditions for an optimal solution.

• An example application of the algorithm on a 5-stage serial system is shown, computing the optimal partition and corresponding reorder intervals.

• The key steps of the algorithm are explained, particularly step 3 which drives the partition decisions based on comparing interval ratios between the current subgraph and potential new node.

So in summary, the passage presents an optimization problem for serial inventory systems, defines conditions for an optimal solution, and introduces a string algorithm to computationally obtain that optimal solution. An example demonstrates applying the algorithm.

Here is a summary of the key points about solving the optimization model for a multi-echelon distribution system with one central warehouse and multiple regional warehouses:

• The model is to minimize total holding and ordering costs, with costs charged proportionally to echelon stock levels.

• The reorder intervals at each location must be powers of two multiples of a base planning period.

• A two-step solution approach is used:

1. Relax the power-of-two constraint and solve for optimal (unconstrained) reorder intervals.

2. Use the relaxed solution to determine the optimal power-of-two reorder intervals.

• For the relaxed problem, an ordered partition of the graph is found such that some regional warehouses share the central warehouse’s reorder interval, while others use their own optimal intervals.

• Regional warehouses are added to share the central warehouse’s interval only if using their own interval would make it infeasible.

• This method forms an optimal ordered partition that satisfies the theorems for optimality.

• The relaxed solution is then used to determine the smallest power-of-two multiples that satisfy the nested policy requirements at each location.

So in summary, it uses graph partitioning and the relaxed solution approach to determine the optimal nested and power-of-two inventory policy for the multi-echelon distribution system.

Here is a summary of key points about distribution systems:

• Distribution systems involve the flow of goods from suppliers to customers through multiple levels or echelons, such as manufacturers, warehouses, retail stores, etc.

• They can be modeled as graphs with nodes representing inventory locations and arcs representing the flow of goods between nodes.

• The goal is to determine optimal reorder intervals and policies that minimize total system costs like holding costs and ordering/setup costs.

• Costs are classified as echelon costs, which depend on stock levels at an entire echelon, or installation costs, which depend on individual node stock levels.

• Optimal reorder interval policies balance ordering/setup costs with holding costs to minimize total cost.

• Solutions involve determining the reorder intervals that satisfy certain optimality conditions derived from the relaxation of this problem.

• Power-of-two policies restrict reorder intervals to multiples of a base period, making them easier to implement in practice compared to unrestricted policies.

• Multi-echelon systems add complexity but can be addressed using similar optimization techniques as single-echelon systems.

• Joint replenishment problems involve jointly determining reorder intervals for multiple products ordered through a common supplier.

Here are the steps to solve this problem:

1. The demand per period is 30 folders. Let’s denote this by D.

2. There are 26 periods in a year.

3. A power-of-three policy orders every 3 periods. So the order quantity is Q = 3D = 3 * 30 = 90 folders.

4. The average inventory is (Q/2) = (90/2) = 45 folders.

5. The holding cost per folder per period is h.

6. The total holding cost per year = Average inventory * Holding cost per period * Number of periods = 45 * h * 26 = 1170h

7. The ordering cost per order is K.

8. Number of orders per year = Total demand / Order quantity = 26 * 30 / 90 = 10

9. Total ordering cost per year = Number of orders * Ordering cost per order = 10 * K = 10K

10. Total cost per year = Holding cost + Ordering cost = 1170h + 10K

11. From the economic order quantity (EOQ) formula, the optimal (minimum) total cost is: = √(2KD/h)

12. Ratio of power-of-three cost to optimal cost is: Total cost with power-of-three policy / Optimal total cost = (1170h + 10K) / √(2KD/h) = 1.1547 < 1.1547 * 100% = 15.47%

So the cost of the power-of-three policy is at most 15.47% more than the optimal policy cost.

3.3. A retail store stocks four products A, B, C and D. The relevant data are:

Product Demand (units/week) Holding Cost ($/unit/week) Ordering Cost ($/order) A 30 2 10 B 20 3 10 C 50 1 10 D 15 4 10

All demands are independent. Compute the economic order quantity (EOQ) and the total cost per week for each product assuming that the products are ordered individually. Then compute the optimal joint order quantities and the total cost if the products are ordered jointly.

Here is a summary of the key points about the town. The deliveries from the suppliers:

• First arrive in the warehouse located in town. This is where materials from suppliers are initially received and stored.

• Materials are then repackaged in the warehouse as needed for shipment to the individual store locations.

• Items are transferred from the warehouse to stores within a few hours, so transfer time is negligible.

• The fixed cost to place an order with a supplier is $20 per order.

• The fixed cost to transfer items from the warehouse to a store is $10 per transfer/order.

• The holding cost at the warehouse is $0.30 per unit per year.

• The holding cost at each individual store is $0.50 per unit per year.

• The problem is to determine the optimal replenishment policy for placing orders with suppliers and transferring inventory to stores, using a periodic order quantity (POQ) approach with a weekly base planning period.

• The dynamic lot sizing problem involves determining optimal order quantities over multiple time periods to minimize total costs, which include fixed ordering costs, variable purchasing costs, and inventory holding costs.

• The objective is to find the order quantities y1, y2, …, yT that minimize the total cost over T periods, subject to demand d1, d2, …, dT and inventory balance constraints.

• Variable purchasing costs can be removed from the objective function since the total purchases only depend on exogenous demand values and not the decision variables.

• In any optimal solution, either the order quantity or beginning inventory in each period must be zero. Further, if demand in a future period is met by an order, it will meet demands sequentially in consecutive future periods.

• The problem can be solved recursively by considering truncated T-period problems and determining period-by-period optimal ordering policies, assuming zero ending inventory each period.

• The solution approach is to compute costs F(t) and Cs recursively to determine optimal order-up-to levels period-by-period.

• The Wagner-Whitin algorithm uses dynamic programming to solve the dynamic lot sizing problem with deterministic demand over multiple periods.

• It works by progressively considering problems over larger time horizons, starting with just the first period and building up to the full planning horizon.

• In each iteration, it computes the minimum cost for satisfying demand from period 1 up to the current period t, given an order is placed in period v.

• It tracks the optimal cost F(t) and period v in which to place the order for each iteration.

• Using the planning horizon theorem, it limits the possible periods to place an order for period t+1 to be v or later, based on the solution for period t.

• This avoids recalculating costs for all prior periods in each iteration, making it more efficient than the shortest path network formulation.

• The algorithm outputs the optimal production plan, including the periods in which to place orders to satisfy all demand at minimum total cost.

So in summary, it’s a dynamic programming approach that recursively solves subproblems from smaller to larger time horizons to find the optimal lot sizing solution.

• The WHK algorithm solves the dynamic lot sizing problem backwards, starting from period T and iteratively adding previous periods. This is in contrast to the WW algorithm which works forwards.

• Working backwards one period at a time, the algorithm focuses only on that period and ignores previous ones. It then incrementally incorporates previous periods as it moves backwards.

• The worst-case computational time of the WHK algorithm is O(T log T) for problems with time-varying costs, compared to O(T^2) for the WW algorithm. It is O(T) for problems with stationary costs.

• Efficient periods are identified which reduce the number of periods that need to be considered at each step. Only efficient periods from the previous step need to be checked.

• The algorithm uses dynamic programming to recursively compute the optimal cost Zt for each period t by determining the optimal period s to carry demand from t to.

• It works by identifying efficient periods algebraically rather than graphically as the number of periods increases backwards from T to 1.

So in summary, the WHK algorithm improves upon the WW algorithm by working backwards period-by-period and leveraging the concept of efficient periods to achieve better worst-case computational performance.

• The Silver-Meal heuristic tries to find an order quantity that minimizes the average cost per period, based on the EOQ formula.

• It starts with an order in period 1 to satisfy demand in period 1 only. Then it iteratively expands the order quantity to include subsequent periods’ demand, as long as this decreases the average cost.

• If adding a period’s demand causes the average cost to increase, then that period starts a new order with its demand only.

• The heuristic calculates the average cost C(s,t) for different order quantities from period s to t, and expands the order up until the point where adding another period causes the average cost to rise.

• This gives a heuristic solution that aims to minimize average cost locally at each step, without guaranteeing an optimal solution overall. But it provides a reasonable policy with fewer computations than exact algorithms.

• An example application of the Silver-Meal heuristic on a sample data set is provided to illustrate how it works step-by-step.

Here is a summary of ot sizing with deterministic demand:

• Deterministic demand is known with certainty for each period in the planning horizon.

• The objective is to determine the optimal production/ordering quantities and timing to minimize total setup and holding costs.

• The Wagner-Whitin model formulates this as a dynamic programming problem where the optimal policy satisfies the principle of optimality.

• The optimal policy can be found using the Wagner-Whitin algorithm, which solves the dynamic program in a forward recursion.

• Heuristic methods like the Silver-Meal heuristic and least unit cost heuristic provide near-optimal solutions with less computation. They determine when to place a new order based on average cost measures.

• Properties of the optimal policy include lot-for-lot production, where demand for multiple periods is satisfied in one production lot. There is also a planning horizon property where the optimal solution depends only on a finite planning horizon.

• Variations include stochastic demand, capacity constraints, sequence-dependent setup costs, and time-dependent costs. But the basic model captures the tradeoff between setup and holding costs.

• This is a single-period inventory problem where a decision needs to be made each week on how much stock to place on an offshore oil rig to meet demand for a critical part.

• Demand is uncertain and represented by a random variable, since the number of part failures per week cannot be predicted precisely in advance.

• The objective is to determine the optimal stocking level that balances the costs of stockouts versus excess inventory.

• Historical demand data over the past 100 weeks is available and shows that demand is stationary and independent from week to week.

• This data will be used to fit a demand distribution that can model the uncertainty in demand.

• Cost data is also needed, such as the cost of airlifting parts for stockouts versus refurbishing excess inventory.

• The optimal stocking level will be determined by analyzing the tradeoff between these costs under the assumed demand distribution. This will provide a policy for setting weekly stock levels to meet uncertain demand at minimum expected total cost.

• The passage discusses modeling uncertainty in demand rather than using the actual historical demand data as the sole predictor of future demand. It notes some limitations of using just the historical data.

• It constructs a histogram of the historical demand data and finds that it resembles a normal distribution. So it will model demand as a normally distributed random variable.

• It provides some cost parameters: $1000 per unit for refurbishing unused inventory, $9000 per backorder.

• It builds a decision model to determine the optimal stocking level each week. The objective is to minimize expected weekly operating costs.

• It simulates stocking levels of 30, 40 and 100 units and graphs the resulting costs and demand over 100 weeks to compare the outcomes.

• It develops a mathematical expression for the expected weekly cost function G(s) given a stocking level s, modeling demand as a normally distributed random variable. This allows calculating the optimal s to minimize expected costs.

So in summary, the passage discusses modeling uncertainty in demand as a normal distribution rather than using just historical data, and develops an expected cost framework to determine the optimal stocking level.

• The article describes single-period inventory models for determining the optimal order quantity that minimizes total expected costs when demand is uncertain.

• Total costs include setup, holding, backordering/lost sales costs. The optimal quantity balances these factors.

• Demand is assumed to follow a known probability distribution like normal or Poisson.

• The newsvendor model provides a solution for the optimal quantity when costs are linear and demand is random.

• The solution sets the stockout probability equal to the ratio of unit backorder cost to total unit cost.

• The models are extended to handle multiple products with finite storage capacity, discrete demand distributions, and demand forecast updates.

• Excel can be used to compute solutions numerically given distribution parameters of demand.

• Higher demand uncertainty increases optimal quantities and expected costs to hedge against stockouts.

So in summary, it presents single-period inventory models to determine optimal order quantities balancing costs under uncertain demand described probabilistically. The newsvendor model and extensions are key solutions.

• The problem involves finding the optimal stock levels for two part types (denoted part 1 and part 2) to minimize the total expected weekly operating costs while satisfying a space constraint.

• The expected weekly operating cost functions Gi(si) for each part type i depend on the stock level si and factors like holding and shortage costs.

• The space constraint is that the total stock levels s1 + s2 cannot exceed the available storage space V.

• An algorithm called marginal analysis is used to find the optimal stock levels. It considers incrementally increasing the stock level of the part that reduces expected costs the most at each step, until the space constraint is reached.

• Tables are provided showing the expected cost functions Gi(si) and the cost reductions ΔGi(si) from incrementing each stock level.

• For the given example where V=40, the optimal unconstrained solution is s1*=14 and s2*=26, which satisfies the space constraint of V≥40.

• The algorithm is also explained for situations where V<40, requiring the stock levels to be adjusted down from the unconstrained solution to respect the tighter space constraint.

• Finally, a more general multiple-item single-period model is formulated as an optimization problem minimizing the total expected costs subject to a resource constraint on the total stock levels. Solution methods like Lagrangian relaxation are discussed.

• The passage describes how to model and solve a single-period inventory problem with multiple items and multiple constraints.

• It introduces constraints on total resources (like budget or storage space) that link the purchase decisions for multiple items. These are represented as linear constraints on the total resources consumed.

• The objective function for profit from each item is non-linear due to the expected shortage term. This is transformed into a linear function using auxiliary 0-1 variables.

• This transformation allows the overall multi-item, multi-constraint problem to be formulated as a linear program. Solving the linear program provides an optimal inventory quantity for each item.

• Fractional values in the linear program solution can be resolved using rounding or setting to 0, since resource coefficients are typically small relative to available quantities.

• The method shows how more complex single-period inventory problems with probabilistic demand can often be modeled and efficiently solved as linear programs through variable transformations.

So in summary, it presents an approach to formulate and solve a multi-item, multi-constraint probabilistic inventory problem as a linear program through linearizing the objective functions.

Here is a summary of the key points in the passage:

• The chapter discusses inventory planning over multiple time periods when costs are linear (fixed ordering costs are negligible).

• It will establish that a base-stock or order-up-to policy is optimal in several environments, including finite horizons, deterministic lead times, and linear costs (purchasing, shipping, holding, backordering).

• A dynamic programming model will be developed that can find optimal stocking decisions over multiple periods.

• Properties of optimal solutions will be derived from the dynamic programming model.

• The impact of production capacities on stocking decisions is analyzed.

• Finally, it discusses multi-echelon inventory systems and how the role of each location differs within a supply chain.

The main topics covered are determining optimal policies for multi-period inventory planning when costs are linear, developing a dynamic programming model, analyzing properties of optimal solutions, incorporating capacities, and discussing multi-echelon inventory systems. The chapter provides an analytical framework for optimizing inventory decisions over multiple time periods.

• The approach considers a single-stage, periodic review inventory system with a single location and stochastic, independent demand across periods. Orders face a fixed lead time.

• The system can be decomposed into independent subsystems, each consisting of a single unit-customer pair.

• It is shown that each subsystem can be optimized independently using a “critical distance” policy, where a unit is ordered if the corresponding customer’s distance is below a critical threshold.

• When each subsystem follows its optimal critical distance policy, the overall system implements a base-stock policy, where the base-stock level is determined by the critical distances.

• This establishes that base-stock policies are optimal for this single-stage, single-location system with fixed lead times and stochastic demand.

The approach can be extended by making some of the basic assumptions more general, such as allowing:

• Random/stochastic lead times

• Multiple stages in a supply chain

• A more complex demand model than independent periods

• Limited production capacities in each period for serial systems

• An infinite planning horizon rather than a finite number of periods

• S denotes the overall system, and S_w denotes subsystem w. The group of all subsystems is denoted S_.

• When only monotone and committed policies are considered, the constraints in Definition 6.2 apply to S while the constraint in Definition 6.3 applies to S_.

• “The (optimal) expected cost for S_” means the sum of the (optimal) expected subsystem costs.

• While the subsystems are dependent through demand, they can be managed independently without affecting each other’s policies.

• The state for subsystem S_w is x_w = (z_w^n, y_w^n), which is sufficient for determining its optimal policy.

• An optimal policy for S can be found by independently managing each subsystem S_w using its state x_w. This decentralized policy is also optimal for the overall system S.

• The thesis then proves this result about the equivalence of optimal policies and costs for S and S_. It also shows the optimal policy for each subsystem S_w has a “critical distance” structure.

• The problem of finding optimal stock levels over a finite planning horizon is modeled as a dynamic programming problem.

• The horizon is divided into periods n=1,…,N. The cost function gn(y) represents the total expected discounted cost from period n to the end, given an inventory position of y at the start of period n.

• gn(y) satisfies a recursion involving the one-period cost Cn(s) of ordering to level s, plus the expected future cost gn+1(s-Dn) discounted by α.

• The base case is gN+1(y) which represents costs of overage/underage at the end of the horizon.

• Solving the recursion backwards gives the optimal stock levels sn* that minimize gn(y) in each period.

• The model assumes iid demand over time but can be extended to more complex demand processes like Markov-modulated demand.

• The analysis presented is for a zero lead time case but can be extended to positive lead times in a similar way.

• The problem describes finding optimal order-up-to levels (stock levels) for inventory decisions over multiple periods, taking into account how decisions in one period impact future costs.

• A recursive method called dynamic programming can be used to solve this optimization problem. It represents the problem as a set of “functional equations” relating the cost functions across periods.

• For the oil rig inventory problem, the approach initially taken only looked at single period costs (myopic policy), ignoring cross-period impacts.

• Assuming all inventory gets used or refurbished each period, the myopic policy is actually optimal when the discount factor is 1.

• When the discount factor is <1, the multi-period formulation needs to account for inventory remaining at the end of each period.

• The myopic order-up-to levels provide upper bounds on the true optimal levels, due to consideration of future costs.

• Examples are presented comparing myopic and optimal levels under different demand distributions. When demand trends are consistent, the myopic levels are often close to optimal. Heavier-tailed distributions like geometric can lead to larger differences.

• In general, the myopic approach provides a practical approximation when true optimal levels are difficult to compute, especially if demand patterns don’t change significantly over time.

• The optimal order-up-to levels differ from the myopic order-up-to levels in periods 2 and 3, due to considering the impact on future periods.

• Following a myopic policy rather than the optimal policy increases costs by 8.205% in this example. However, the cost difference is not too large.

• With a positive lead time τ, inventory ordering decisions affect costs over periods n through n + τ - 1. The optimal order-up-to level reflects demand needs over periods n through n + τ plus future impacts.

• The myopic approach only considers demand over periods n through n + τ.

• With stationary demand and costs over an infinite horizon, the myopic order-up-to level s* is also the optimal level.

• With non-stationary demand, s* values are stationary but represent different demand distributions over time.

• Estimating future demand distributions can help address end-of-horizon effects on choosing s*.

• The lost sales case is more complex, as the optimal order quantity depends on on-hand and on-order inventory levels, not just the inventory position.

• The passage introduces a capacity limited production system with a single item produced each period to meet demand. Production capacity is limited to C units per period.

• If demand exceeds available inventory plus production capacity, the unsatisfied demand is backordered. If demand is less, all demand is satisfied. Holding and backorder costs are incurred.

• The optimal policy is a modified base-stock policy where production aims to reach a target inventory level s, but is capped at C if capacity is insufficient.

• The “shortfall” is defined as s minus the actual ending inventory. Its distribution, called the shortfall distribution, is important to analyze as it determines inventory requirements.

• The shortfall follows a Markov process and its steady state distribution can be derived by modeling it as a Markov chain with transition probabilities.

• Tables and graphs show how the mean and variance of the shortfall distribution change with demand variance and capacity utilization. Higher variance or lower capacity increase shortfall.

• The passage uses an oil rig inventory example to illustrate how the shortfall distribution would apply in a capacity constrained setting.

• The system involves an oil rig (facility 1) that places replenishment orders with an onshore stocking facility (facility 2). Facility 2 in turn places orders with an external supplier.

• The lead time from facility 2 to facility 1 is assumed to be negligible.

• An echelon inventory approach is used to analyze the system, where inventory positions consider stock at both facilities.

• A cost model is developed to minimize total expected holding and shortage costs over the planning horizon. Costs include:

  • Holding cost h1 per unit at facility 1
  • Shortage cost b per unit at facility 1
  • Echelon holding cost h’2 per unit for facility 2

• The goal is to determine the optimal echelon base stock levels for each period that minimize total expected costs over the planning horizon.

• A dynamic programming-like approach is outlined to compute the optimal stocking policy by considering costs period by period.

So in summary, it presents an echelon inventory approach to analyze and optimize a two-facility serial supply chain system with the objective of minimizing total expected holding and shortage costs.

Here is a summary of the key steps in the algorithm presented for computing optimal echelon stock levels in a serial system with stationary demand:

1. Find the optimal stock level y1* for the onshore facility by minimizing the expected cost function C1(y). This is done by working backwards from the final period N.

2. Given y1*, calculate the incremental cost function Δ2(I2) for the oil rig. If the oil rig’s stock level I2 is below y1*, there is an incremental cost. Otherwise the cost is 0.

3. Determine the optimal stock level y2* for the oil rig by minimizing its expected cost function C2(y) plus the expected incremental cost E[Δ2(y - D1)]. D1 is demand over the lead time plus one period.

4. The algorithm is extended to systems with more than two facilities by recursively determining the optimal stock level for each facility working upstream from the end customer, and accounting for incremental costs imposed on upstream facilities by stocking decisions further downstream.

The key points are: 1) determining stock levels recursively working backwards in time and upstream in the serial system, 2) representing incremental costs imposed on one facility by another, and 3) finding optimal stock levels by minimizing expected costs incorporating incremental costs.

• The (s-1, s) policy means that an order is placed whenever inventory position drops to s-1, to raise it back up to s.

• The analysis focuses on the stationary/steady state distribution when costs and demand are constant over an infinite horizon.

• A key variable is the quantity in resupply, which is the amount on order at a random point in time. Its distribution determines performance measures.

• For Poisson demand, the time of each demand arrival is uniformly distributed. This allows calculating the distribution of quantity in resupply.

• The distribution is derived by modeling the order arrivals as a Poisson process and using properties like stationary and independent increments.

• The calculation shows the quantity in resupply follows a discrete distribution that can be used to determine other variables like on-hand and backordered inventory.

• The analysis is presented first for the backorder case, then extended to lost sales. Key measures like average inventory and backorders can then be determined.

• Optimization models are developed later to determine the optimal s level minimizing costs for a given demand distribution and (s-1, s) policy.

So in summary, it develops the theoretical foundation for analyzing steady state behavior and optimization of the (s-1, s) policy under Poisson demand through the distribution of the quantity in resupply.

• The fill rate is defined as the expected fraction of demands that can be satisfied immediately from on-hand stock, given a stock level s. As s increases, the fill rate also increases intuitively.

• The ready rate is the probability that an item observed at a random point in time has no backorders, meaning its net inventory is non-negative. It measures whether there are any backorders or not.

• Expressions for the fill rate F(s) and ready rate R(s) will be derived based on the steady state probabilities developed earlier for the number of units in the resupply system.

• These measures (fill rate and ready rate) relate to performance at a single location for a single item, assuming the backorder case.

• Subsequent discussion will cover how to compute steady state probabilities and performance measures in multi-echelon situations.

So in summary, the key performance measures of fill rate and ready rate are defined, and expressions for them will be derived based on the previous steady state analysis, for a single item/location assuming backorders.

• When computing fill rate or ready rate, we are only concerned with what fraction of demands can be satisfied immediately, not how long backordered demands have to wait.

• A high fill rate does not necessarily mean customer needs are fully satisfied, as some items’ demands could be filled immediately while others have 0% fill rate.

• The ready rate is always at least as high as the fill rate. When stock level is 0, fill rate is 0 but ready rate could be close to 1 if demand is low and lead time is short.

• Backorders outstanding (B(s)) accounts for length of time backorders exist. It measures expected number of backorders at a random time and is a response-time focused measure related to demand rate and waiting time by Little’s Law.

• For multiple item types, fill rate is calculated as a weighted average of individual item fill rates based on demand probabilities. Backorders outstanding is the sum over all item types.

• Ready rate is modified to operational rate, which is the probability all item types have stock levels at or above requirements.

• Cannibalization allows utilizing spare parts across systems, like aircraft, to keep more systems operational. Performance measures can then be applied to the system level rather than individual items.

• Fill rate, ready rate and backorders outstanding are useful for optimization but have different properties that must be considered, like fill rate not being a concave function.

This passage describes using a Lagrangian relaxation approach to solve an optimization problem of minimizing expected backorders subject to an inventory investment constraint.

The key steps are:

1. Formulate the optimization problem involving minimizing expected backorders subject to a budget constraint on average inventory investment.

2. Introduce a Lagrangian multiplier θ for the budget constraint, yielding a relaxed problem that decomposes by item type and can be solved separately for each item.

3. For each item, the relaxed problem involves minimizing a convex function f(s) of the stock level s. The optimal s is the smallest s for which the discrete difference Δf(s) is non-negative.

4. Varying the multiplier θ yields different optimal stock levels s* and a relationship between the cost C(θ) and θ, with C(θ) non-increasing as θ increases.

5. The goal is to find a θ such that C(θ) is close to the budget b, by constructing the graph of minimum expected backorders versus average inventory investment obtained by solving the relaxed problems over a range of θ values.

So in summary, it uses Lagrangian relaxation to decompose the problem and gradient-based search over the Lagrangian multiplier to approximate the efficient frontier tradeoff between backorders and inventory investment.

Here is a summary of the key points about a tactical planning model for managing recoverable items in multi-echelon systems:

• The model deals with managing recoverable or repairable items across multiple inventory locations or echelons (e.g. depot, zone, local). These items are repaired/refurbished after failure rather than being discarded.

• The goal is tactical planning over a finite planning horizon, typically months to a year in the future. This is different than operational decision making with shorter timeframes.

• The model considers items moving between echelons due to demand as well as returns for repair/refurbishment. It keeps track of inventory levels and status (serviceable vs. repairable) across locations.

• It aims to determine replenishment, repair, and redistribution policies/quantities to minimize total system costs over the planning horizon. Costs include holding, backorder, repair/refurbishment, transportation, etc.

• Forecasts are made for demands, returns, and repair lead times. A probabilistic approach may be used to model uncertainties.

• Linear programming or other quantitative optimization techniques are commonly applied to solve the multi-location, multi-period model and determine optimal policies.

• The output provides plans and targets for inventory levels and movement of items between locations over the planning horizon to effectively support item availability goals.

So in summary, it is a quantitative, optimization-based model to determine tactical plans for managing recoverable items across the supply chain over a multi-period planning horizon. The goal is to minimize total costs while meeting availability targets.

• The paper describes METRIC, a tactical planning model developed by Sherbrooke to manage inventory of repairable items like parts for the US Air Force.

• The model considers a multi-echelon system with bases and a central depot stocking and repairing items. When an item fails at a base, it is either repaired locally or sent to the depot.

• The goal is to determine optimal stocking levels (sij) at each location to minimize total backorders, approximating the goal of maximizing operational aircraft.

• Key assumptions include Poisson failures, (s-1,s) review policies, and no lateral resupply between bases.

• The expected resupply time Tij factors in probabilities of local vs. depot repair and associated repair times.

• Depot delay δ(si0) is estimated from expected backorders using Little’s Law. Approximations show δ decreases rapidly as si0 exceeds demand rate λi0.

• The paper summarizes Sherbrooke’s original METRIC model and important improvements by Graves, Sherbrooke, Muckstadt and O’Malley to make it more tractable for practical applications.

Here is a summary of the tables and equations presented:

• Tables 8.3, 8.4, 8.5, and 8.6 show values of δ(si0) for different values of λi0 and Di. δ(si0) represents the expected depot delay.

• As λi0Di remains constant but Di increases, δ(si0) increases proportionately to the increase in Di.

• The range of values needed to evaluate si0 explicitly can usually be limited to between ⌊λiDi⌋ and ⌈2 · (λiDi)1/2⌉ + ⌊λiDi⌋.

• The stationary distribution of the number of LRUs in resupply (Xi j) is approximated as a negative binomial distribution with parameters:

Mean (μi j): ri j λi jBi j + (1 − ri j)λi jAi j + (1 − ri j)λi 0D(si 0)/λi 0

Variance (σ2): ri j λi jBi j + (1 − rij)λijAij + ((1 − ri j)λi j)2Var(ND|si 0)/λi 02

• Formulas are provided to calculate the expected number and variance of depot backorders (ND) recursively based on the Poisson demand process assumptions.

• The variance of demand at base j (Yj) over the supply lead time is calculated based on the variance of demand during supply lead time and variance of backordered demand due to depot stock level si 0.

• The problem seeks to determine optimal stock levels (sij) for each LRU type i at each base j to minimize expected backorders, subject to a budget constraint on total inventory investment.

• A marginal analysis approach is used to iteratively allocate inventory units to minimize expected backorders. Convex minorants are constructed to deal with non-convexity.

• A Lagrangian relaxation is also used, relaxing the budget constraint and allowing the problem to be decomposed into independent subproblems for each LRU type. The Lagrangian multiplier is varied to approximate the original problem.

• For each method, expected backorder functions are constructed based on demand distributions to evaluate stocking levels. The marginal analysis approach incrementally allocates inventory, while the Lagrangian method solves independent problems for each LRU type for different multiplier values.

• Both aim to determine stock levels that minimize expected backorders across all LRU types within the given budget constraint. The marginal analysis does so directly, while the Lagrangian method approximates a solution.

• The problem formulates the optimization of depot and base stock levels to minimize expected base-level backorders for items.

• It uses a Lagrangian relaxation approach where we introduce a coefficient θ on the backorder terms and solve the resulting relaxated problem.

• Given a value of θ, we can find the optimal depot and base stock levels for each item using an exhaustive search approach.

• Two algorithms are presented - one uses bisection search to find the optimal θ value that meets the budget constraint, while the other pre-specifies a set of θ values to evaluate possible solutions.

• The second algorithm is more efficient as it only needs to solve each sub-problem once rather than repeatedly as θ values are updated in the bisection search.

• It also derives the distribution of waiting times for an item request, taking into account the current inventory, items in repair, and arriving completions to determine the probability a request takes over a given time u to fulfill.

So in summary, it formulates the stock level optimization problem, solves it using Lagrangian relaxation and presents two algorithms for finding the optimal solution subject to a budget constraint, and also analyzes the distribution of waiting times.

• This chapter examines reorder point, lot size models where there is a continuous review system and significant fixed ordering costs.

• Under these models, a constant lot size Q is ordered whenever the inventory position (on-hand inventory + on order - backorders) reaches the reorder point r.

• The reorder point and lot size are chosen to minimize the average annual costs, which include holding costs and fixed ordering costs.

• The analysis assumes the demand process is stationary and memoryless, such as a Poisson process. This allows the reorder decision to depend only on the current inventory position, not on when the last demand occurred.

• The (Q, r) models assume the inventory position is never allowed to fall below the reorder point before an order is placed. There is no overshooting of the reorder point.

• Approximations are made to simplify the probability distributions and calculation of the objective function. The results are approximations but aim to provide insights into how to determine optimal reorder points and lot sizes.

So in summary, this chapter examines models where fixed ordering costs are significant and demand is reviewed continuously to decide when to place orders of a fixed lot size Q, based on the inventory position reaching the reorder point r.

• This discusses an (Q, r) inventory control model where demand occurs as a Poisson process, usually for single units.

• With integer reorder points and single unit demands, the reorder point will be reached each time an order is placed. But when demand sizes are random, the reorder point may not be reached each cycle.

• Real inventory systems are seldom strictly continuous review or transaction-based, but use periodic order points like daily even if demands are recorded continuously.

• The goals are to develop (Q, r) models minimizing average annual costs, first using a simple approximate model commonly used in practice, then a more exact Poisson process model, and discuss how the results differ.

• The models find optimal Q and r values for a single item at one location. Approximations also exist for managing multiple items across a system.

• When demand sizes are non-unit, reorder point overshoot is possible requiring alternate policies.

So in summary, it introduces common (Q, r) inventory models, discusses their assumptions and limitations, and aims to find optimal values through both approximate and exact formulations considering key cost factors.

• The passage describes an approximate model for computing the optimal lot size Q* and reorder point r* when backordering is permitted for a periodic review inventory system with stochastic demand.

• The goal is to minimize the expected total cost per unit time, which includes holding, ordering, and backorder costs.

• The model assumes demand during lead time is represented by a probability distribution function f(x).

• Equations are derived for Q* and r* by setting partial derivatives of the cost function equal to zero.

• This results in two non-linear equations that must be solved numerically using an iterative algorithm.

• The algorithm starts with an initial guess for Q based on the economic order quantity formula, and iteratively updates Q and r until convergence.

• For the objective function to be convex, certain conditions on f(x) must hold, such as being non-increasing past some point.

• When demand is normally distributed, closed form expressions can be obtained and the objective function is convex when r is greater than the mean demand.

• An example application of the algorithm is shown for the normal demand case using tables to calculate values needed at each iteration.

So in summary, the passage presents an approximate model for determining optimal lot sizing and reorder points when backordering is allowed, and describes how to solve it numerically, particularly for the case of normal demand distribution.

• An exact model for determining reorder points and lot sizes is developed based on assuming demand follows a Poisson process. This allows for the possibility of more than one order being outstanding at a time.

• The reorder point will be based on inventory position (on-hand inventory plus on-order inventory minus backorders) rather than just on-hand inventory, since multiple orders could be outstanding.

• The exact representation is developed using the stationary probability distribution of net inventory levels. This allows expectations in the objective function to be computed precisely rather than using approximations.

• Important assumptions include demand being a Poisson process, constant lead time measured in years, and Q and r being integer-valued to match the discrete demand.

• The exact model relaxes some of the restrictive assumptions of the approximate model, like assuming at most one order outstanding and inventory levels being a good proxy for inventory position.

So in summary, it develops a more precise model by using an exact representation of net inventory levels and inventory position rather than approximations, to determine optimal reorder points and lot sizes.

• The system places a new procurement order when inventory position drops to the reorder point r due to a customer demand. This raises inventory position to r + Q.

• Inventory position assumes values from r+1 to r+Q after an order is placed. A cycle is the time between entering state r+Q and the next time in that state, which has an expected length of Q/λ.

• The analysis will first derive the stationary probability distribution of inventory position, then use this to determine the distributions of net inventory and various performance measures like stockouts, backorders, and on-hand inventory levels.

So in summary, when a demand occurs that reduces inventory position to r, a new order for Q units is immediately placed to restore inventory position to between r+1 and r+Q. This cycle then repeats until the next time inventory position reaches r+Q again.

• Equations (9.36) and (9.37) provide formulas for calculating E[N] and E[N^2] respectively by evaluating the Laplace transforms of the waiting time distribution at z=1.

• Var(N) = E[N^2] - E[N]^2, so the variance of N can be found using (9.36) and (9.37).

• (9.36) evaluated at z=1 yields E[N], which equals λE[W] by Little’s law.

• (9.37) evaluated at z=1 yields E[N^2], which can be used to calculate Var(N) = E[N^2] - E[N]^2.

• Var(W) can then be found using the relationship Var(N) = λE[N] + λ^2Var(W) shown in (9.47).

So in summary, (9.36) and (9.37) provide the means to calculate key performance metrics like E[N], Var(N), and Var(W) for the continuous review inventory model by utilizing the Laplace transforms of the waiting time distribution.

• The passage discusses optimization of the reorder point and lot size models under different demand distribution assumptions (normal, Laplace, Poisson).

• For the normal demand model, closed-form expressions can be derived for the optimal Q* and r* under certain simplifying assumptions like no backorder costs.

• For the Laplace demand model, the optimal Q* and r* can be found without the need for numerical search by exploiting properties of the Laplace distribution. This provides simplicity compared to the normal model.

• For the exact Poisson demand model, the optimal Q* and r* cannot be found in closed form and must be approximated numerically by constructing first difference functions of the cost function C(Q,r) and finding where the differences equal zero.

• In general, the Laplace and Poisson models are considered more accurate than the normal model for modeling demand distributions in practice. But the normal model allows more tractable optimization, while the Laplace provides a simpler solution than the normal without numerical search. The Poisson model requires numerical optimization.

• The passage presents models for determining reorder points and lot sizes when demand follows a compound Poisson process (i.e. arrivals are Poisson but order quantities are random).

• Two types of policies are considered: (nQ, r) and (s,S).

• For an (nQ, r) policy, the distribution of the inventory position random variable I is derived and shown to be uniformly distributed between r+1 and r+Q.

• For an (s,S) policy, two methods are presented to find the distribution of I: 1) using the transition matrix modeling the change process, and 2) considering cycles between crossing s.

• Computing the distribution of I allows determining the distribution of net inventory N, which is needed to calculate performance measures and average annual costs.

• The (s,S) policy is noted to be optimal for the cost model, while (nQ, r) is generally suboptimal.

So in summary, it presents analytical models and methods for determining optimal reorder points and lot sizes when demand follows a compound Poisson process rather than simple Poisson, by deriving the distribution of related random variables.

Here is a summary of the key points about constructing a transition matrix for an (s,S) inventory policy and using it to develop an exact multi-item inventory model:

• The transition matrix P represents the possible transitions between inventory position states for an item under an (s,S) policy, where S is the reorder point and s is the order-up-to level.

• P depends only on Q = S - (s-1) and the probability distribution of demand sizes, not the specific values of S and s.

• The steady state probability distribution π can be found by solving the equations πP = π, ∑πi = 1.

• This gives the long-run probability of being in each inventory position state.

• An alternative is to use a renewal equation to compute Mj, from which the π’s can also be derived.

• Given π and the demand distribution, performance measures like stockout probability, expected stockouts, and expected inventory can be computed.

• The same approach can be extended to a multi-item model that sets reorder points and quantities across many different inventory items to optimize overall costs and meet a service level target.

• The Presutti-Trepp model is given as an example used by the Department of Defense to manage hundreds of thousands of consumable inventory items.

• The inventory position approach considers the inventory on-hand plus all inventory in the pipeline (orders that have been placed but not received yet). This better reflects the total amount of money tied up in inventory.

• Computing stock levels is easier when using average inventory position rather than average on-hand inventory as the basis for calculating holding costs.

• Two types of performance constraints are considered: (1) Limiting the total expected number of backordered units per year (incident-oriented measure) and (2) Limiting the total average number of outstanding backorders at a random point in time (time-weighted measure).

• Four models are presented that consider different combinations of holding cost calculations (based on average inventory, average inventory position, or on-hand inventory) and different types of performance constraints.

• Expressions are provided for how to calculate the reorder point, safety stock level, average on-hand inventory, average backorders, and expected backorders under each model’s assumptions.

• An optimization formulation is given for each model to determine the reorder quantities and reorder points that minimize total costs subject to the performance constraint.

• Lagrangian methods are used to solve the optimization problems and obtain expressions for the reorder points and quantities under each model.

• A search method may be needed to find the optimal reorder quantities if a closed-form solution is not available.

So in summary, the passage presents a multi-item inventory model that considers different approaches to holding cost calculation and performance measurement, and provides mathematical optimization formulations to determine optimal reorder parameters under each approach.

• The goal is now to minimize average annual fixed ordering costs, holding costs, and lost sales costs. Lost sales are assumed when inventory is not available to meet demand.

• A new cost parameter π is introduced, which represents the cost of a lost sale.

• The model needs to account for both expected cycle length and expected number of orders per year. This is because some demand will now be lost, so demand during a cycle will equal Q plus expected lost sales.

• An assumption will need to be made about how to deal with expected lost sales in constructing the model and solution methodology. It will be assumed lost sales are a small fraction of total sales and that Q is large enough such that no more than one order is outstanding at a time.

• The model and solution procedure will need to determine the optimal Q* and r* values that minimize the total average annual costs considering fixed ordering costs, holding costs, and lost sales costs.

• Key steps will be to determine the expected cycle length, expected number of orders per year, and expected number of lost sales per cycle to fully specify the cost model and develop a solution approach.

This passage describes an approximation algorithm for solving a periodic review inventory control problem with fixed ordering costs and lead times of zero periods.

The key aspects of the problem and algorithm are:

• The objective is to minimize total costs over a finite planning horizon of T periods, considering fixed ordering costs, holding costs, and backordering costs.

• The optimal policy takes the (s,S) form, with reorder point s and order-up-to level S.

• The algorithm computes approximate values of s and S rather than obtaining the optimal values.

• Demand in each period is initially assumed to be unknown, but then reformulated to be known to simplify the analysis.

• The algorithm iteratively computes the total backordering cost from the last order period up to the current period, given known demands.

• It triggers an order if this backordering cost exceeds a threshold, and sets the order-up-to level S to cover demand through the current period.

• This process is repeated period-by-period over the planning horizon to determine an approximate policy.

• Bounds are established on how much worse the approximate solution may be than the true optimal solution.

So in summary, it presents an iterative, myopic approximation algorithm to solve the periodic review inventory problem with known demands, and analyzes its performance bounds.

• The algorithm is an approximation algorithm for computing a periodic review policy for an inventory system with stochastic demands, fixed ordering costs, and linear holding and backordering costs.

• It works on a concurrent basis, evaluating whether to order at the beginning of each period based on costs incurred so far.

• In each period, it first calculates the backordering cost if no order is placed. If this is above the fixed ordering cost K, it calculates the maximum order quantity such that the expected holding cost till end of horizon is ≤ K.

• The order quantity must be at least demand for the current period plus any backlog. Holding costs are calculated based on future demand distributions.

• The algorithm returns a policy that is within a factor of 3 of the optimal cost in expectation. It aims to balance holding, backordering and fixed ordering costs over time.

• Examples show how it computes costs and determines order quantity in a period given demand realizations and distributions of future demands.

Here are the key steps of the algorithm to compute the optimal (s,S) policy for infinite horizon inventory management:

1. Formulate the problem as a dynamic program with average cost per period as the objective. The state variables are the beginning inventory levels x and y.

2. Define the one-period cost function V1(y) which depends on holding and backorder costs.

3. Define two intermediate functions - the expected time until next order t(y) and expected cost until next order Cs(y). These satisfy recursive equations similar to V1(y).

4. Compute t(y) and Cs(y) for all relevant y values starting from s+1 upwards, using the recursive equations and given demand distribution probabilities.

5. Use the optimality equation g(y) = V1(y) - va* + Σpjg(y-j) with boundary condition g(S) = 0 to solve for the optimal average cost va*.

6. Check all possible combinations of s and S to find the ones that minimize va*. The optimal policy is (s*,S*) where s* and S* achieve the minimum va*.

7. The algorithm exploits the renewal property and stationarity to recursively compute the intermediate values needed to solve the optimality equations. It finds the optimal policy by enumeration over the state space.

The theorem and proof can be summarized as follows:

Theorem 10.1 states that there exists an optimal stationary policy for the periodic review inventory control problem with lost sales and lead times.

The proof has four main parts:

1. It shows that if inventory system 1 operates closer to the one-period cost minimizer compared to system 2, then the cost difference between the two systems is at most K. This relies on the convexity of the relevant cost functions.

2. It defines an ordering policy that is non-anticipating and feasible.

3. It uses induction to show that following this policy yields costs less than or equal to the optimal costs under any other feasible policy for all periods up to T.

4. It then concludes that the proposed non-anticipating policy must be optimal.

The key aspects of the proof are: defining appropriate cost functions, using convexity properties, defining a non-anticipating feasible policy, and applying induction to establish optimality of the proposed policy. Convexity and non-anticipation are important concepts used in the analysis.

• The algorithm deals with determining the order point s and order-up-to level S in an inventory system with independent periodic demand, fixed ordering costs, holding costs, and backorder costs.

• Demand over the lead time is assumed to be normally distributed. The lead time is constant at τ periods.

• It uses a heuristic approach to calculate s and S, as the exact algorithm is complex.

• s is set such that the inventory position meets demand over τ+1 periods with sufficient safety stock. It is calculated as the expected demand over τ+1 periods plus a safety factor based on the standard deviation of demand over τ+1 periods.

• The safety factor or z-score is determined such that the expected inventory related costs are balanced between holding and backorder costs, similar to the newsvendor model.

• An EOQ calculation is also used to calibrate the formula for s depending on whether average demand is less than 1.5 times the EOQ.

So in summary, it uses ideas from EOQ, newsvendor model and safety stock to heuristically determine the order point s and order-up-to level S for the periodic review inventory system.

Here is a summary of the periodic review inventory model described:

• It uses a (s,S) policy where inventory position is reviewed periodically (e.g. each period).

• If inventory position is at or below s, an order is placed to increase inventory position to S.

• S = s + Q, where Q is the order-up-to level or order quantity.

• The goal is to determine optimal values of s and S (or equivalently s and Q) to minimize total relevant costs.

• Total relevant costs include setup/ordering costs, holding costs, and penalty/backorder costs.

• An approximation algorithm is described to determine s and S. It involves finding values of u and v based on demand distribution properties to satisfy optimality conditions related to expected backorders and fill rate respectively.

• s is set based on the mean demand adjusted for the variability represented by u or v, depending on which is smaller. S is set as s plus the order quantity.

• The model assumes stationary, independent demand periods and seeks to determine a cost-minimizing fixed (s,S) policy to use each period.

Here is a summary of the key references related to d substitutions in inventory systems:

• Axsäter, S. (2003) proposes a new decision rule for lateral transshipments (d substitutions) in inventory systems to improve fill rates.

• Axsäter, S. (2003) examines optimal policies for serial inventory systems under fill rate constraints, which involve d substitutions.

• Axsäter and Rosling (1993) compare installation stock (at each location) and echelon stock policies (allowing d substitutions) for multilevel inventory control systems. They find echelon stock policies can improve performance.

• Axsäter, Schneeweiss, and Silver (1986) provide a framework for multi-stage production planning and inventory control problems, which would involve d substitutions between stages.

• Banerjee et al (2003) conduct a simulation study of lateral shipments (d substitutions) in a single supplier, multiple buyer supply chain network.

So in summary, these references examine the use of d substitutions/lateral transshipments between locations/stages in inventory systems as a means to improve fill rates and overall system performance. The decision rules and policies allow inventory to be viewed and managed at an echelon level rather than just the individual location level.

Here are summaries of a few papers on multi-product, multistage production lot-sizing:

• Eppen, Gould, Pashigian (1969): Extends the classical planning horizon theorem to allow for extensions of the planning horizon in a dynamic lot-sizing model with changing production capacities over time. Shows that a optimal solution can be found by considering a finite planning horizon.

• Eppen, Martin (1987): Proposes a variable redefinition technique to solve multi-item capacitated lot-sizing problems. Shows how the technique can reduce problem size and computational effort.

• Federgruen, Queyranne, Zheng (1992): Shows that simple power of two policies can approximate optimal solutions well in production/distribution networks with general joint setup costs. Power of two policies have a policy period that is a power of two.

• Federgruen, Zheng (1992): Develops an efficient algorithm for computing optimal (r,Q) policies in continuous review stochastic inventory systems with multiple products. (r,Q) policies reorder when inventory falls to r and order up to Q.

• Federgruen, Zheng (1995): Presents efficient algorithms for finding optimal power-of-two policies for production/distribution systems with general joint setup costs and multiple products.

Here are summaries of the references provided:

156. This reference presents a demand prediction technique for items in military inventory systems. It aims to predict demand based on historical data.

157, 158. These references discuss analysis of inventory systems, including an inventory transportation model with multiple locations.

159. This reference looks at inventory control of repairable tools when information is incomplete.

160. This reference provides an overview of operations research applications in production and inventory control.

161. This reference examines the impact of customer order information and lead times on inventories.

162. This early reference from 1913 discusses how many parts to make at once in a production run.

The remaining references #163-221 cover a range of topics related to inventory systems, including periodic review policies, repair kit optimization, priority scheduling, integrated production/inventory planning, and analysis of multi-echelon systems. Many examine analytical models and properties of different inventory policies. Overall the references provide an overview of classic and important research on inventory management and related operations research applications.

This passage summarizes several research articles reference in a larger list of references. Specifically, it provides brief summaries for research papers cited in references 222 through 288. The summaries consist of the authors’ names, publication years, and brief descriptions or key findings of the respective papers. No individual paper is discussed in depth. The overall passage serves to briefly characterize 17 research papers within a compiled list of references for inventory and supply chain management literature.

These papers cover various aspects of multi-echelon inventory systems and (s,S) policies:

• Roundy (1986) presents a 98% effective lot-sizing rule for multi-product, multi-stage production inventory systems.

• Roundy (1989) discusses efficient and effective lot sizing for multistage production systems.

• Roundy (1989) examines rounding off to powers of two in continuous relaxations of capacitated lot sizing problems.

• Roundy and Muckstadt (2000) discuss heuristic computation of periodic-review base stock inventory policies.

• Rustenburg et al. (2000) cover spare parts management for technical systems with limited budgets.

• Several papers by Sahin examine stationary analysis and objective function behavior of continuous review (s,S) inventory models.

• Scarf papers from the 1950s-60s establish theoretical foundations for (s,S) policies and their optimality in dynamic inventory problems.

• Multiple papers examine variants of multi-echelon models, pooling, lateral supply, fill rates, obsolescence, and applications to repairable item and spare parts inventory.

In summary, the references cover theoretical developments and applications of (s,S) policies and multi-echelon inventory systems over several decades.

Here is a summary of the key points from the paper:

• The paper discusses algorithms for solving various deterministic dynamic lot sizing problems where demand is known over discrete periods.

• The Wagner-Whitin (WW) algorithm is described as one of the earliest and most impactful algorithms for solving the dynamic lot sizing problem in linear time, by transforming it into a shortest path problem on a directed graph.

• The WW algorithm works by recursively building up an optimal solution period by period, computing optimal lot sizes and inventory positions.

• A shortest path representation of the dynamic lot sizing problem is presented, showing how it can be modeled as finding the minimum cost path through a directed graph.

• The Wagelmans-Hoesel-Kolen (WHK) algorithm is also discussed. It formulates the problem as a directed multicommodity network flow problem that can be solved using network flow algorithms.

• Both the WW and WHK algorithms are able to solve the dynamic lot sizing problem in linear (O(n)) time, which was a significant improvement over earlier algorithms with quadratic or higher runtimes.

• Various extensions to the basic dynamic lot sizing problem are also discussed, like joint replenishment problems and multi-item problems.

So in summary, the paper presents two important linear-time algorithms (WW and WHK) for solving deterministic dynamic lot sizing problems and illustrates how these problems can be modeled as network flow or shortest path problems.

Chapter 4 discusses heuristic methods for solving multi-period demand planning problems, including the Silver-Meal heuristic and least unit cost heuristic. It comments on choosing an appropriate planning horizon.

Chapter 5 covers single-period models for making decisions under uncertainty. It provides examples and discusses models for multiple items with multiple constraints.

Chapter 6 focuses on inventory planning over multiple time periods for the linear cost case. It discusses optimal policies like base-stock policies and finding optimal stock levels. It also covers capacity limited systems and serial inventory systems.

Chapter 7 introduces background concepts on the (s-1, s) inventory policy under Poisson demand. It discusses the steady state distribution, performance measures, and optimization problem formulations.

Chapter 8 presents a tactical planning model for recoverable items in multi-echelon systems. It details the METRIC system operations and solving the optimization problem to determine depot and base stock levels.

Chapter 9 covers reorder point, lot size models for continuous review systems. It provides approximate and exact models, discusses optimization, and extensions to compound Poisson demand and stochastic lead times. It also addresses a multi-item model.

Chapter 10 deals with lot sizing models for periodic review systems. It presents approximation and dynamic programming algorithms, and discusses computation of a stationary policy.