# SUMMARY - 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 from section 2.3 of the passage:

It considers a more realistic scenario where the unit purchasing cost depends on the order size, as larger orders may receive quantity discounts.

Previously in section 2.1, the purchasing cost per unit was assumed to be constant, but this is relaxed here.

When the purchasing cost varies with order size, it requires modifying the EOQ cost model from section 2.1, as the purchasing cost term Cλ would now be a function of the decision variable Q rather than a constant.

The goal is still to determine the optimal order quantity Q to minimize total inventory costs, but now the purchasing cost component is variable based on the order size rather than fixed.

This captures a more realistic scenario where bulk discounts are offered for larger orders, changing the economic trade-off in the EOQ model.

So in summary, section 2.3 generalizes the EOQ model by allowing the purchasing cost per unit to vary with order size, as is often the case in practice with volume-based discounts. This requires modifying the underlying cost model.

Here is a summary of the key points:

The model considers a multi-echelon inventory system with one central warehouse that supplies multiple regional warehouses.

The regional warehouses face stochastic demand and place orders to the central warehouse to replenish inventory.

The objective is to determine optimal order-up-to levels and reorder points for each warehouse to minimize total system-wide costs (holding, ordering, shortage).

Dynamic programming is used to solve this optimization problem in a backwards recursive manner, starting from the demand points (retailers) and moving up the supply chain.

The value/cost functions are defined recursively at each stage/warehouse based on costs above and demand/lead times below that stage.

The optimal policy balances inventory being held at each stage against expected inventory and shortage costs at downstream stages.

Computational steps are provided to calculate the value functions moving from retailers to regional warehouses to the central warehouse.

Optimal order-up-to levels and reorder points can then be determined from the value functions at each stage of the supply chain.

This multi-echelon approach coordinates inventory decisions across the entire supply chain for improved efficiency over location-by-location optimization.

So in summary, dynamic programming is applied to determine optimal inventory policies across all stages of a multi-echelon distribution system in a computationally efficient way. The value/cost functions are defined recursively to calculate optimal solutions.

Here is a summary of the key points about the Silver-Meal heuristic for solving the dynamic lot sizing problem:

It is a heuristic/approximation algorithm that aims to find a near-optimal solution more quickly than exact algorithms like Wagner-Whitin.

It starts by ordering enough in period 1 to satisfy just period 1 demand.

It then iteratively expands the order quantity to include subsequent periods, as long as this decreases the average cost per period based on the EOQ formula.

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

At each step it 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 increases average cost.

This gives a solution that locally minimizes average cost at each step without guaranteeing global optimality.

It provides a reasonable solution with fewer computations than exact algorithms at the expense of solution optimality.

The key idea is to quickly find a near-optimal solution by greedily minimizing average cost in a step-wise manner.

Here is a summary of the key points:

The passage discusses modeling inventory systems with stochastic demand over multiple time periods.

It presents a dynamic programming formulation to determine optimal inventory policies that minimize expected total costs over the planning horizon.

The dynamic program considers the system state (inventory level) in each period and calculates the optimal decision (order quantity) based on the costs incurred in the current and future periods.

Properties of the optimal policy are derived from analyzing the structure of the dynamic program, including that a base-stock policy is optimal when costs are linear.

A base-stock policy specifies a target inventory level and places orders to replenish up to this level after demand is observed.

The impact of production capacity constraints is analyzed, showing capacities may force a modification of the simple base-stock policy.

Finally, it briefly discusses multi-echelon inventory systems in supply chains and how individual locations operate under different types of policies depending on their position in the chain.

In summary, it presents analytical modeling and optimization of multi-period inventory systems using dynamic programming to determine optimal ordering policies under uncertainty.

Here is a summary of the key points:

The passage examines a single-period inventory model where demand is uncertain. The goal is to determine the optimal order quantity Q to minimize expected costs.

Costs include holding cost h per unit carried in inventory at the end, as well as shortage cost b per backordered unit.

Demand D is represented as a random variable with a known probability distribution (e.g. normal, uniform, etc.).

If Q > D, the holding cost is h(Q - D). If Q < D, the shortage cost is b(D - Q).

Taking the expected value of these costs yields an expected cost function EC(Q).

Taking the derivative and setting it equal to 0 allows finding the optimal order quantity Q* that minimizes EC(Q).

Examples are provided to illustrate the calculations for common demand distributions like normal and uniform.

The optimal Q* balances the tradeoff between expected holding and shortage costs based on the parameter values.

This basic model provides insights into the inventory optimization problem and serves as a building block for more complex multi-period models.

Here is a summary of the key points:

The paper proposes a tactical planning model called METRIC to determine optimal stocking levels of repairable items like aircraft parts across multiple bases and a central depot.

The goal is to minimize expected backorders at the bases while considering stochastic demand, repair timelines, and restricted budgets.

Key assumptions include Poisson failures at bases, (s-1,s) review policies, and no lateral resupply between bases (items go to depot for repair if needed).

The model calculates expected resupply times and backorders based on probabilities of local vs depot repair and associated repair times.

Approximations are used to estimate depot delay and expected inventory levels based on demand rates and stock levels.

Optimal stocking levels (sij) for each item type i at each base j are determined to minimize total expected backorders across all items and bases.

Improvements are discussed to make the METRIC model more tractable for practical tactical planning applications in multi-echelon repairable item supply chains.

Here is a summary of the key points:

The passage presents a method for constructing a transition matrix model of an (s,S) inventory system with stochastic demand.

The transition matrix P captures the possible state transitions between inventory position levels based on the demand distribution and reorder quantities.

Solving the equations πP = π and ∑πi = 1 yields the steady-state probability distribution π of being in each inventory position state.

This π distribution allows calculating important performance measures like stockout probability, expected stockouts, and expected inventory levels.

The same transition matrix approach can be extended to model multi-item inventory systems and optimize reorder points/quantities across multiple items simultaneously.

The transition matrix framework provides an exact representation that can be used to evaluate (s,S) policies and optimize parameters for single or multi-item inventory control problems.

So in summary, it introduces using a transition matrix model to derive the π distribution for (s,S) policies, enabling exact analysis and optimization for single or joint multi-item inventory systems.

Here is a summary of the key points about the periodic review inventory model:

It uses a (s,S) policy where inventory position is reviewed periodically (e.g. each period). An order is placed if inventory position drops to or below the reorder point s to raise it to the order-up-to level S.

S is defined as s + Q, where Q is the order quantity. The goal is to determine optimal values of s and Q (or equivalently s and S) to minimize total relevant costs.

Total relevant costs include fixed/setup ordering costs, holding costs for inventory carried, and penalty/backorder costs for unsatisfied demand.

An approximation algorithm is described to help find the optimal s and S values. It involves setting parameters u and v based on demand distribution properties to satisfy optimality conditions related to expected backorders and fill rate respectively.

The reorder point s is then set based on the mean demand adjusted for variability in demand using parameters like the standard deviation or appropriate z-value from the demand distribution.

This periodic review model aims to balance ordering, holding, and stockout costs over time through optimizing the reorder point and order-up-to level.

Here is a summary of the key points:

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

The Wagner-Whitin algorithm transforms the problem into a shortest path problem that can be solved in linear time using a recursive approach.

The Wagelmans-Hoesel-Kolen algorithm formulates the problem as a directed multicommodity network flow problem that can also be solved in linear time using network flow algorithms.

Both algorithms represent significant improvements over earlier quadratic time algorithms for this class of problems.

Extensions of the basic problem are also discussed, like joint replenishment and multi-item inventory problems.

The algorithms illustrate how deterministic dynamic lot sizing problems can be modeled as network flow or shortest path problems, enabling efficient linear-time solution approaches.