Über die Autorin bzw. den Autor

Evan L. Porteus is the Sanwa Bank Professor of Management Science at the Stanford Graduate School of Business.

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Foundations of Stochastic Inventory Theory

Stanford Business Books

Contents

Preface...................................................................xvConventions...............................................................xviii1 Two Basic Models........................................................12 Recursion...............................................................273 Finite-Horizon Markov Decision Processes................................414 Characterizing the Optimal Policy.......................................575 Finite-Horizon Theory...................................................776 Myopic Policies.........................................................917 Dynamic Inventory Models................................................1038 Monotone Optimal Policies...............................................1199 Structured Probability Distributions....................................13310 Empirical Bayesian Inventory Models....................................15111 Infinite-Horizon Theory................................................16712 Bounds and Successive Approximations...................................18113 Computational Markov Decision Processes................................19314 A Continuous Time Model................................................209Appendix A Convexity......................................................223Appendix B Duality........................................................241Appendix C Discounted Average Value.......................................261Appendix D Preference Theory and Stochastic Dominance.....................279Index.....................................................................293

Chapter One

This chapter introduces two basic inventory models: the EOQ (Economic Order Quantity) model and the newsvendor model. The first is the simplest model of cycle stocks, which arise when there are repeated cycles in which stocks are built up and then drawn down, both in a predictable, deterministic fashion. The second is the simplest model of safety stocks, which are held owing to unpredictable variability.

1.1 The EOQ Model

A single product is used (or demanded) at a continuous fixed and known rate over time, continuing indefinitely into the future. Shortages or delayed deliveries are not allowed. Replenishments are received as soon as they are requested. The ordering cost consists of a setup (fixed, redtape) cost that is incurred each time an order of any size is placed plus a proportional cost incurred for each item ordered. A holding cost is incurred on each item held in inventory per unit time. See Figure 1.1 for a typical plot of the inventory level as a function of time.

The product may be an item that is used internally by your organization and the issue is how many items to order each time an order is placed (equivalently, how often to replenish the stock of the item). The product may also be a manufactured product that is used at a uniform rate at downstream stations, in which case the issue is how long the production runs should be. In a manufacturing setting, the order quantity is called either the batch size, the lot size, or the production run, and the setup cost is the fixed cost to set up the equipment to produce the next run. It can include direct costs, the opportunity cost of the time it takes to carry out the setup, and the implicit cost of initiating a production run because of learning and inefficiencies at the beginning of a run. Technically, we assume that all completed units become available at the same time, such as when all units are processed simultaneously and completed at the same time. However, we shall see that the case of sequential manufacturing, under a fixed manufacturing rate, reduces to a form of this model.

There are economies of scale in ordering in that the average order cost (per unit ordered) decreases as the size of the order increases. This effect favors large orders, to average the fixed order cost over a large number of units ordered, leading to long cycles, with infrequently placed orders. However, the holding costs favor lower inventory levels, achieved by short cycles, with frequently placed orders.

Notation

K = fixed (setup) cost. c = direct unit ordering cost. [lambda] mean demand rate per week. h = unit holding cost incurred per unit held per week. Q = order quantity (decision variable). C(Q) = cost per week (objective function).

Formulation of Objective Function

A cycle is the amount of time T that elapses between orders. We first determine the cost per cycle. Dividing by the time length of the cycle yields the cost per week, which we seek to minimize. Exercise 1.1 verifies that we should assume that the cycle length T is the same for every cycle and that replenishment takes place when the stock level drops to zero.

Suppose the first cycle starts at t = 0 with the inventory level at zero and ends at t = T. An amount Q is ordered and is received immediately, incurring an ordering cost (per cycle) of K + cQ. Let x(t) denote the inventory level at time t. Then, for 0 [is less than or equal to] t < T, x(t) = Q - [lambda]t. (See Figure 1.1.) Thus, to calculate T : we set Q - [lambda]T = 0, so T = Q/[lambda]. The inventory cost per cycle is therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the total cost per week is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first term gives the average setup costs per week. The second gives the average holding cost per week, which is simply the product of the unit holding cost per week, h, and the average inventory level over time, Q/2. The last term, c]lambda], is the required direct cost per week. Because it is independent of the replenishment policy, we follow the usual practice of ignoring it and write our objective function, consisting of the remaining relevant costs, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Economic Order Quantity (EOQ)

To find the optimal order quantity, we differentiate the objective function C(Q) with respect to decision variable Q and set the result equal to zero. The resulting first-order conditions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Checking the second-order conditions, we find that the second derivative is 2K]lambda]/[Q.sup.3], which is strictly positive for Q > 0, so the objective function is (strictly) convex on (0, [infinity]). Hence, the first-order conditions are sufficient for a minimum. The resulting solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is the famous square root EOQ formula, introduced by Harris (1913) and popularized by Wilson (1934).

Interestingly, this problem represents a special case, illustrated in Figure 1.2 below, in which, to minimize the sum of two functions, we set the two functions equal to each other. An empirical consequence for a practical setting in which this model applies is this: If holding costs for cycle stocks are more [less] than setup costs per week, then order quantities are too large [small].

Optimal Cost per Week

Plugging the EOQ back into the objective function leads to

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as the optimal level of setup costs per week. The optimal level of holding costs per week is exactly the same. The resulting, induced optimal setup and holding cost per week is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is an increasing concave function of each of the parameters. (Its first derivative is positive and its second derivative is negative.)

Comparative Statics

If volume (demand rate) is doubled, you don't double the order quantity. You multiply by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.414, so the order quantity gets increased only by about 40%. Similarly, the total cost per week goes up by about 40%, an economy of scale. Each additional unit per week of volume requires a smaller additional cost. Another way to view this phenomenon is to compute the elasticity of the optimal cost as a function of the demand rate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which can be interpreted as (roughly) the percentage increase in optimal costs for each 1% increase in the demand rate. Thus, for every 2% increase in the demand rate (or setup cost or holding cost, for that matter), the optimal cost will increase by about 1%.

A central feature of many Just-In-Time (JIT) campaigns in manufacturing practice is setup reduction. Suppose such a campaign is implemented in an environment in which the assumptions of the EOQ model hold. If you cut the setup cost in half, you don't cut the order quantity in half. You multiply by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0.707, so the order quantity gets cut by about 30%. Similarly, the induced optimal cost per week goes down by about 30%. Each additional reduction of $1 in the setup cost yields a larger cost savings. Thus, there are increasing returns to setup reduction, which can lead to underinvestment in setup reduction if only local marginal returns are taken into account. However, if the analysis is done in elasticity terms (percentage changes) instead of absolute terms, then each 2% decrease in the setup cost will yield about a 1% decrease in the order quantity and the total cost. See Porteus (1985) for examples where the optimal amount of setup reduction can be found analytically.

Robustness of Cost to Order Quantity

Suppose, for a variety of possible reasons, such as misestimated parameters, the implemented order quantity is a times the (optimal) EOQ. The resulting cost is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, if the order quantity is either 50% higher than the EOQ (a = 3/2) or the EOQ is 50% higher than the order quantity (a = 2/3), then the resulting cost is (13/12)ITLITL*, which means that actual cost is about 8% higher than optimal. The EOQ is famous for being robust in this sense: large errors in the implemented order quantities lead to small cost penalties.

Extension to Sequential Manufacturing

It usually takes time to produce something. If all completed units become available at the same time, after a fixed time for processing/production, then the production run can be scheduled in advance, so that all completed units become available exactly when needed (when the inventory level drops to zero). However, under sequential manufacturing, there will be a buildup of completed units until the end of the production run, at which time the inventory level will start to decrease again. Figure 1.3 illustrates that each cycle consists of the time to complete the production run plus the time for the inventory level to drop to zero. This subsection shows that the essential structure of the EOQ model is maintained in this case.

Let r denote the (finite) rate at which units are produced. (Note that we must have [lambda] < r for this analysis to work: If [lambda] = r, then there will be only one production run, which will continue for the lifetime of the product. If [lambda] > m, then all demands cannot be met, as we assume they must be.) In this case, the maximum inventory level is less than the run size Q, namely, Q(r - [lambda])/r, so the average holding cost per week is hQ(r - [lambda])l(2r). The resulting optimal lot size is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is sometimes called the Economic Production Quantity (EPQ). By replacing the original holding cost parameter h by h := h(1 - [lambda]/r), the (smaller) effective holding cost parameter, we can retain the original EOQ notation, terminology, and results.

1.2 The Newsvendor Model

An unknown quantity D of a single product will be demanded (requested for use) during a single period. While the probability distribution of demand is known, the actual number demanded will not be known until after a decision y, how many to order (or make), is made. In short, order y first, observe the stochastic demand D second. The amount sold/delivered will be min(D, y) You can't sell more than you have or than are demanded. Under some demand distributions, such as the normal distribution, it is impossible to always meet all demand, so the prospect of unmet demands must be accepted. The ordering cost consists of a proportional cost incurred for each item ordered. Units are sold for more than they cost. Units purchased but not sold become obsolete by the end of the period and have no salvage value.

The name of the model derives from a newsvendor who must purchase newspapers at the beginning of the day before attempting to sell them at a designated street corner. However, the model applies to products that must be produced to stock (commitment to the quantity, whether procured or produced, is required before observing demand) under the following two conditions: (1) Only one stocking decision is allowed (with no opportunity for replenishment), and (2) The financial consequences can be expressed as a function of the difference between the initially chosen stock level and the realized demand. Actually, we shall see later that there are conditions when it also applies to products that can be replenished infinitely many times.

The tradeoff is between too many and too few. If we make too many, then we have some left over, and we paid for more than we need. If we make too few, then we could have sold more if we had bought more. Inventories held, as they are in this setting, because of unpredictable demand are called safety stocks. If there is no variability in the demand, then we would stock the mean demand. Any more than that is safety stock. (Safety stock can be negative.)

Notation

p = unit sales price (p > 0). c = unit cost (0 < c < p). D = unit demand in the period (D [is greater than or equal to] 0) (continuous random variable). [phi] = cumulative distribution function of demand. [??] = 1 - [phi]. [phi] = probability density of demand. [micro] = mean demand. [sigma] = standard deviation of demand. y = stock level after ordering (available) (decision variable). v(y) = expected contribution (objective function).

Formulation of Objective Function

The contribution in a period equals the dollar sales from the units sold less the cost of the units purchased: p min(D, y) - cy. We assume henceforth, unless indicated otherwise, that [phi](0) = 0 : demand will be strictly positive with probability one. Using Lemma D.1 (Lemma 1 of Appendix D) and [??](y) = 1 - [phi](y), the expected contribution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

Expanding the expression for the expected units sold leads to the alternative representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Recalling that [x.sup.+] := max(x, 0) we can also write (Exercise 1.5)

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which are dollar sales on expected demand less cost of the stock purchased less opportunity losses (revenues not received for lack of product),

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which is the profit margin on the units stocked less unrealized revenues on unsold units, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where

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which is called the expected one period holding and shortage cost function and also the expected loss function. The contribution therefore consists of the profit margin on the mean demand less the expected holding and shortage costs. Thus, since (p - c) is fixed, we can equivalently minimize the expected holding and shortage costs. It is also convenient to express L(y) as follows:

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where

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The Critical Fractile Solution

To find the optimal order quantity, we differentiate the objective function with respect to the decision variable y and set the result equal to zero. By Exercise 1.7, the resulting first-order conditions are

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Checking the second-order conditions, we find that the second derivative exists and equals -p]phi](y), which is negative, so the objective function is concave. Hence, the first-order conditions are sufficient for a maximum. Let S denote an optimal stock level after ordering. The first-order conditions lead to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

which is the famous critical fractile solution, first appearing in Arrow, Harris, and Marschak (1951), that is illustrated in Figure 1.4. Arrow (1958) attributes the newsvendor model itself to Edgeworth (1888). To avoid certain technical details, assume henceforth, unless indicated otherwise, that an optimal solution to the newsvendor problem exists and is unique.

It is useful to define [Zeta] := c/p, which is the optimal stockout probability (OSP)

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Interpretation

If p = 2c, which corresponds to a 100% markup over cost (but is called a 50% markup by retailers who like to think of the markup as a percentage of the sales price), then it is optimal to stock the median demand. In this case, it is optimal to stock out (run out) half the times the product is stocked and to have leftover stock half the times. The lower the wholesale cost or the higher the retail price, the less likely you are to run out. Thus, if your retailer of perishable goods rarely runs out, you are probably paying a big markup.

Note, however, that while the stockout probability may be fairly high, the optimal probability that a randomly selected demand is unsatisfied is typically much smaller. For example, the ratio of expected unsatisfied demand to expected total demand, namely, E[(D - S).sup.+]/E(D), can be substantially smaller than the stockout probability. In particular, it will be illustrated shortly that the optimal fill rate, defined here as the ratio of expected satisfied demand to expected total demand when stocking optimally, namely, E min(D, S)/E(D), can be substantially larger than the critical fractile.

Discrete Demand

This subsection explores the consequences of dropping the assumption that demand is a continuous random variable. In particular, suppose that demand D is discrete in the sense that there exists a sequence of real numbers, {[x.sub.1], [x.sub.2], ...}, representing all potential demand quantities, and 0 [is less than or equal to] [x.sub.1] [is less than or equal to] [x.sub.2] [is less than or equal to] .... Furthermore, suppose all these quanities are admissible stock levels. These assumptions hold if the stock level must be an integer and demands arise only for integer quantities. These assumptions also hold if the admissible stocking units are continuous and demands are discrete but not necessarily integers. For example, the product might be gravel that can be produced to reach (essentially) any fractional level requested but is sold in full containers of various specific sizes, which are not necessarily integer multiples of the smallest size.

(Continues...)

Excerpted from Foundations of Stochastic Inventory Theoryby EVAN L. PORTEUS Copyright © 2002 by Board of Trustees of the Leland Stanford Junior University . Excerpted by permission.
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