Periodic Review Inventory Control With Lost Sales and Fractional Lead Times

Transcription

1 Periodic Review Inventory Control With Lost Sales and Fractional Lead Times Ganesh Janakiraman IOMS-OM Group, Stern School of Business, New York University. John A. Muckstadt School of Operations Research and Industrial Engineering, Cornell University. October 2004 Subject Classification: Inventory/Production - periodic review, lost sales, lead times.

2 Abstract We introduce a single location periodic review inventory control problem with lost sales and fractional lead times. We model the optimal inventory control problem as a stochastic dynamic program and analyze properties of the objective function as well as the optimal policy. We present upper and lower bounds on the optimal policy. These bounds can readily be used in easily computable heuristics. In addition to these heuristics, we also analyze properties of the system when order-up-to S policies are used. We prove the convexity of the cost function with respect to the order-up-to parameter S. We use this convexity property to determine the best order-up-to levels for this system using bisection search. Our computational investigation reveals that the deviation from the optimal cost is less than 1.5% on an average for both the upper-bound heuristic and the best order-up-to S policy.

3 1 Introduction Consider the following replenishment problem that retail and grocery stores face. Inventory of a product is reviewed periodically (say weekly or daily) and replenishment orders can be placed at the beginning of each period (week or day). In many cases, the replenishment lead times are shorter in length than a period; for example, grocery stores might order replenishment stock late each afternoon and this stock might arrive the next day. In these environments, being out of stock of a product when it is demanded typically leads to a lost sale. In this paper, we study an inventory control problem that represents such environments. We study a single-item, single-location system served by a supply source with infinite availability of inventory. The system is reviewed periodically and orders can be placed at the beginning of every period. The replenishment lead time is random but shorter than the length of a period. In both parts of the period, that is, before and after delivery, any demand in excess of available inventory is lost. Correlation between demands in the two parts of the period is permitted. We model this control problem as a dynamic program. We prove important properties of the cost function and the optimal ordering quantity function. Using these properties, we derive functions that bound the optimal ordering quantity function on both sides. We also present results from computational experiments that indicate the effectiveness of using these bounds as heuristic policies. In addition, we present some analysis of order-up-to policies and investigate their performance in this setting. Apart from the example of retail and grocery stores given in the opening paragraph, similar problems also arise in service parts supply chains, which motivate this study. For example, the service parts supply chain of a major automobile manufacturer consists of three tiers or echelons. Automobile and service parts dealers form the bottom echelon, parts distribution centers (PDCs) comprise the next echelon, and, parts plants form the top echelon. Each dealer s source of supply is a unique regional PDC. Each dealer sends out orders to its regional PDC every afternoon. The PDC processes these orders and sends a truck 1

4 with the ordered inventory to the dealer. This inventory reaches the dealer the subsequent morning. Service level expectations are very high in this business. Consequently, if a dealer and its regional PDC do not possess the required amount of inventory to meet customer demand on some day, the demand at the dealer is met by an emergency supply source. This supply source is usually another dealer in the same region or the PDC closest to the regional PDC or a parts plant itself. Needless to say, the emergency supply activity is significantly more expensive compared to the regular supply. Every unit demanded at a dealer in excess of regular supply (dealer s inventory and regional PDC s inventory) is a lost sale in the sense that this demand does not reduce the inventory at the dealer or the regional PDC. In some sense, the cost of each unit lost is the incremental cost to the system as a result of resorting to the more expensive emergency supply activity. In both the cited examples (service parts supply chains and the retail industry), thousands of parts or stock keeping units are stored in inventory. Further, demand forecasts are created periodically and these forecasts change through time. These two facts imply that any useful heuristic for inventory procurement would necessarily have to be computationally efficient and capable of being executed in virtually real-time. We develop computationally attractive heuristics based on the inverse of the CDFs (cumulative distribution functions) of random variables representing demand; for commonly used distribution functions like the normal and gamma, these functions are available in standard spreadsheet software like Excel. These direct non-recursive heuristics perform very well when compared with the optimal solutions obtained using the dynamic programming model. We present a review of related literature in section 2. The optimal control problem is formally defined as a stochastic dynamic program in section 3. Section 3.1 contains a list of assumptions and section 3.2 contains a discussion of myopic policies in order to build some intuition for structural results that are derived subsequently. The structural results for the optimal control problem are stated in section 3.3. In section 4, base-stock policies are 2

5 considered and some properties of the system under such policies are discussed. We present computational results in section 5 and present concluding remarks in section 6. In the following sections, we use policy to mean a function that prescribes an order quantity based on the current inventory status. We use increasing and decreasing in the weak sense. 2 Literature Review The classical newsvendor problem is a special case of our problem where all the lead times are zero. It is well known that a base-stock policy is optimal for such a problem and that the policy is such that the probability of not stocking out is a fractile (that depends on the cost parameters) of the one-period demand distribution. However, there is no such nice structure for those problems with lost sales and positive lead times; lost sales problems with lead times are far less tractable analytically than their backorder counterparts. Consequently, there are very few papers that contain analytic results for periodic review problems with lost sales and positive lead times. In their seminal paper, Karlin and Scarf (1958) analyzed such a problem for a single location system. They proved some basic properties about the optimal policy with the assumption that the lead time of all orders is exactly one period. However, they did not provide a method for computing the optimal order quantities. Later, Morton (1969) generalized these basic results to periodic review lost sales problems with arbitrary, but fixed lead times that are integer multiples of the period s length. In addition, he developed easily computable bounds on the optimal order quantity for a given inventory vector (the vector of the quantities of inventory in different stages of the pipeline). Subsequently, Morton (1971) proposed and evaluated myopic policies as effective heuristics 3

6 for these problems. We extend the work of Karlin and Scarf (1958) and Morton (1969) by analyzing the optimal inventory control problem for a single item, single location, lost-sales problem with lead times that are random and shorter in length than a period. Our model also has the following features: (i) non-stationarity of demand throughout a period and (ii) correlation of demand within a period. Our development of bounds on the optimal policy follows closely the approach of Morton (1969). Nahmias (1979) considered more general periodic review lost sales problems that include fixed ordering costs, partial backordering and random lead times. He developed myopic policies for these problems and investigated their effectiveness. In addition, for problems with lead times greater than two periods, he proposed using (s,s) policies, or order-up-to-s policies when the fixed cost is zero. Since the analytic determination of the best policy, among the class of such (s,s) or order up to S policies, may not be possible, he used simulation to determine the best policies. Kapalka et al. (1999) analyze a problem that is closely related to the one examined in this paper. They consider the class of (s,s) policies for a single location, single item periodic review inventory model with lost sales and service level constraints. The lead time is a fraction of a period. Fixed ordering costs are present. A simple lower bound for S is obtained. A theoretical upper bound was not obtained; however, they use a safety-stock estimate to get an upper bound that is adequate in practice. The optimal (s, S) pair is determined by searching over the grid formed using the bounds on S. They developed efficient updating procedures of the transition matrix that help in reducing the computational effort. It should be noted that (s, S) policies are not optimal for lost sales problems with fixed ordering costs. In fact, even when fixed ordering costs are zero, order-up-to policies are not optimal. Their 4

7 work was motivated by a potential application for a retail supply chain. In this paper, we also examine the class of order-up-to policies. We prove that the expected finite horizon discounted cost is a convex function of the order-up-to parameter. Due to this convexity result, we can determine the optimal order-up-to parameter using bisectionsearch. The convexity of the cost function has been established for other lost sales problems with positive lead times when an order-up-to policy is followed. Downs et al. (2001) assume an arbitrary but fixed lead time that is an integer multiple of the period s length. They prove this convexity result and use it to develop a linear program to determine optimal order-upto stocking levels for multiple products in the presence of budget constraints. Janakiraman and Roundy (2004) assume random integer lead times with no order-crossing and show the convexity result. All the papers that we have mentioned so far study periodic review inventory control problems. There is some literature on continuous review inventory models with lost sales, for example, Hill (1999), Johansen (2001), Johansen and Thorstenson (1996), Smith (1977) and Posner and Mohebbi (1998). A key fact to be noted is that fractional lead times are not usually considered in the inventory theory literature. As mentioned earlier, several examples with such lead times and lost sales can be found in service parts chains, retail systems and grocery chains. For problems with backorders, it is trivial to extend results obtained with integral lead times to the case of fractional lead times. With lost sales, however, the analysis with fractional lead times is challenging. 5

8 3 The Optimal Inventory Control Problem: Structural Results Assume that there are N periods in the planning horizon which we index in a backward fashion, (i.e.) period N 1 occurs after period N, and period 1 is the last period in the planning horizon. Linear holding and lost sales costs are present; linear purchase costs can be assumed to be zero without loss of generality as shown for general distribution and assembly systems with lost sales and/or backorders in Janakiraman and Muckstadt (2004) when lead times are integers. (See Janakiraman and Muckstadt (2001) for a proof of this result for the model analyzed in this paper. In order to use this result, however, we need the assumption that inventory at the end of the planning horizon is salvaged at the purchase price.) Let us now state the notation that is used in this section. We use = d to denote definitions. α = d discount factor. x n = d on-hand inventory at the beginning of period n. x 0 = d on hand inventory at the end of period 1. q n = d quantity ordered in period n. D n1 = d random variable representing the demand that occurs between the start of period n and the time the order of size q n is received. D n = d random variable representing the total demand that occurs in period n. D n2 = d D n D n1 = the random variable representing the demand that occurs between the time that we receive the order of size q n and the end of period n. D n = d the random vector (D n1, D n2 ). y n = d on-hand inventory just after receiving q n. = (x n D n1 ) + + q n. 6

9 h = d holding cost incurred per unit of inventory (charged at the beginning of a period). b = d lost sales cost incurred for every unit of sales lost during a period (charged at the beginning of a period). Figure 1 goes here The dynamics of the system s operation are depicted in Figure 1. The timing of events and the manner in which costs are incurred are critical to our analysis. Each period is linked to the following one through the evolution of the on-hand inventory at the beginning of a period. The equation that describes this relationship from period n to n 1 is given by x n 1 = ((x n D n1 ) + + q n D n2 ) +. (3.1) Next, we define the one-period cost and the expected discounted cost. Note that all the expectation operators in this section will be subscripted with the set of random vectors or variables over which the expectation is taken. V n (x n, q n ) = d expected period n holding and lost sales costs if we start period n with x n units of inventory on hand and we order q n units = E Dn [ h xn + b (D n1 x n ) + + b (D n2 y n ) +]. (3.2) f n (x n ) = d minimum expected sum of all discounted future costs over the planning horizon if we start period n with x n units of inventory on hand { = min Vn (x n, q n ) + α E q Dn [f n 1 (((x n D n1 ) + + q n D n2 ) + )] }. (3.3) n 0 f n (x n, q n ) = d minimum expected sum of all discounted future costs if we start period n with x n units of inventory on hand and if we order q n units in period n 7

10 = V n (x n, q n ) + α E Dn [f n 1 (((x n D n1 ) + + q n D n2 ) + )]. (3.4) q n(x) = d arg min q 0 (f n(x, q)), and therefore f n (x) = f n (x, q n (x)). In addition to these costs incurred in periods N, N 1,..., 1 there is an end of horizon cost f 0 (x 0 ) which depends on x 0, the inventory on hand at the end of period 1. Specifically, f 0 (x 0 ) = d cost incurred at the end of the horizon, (i.e.) the end of period 1 = h x 0. (3.5) The finite horizon problem is to determine the optimal policy, that is, the function q n(x), for all n {N, N 1,..., 1} and for all x Assumptions The following conditions are assumed throughout the section α The cost parameters satisfy the relation b h. 3. {D n } = d {(D n1, D n2 )} is a sequence of i.i.d. random vectors. Correlation between D n1 and D n2 is allowed. 4. All the random variables representing demand possess positive continuous density functions on [0, ). Let us now define the necessary probability density functions and cumulative distribution functions. φ 1 (u 1 ) = d the probability density of D n1 at u 1. Φ 1 (u 1 ) = d the cumulative distribution function of D n1 at u 1 = P ( D n1 u 1 ). φ 2 (u 2 ) = d the probability density of D n2 at u 2. Φ 2 (u 2 ) = d the cumulative distribution function of D n2 at u 2 = P ( D n2 u 2 ). 8

11 φ(u) = d the probability density of D n at u. Φ(u) = d the cumulative distribution function of D n at u = P ( D n u ). Φ Φ 1 (u) = d P ( D n + D n 1,1 u ). Φ 1 (u 1 ) = d the complementary cumulative distribution function of D n1 at u 1 = 1 Φ 1 (u 1 ). Φ 2 (u 2 ) = d the complementary cumulative distribution function of D n2 at u 2 = 1 Φ 2 (u 2 ). Φ(u) = d P (D n2 + D (n 1)1 u). φ 12 (u 1, u 2 ) = d joint probability density function of D n at (u 1, u 2 ). The randomness in the lead time is captured in the probability density function of D n = (D n1, D n2 ). 3.2 Myopic Policies Before analyzing the optimal control problem, we will briefly study myopic policies. By myopic policy, we mean a policy that minimizes expected costs over only that period where the ordering decision has an immediate cost impact. Such policies are easy to determine and are usually useful in giving a first cut idea about the dynamic program and the structure of its optimal policy. We define the myopic cost of ordering q units when x is the current inventory on hand as C my (x, q) = d be[(d n2 ((x D n1 ) + + q)) + ] + αbe[(d (n 1)1 ((x D n1 ) + + q D n2 ) + ) + ] + αhe[((x D n1 ) + + q D n2 ) + ]. (3.6) The first term is the expected lost sales cost in the second part of a period; the second term is the expected lost sales cost in the first part of the next period; and the last term is 9

12 the expected holding cost at the beginning of the next period. It can be verified that this function is strictly convex and that the optimal myopic ordering quantity q, given x, is the solution to (1 α)bp ((x D n1 ) + + q D n2 ) αbp ((x D n1 ) + + q D n2 + D (n 1)1 ) + αhp ((x D n1 ) + + q D n2 ) = 0, when one exists and 0 otherwise. Let q my (x) denote this myopic policy. Let us now consider two special cases of our problem. In the first case the lead time is exactly zero, and in the second case it is exactly one period. We now have: (i) when lead time = 0, q my (x) = d min{q 0 : P (x + q D n ) b/(b + αh)}, and (ii) when lead time = 1, q my (x) = d min{q 0 : P ((x D n ) + + q D n 1 ) (b h)/b}. Notice that whenever q my (x) > 0 in these two cases, the myopic policy equalizes the probability of not stocking out before the next receipt of inventory to the constants b/(b+αh) and (b h)/b, respectively. There are two reasons why such a structure is useful: (i) it tells us what service level we can achieve while using this policy and (ii) it is fairly easy to construct upper and lower bounds on the myopic policy from this structure. We might hypothesize that the optimal policy for the dynamic program model would also be a policy that maintains a constant probability of not stocking out. Optimal base-stock policies for problems with backordering have this nice property. Unfortunately, the hypothesis is not true for problems with lost sales and positive lead times. This was demonstrated by Morton (1968) for the case when the lead time is exactly one period long. However, we will show in the following subsection that the probability of not stocking out while using the optimal policy is bounded between two constants that depend on the parameters b, h and α. We will use these bounds to develop computationally efficient heuristic approximations for the optimal policy. But first, we establish properties of the value function and the optimal ordering policy. 10

13 3.3 Properties of the Optimal Ordering Function Our goal in this section is to prove several properties of the optimal policy q n and the discounted cost function f n. The results proved are in the following order. In Theorem 1, we demonstrate the strict convexity of f n (x). Further, we show that there is a critical quantity x n such that it is optimal to order nothing if x n exceeds x n and to order a positive amount, otherwise. The optimal policy qn (x) is strictly decreasing in x, as long as it is optimal to order something, and the rate of decrease is smaller than 1. The next important result (Theorem 2) we prove is that the probability of not stocking out while using the optimal policy is bounded between two constants that depend on b, h and α. To prove this result, however, we need to establish some bounds on f n (x) first, and this is done in Lemma 1. The bounds on the probability of not stocking out are then used to establish easily computable bounds on the optimal policy. This is done in Lemma 2. The proofs of these results can be found in the Appendix. Theorem 1 For any period 1 n N, (a) f n(x) > 0, (3.7) (b) lim f x n (x) > 0, (3.8) f n (x, q) (c) F n (x, q) = d increases with q x 0, (3.9) q (d) f n (x) h b, (3.10) (e) x n such that qn(x) > 0 if and only if x < x n ; also, F n ( x n, 0) = 0, (3.11) (f) 1 < dq n(x) dx < 0 0 x x n, (3.12) (g) f n( x n ) = h. (3.13) 11

14 Next, we derive upper and lower bounds on the derivative of f n, which are useful in constructing bounds on the probability of not stocking out while using the optimal policy. Lemma 1 (b + αh h) + (b + αh)φ 1 (x) f n(x) (h b) + (b + αh)φ 1 (x), x (0, x n ), n 1. We can now construct bounds on the probability of not running out of stock in the interval of time between the receipt of two successive orders while following the optimal policy. Let π n (x) = d P ((x D n1 ) + + q n (x) D n2 + D (n 1)1 ) = P (x > D n1, x + q n(x) D n1 D n2 + D (n 1)1 ) + P (x < D n1, q n (x) D n2 + D (n 1)1 ). The following theorem gives bounds on the probability of not stocking out while using the optimal policy. Let L = d Theorem 2 b h b+αh and U = d b. b+αh L π n(x) U, x (0, x n ), n 2. Two interesting observations can be made that link the service levels of the myopic policy and the optimal policy. When the lead time is zero, the myopic policy is also the optimal policy. Intuitively, the service level attained by the optimal policy when the lead time is zero should be an upper bound on the service level attained by the optimal policy when the lead time is positive. This is indeed true as was established in Theorem 2. The link between the lower bound stated in Theorem 2 and the myopic policy is less obvious. Assume that the lead time is one period. Modify the myopic cost to include that part of the expected holding cost two periods from now which is directly caused by the decision q; that is, consider the modified myopic cost αhe[(x D n ) + + q] + αbe[(d n 1 ((x D n ) + + q)) + ] + α 2 he[((x D n ) + + q D n 1 ) + ]. It is easy to verify that minimizing this modified myopic cost leads to a policy that attains the service level L, which is the lower bound established in Theorem 2. The modified 12

15 myopic cost defined above is more conservative than the original myopic cost in the sense that excess inventory is penalized more. Consequently, it is intuitive that the optimal policy attains a higher service level than the policy obtained by minimizing this modified cost. Incidentally, it can be verified that this modified myopic cost function is the same as f 2 (x, q) h x b E(D 2 x) + (recall that we assume the lead time is one period here); so, the order quantity that minimizes this function is q2 (x). Next, we state easily computable bounds on the optimal order quantity qn (x). All these bounds are derived from Theorem 2 in a straightforward way. For example, since (x D n1 ) + 0 and the results of Theorem 2 hold, P (q n (x) D n2 + D (n 1)1 ) U. This gives the first bound. The other bounds are derived in a similar fashion. Lemma 2 (i) q n(x) Φ 1 (U). (ii) x for which 0 x x n, q n (x) (Φ Φ 1) 1 (U) x. (iii) q n(x) Φ 1 (L) x. (iv) x + q n (x) (Φ Φ 1) 1 [ L Φ 1 ( Φ 1 (U))Φ 1 (x) ]. (v) x n (Φ Φ 1 ) 1 (L). Thus we have developed a series of bounds on the optimal order quantity qn (x). These bounds will be used in the heuristics we describe subsequently. Observe that the first upper bound is a constant while the second upper bound defines a base-stock policy. Similarly, the first lower bound defines a base-stock policy while the second is a more complicated function. In our numerical examples, presented in section 5, we observe that for small values of x the base-stock like upper bound is inferior to the other upper bound while the base-stock like lower bound is superior to the other lower bound. For higher values of x, the other bounds dominate. These examples also suggest that the effective upper bound (minimum of the two upper bounds) is much closer to the optimal policy than the effective lower bound (maximum of the two lower bounds). 13

16 4 Base Stock Policies : A Convexity Result We now consider a particular class of policies, the base-stock or order-up-to S policies. Although such policies are not optimal, they are often used in practice because they are easy to implement. One interesting research question is how well do such policies perform? Another equally important and interesting question is how do we determine the value of S, the order-up-to level, that minimizes the cost among this class of policies?. The latter is the issue that we address in this section; the former we study subsequently. In the absence of simple analytic methods to determine the optimal value of S, it is common to use simple search techniques in combination with simulation to decide on a value of S. Recently, infinitesimal perturbation analysis has been advocated as an effective and efficient technique for computing stock levels (see Glasserman and Tayur (1995)). However, these techniques can be guaranteed to yield optimal solutions only if the cost function is known to be convex. With this as the motivation, we next prove the convexity of the discounted cost function for our finite horizon problem in the parameter S. Convexity results for other lost-sales inventory problems using base-stock policies are available in Downs et al. (2001) and Janakiraman and Roundy (2004). Lead times and demands are still stochastic. However, we will prove that the cost is convex with respect to S for any realization of lead times and demands. Let us fix a realization of lead times and demands. Let x n (S) = d on hand inventory at the beginning of period n, q n (S) = d order placed in period n which equals S x n (S), and, l n (S) = d the amount of lost sales in period n, when an order-up-to policy with parameter S is used. Assume x N (S) = S, that is, the system starts with S units of inventory on-hand. Now we write recursive equations describing the evolution of the on-hand inventory through time. Subsequently, we show that the on-hand inventory at the beginning of any period can be 14

17 seen to be a piecewise linear function of S. Then we show that the cost incurred in period n is a linear combination of S and x n 1 (S). The combination of these results implies that the discounted cost function is a piecewise linear function in S. Lastly, these results will be shown to imply the desired convexity result. At the beginning of a period, just prior to the time an order is placed, all the inventory in the system is on hand. As soon as the order is placed, the inventory position takes the value S. Hence, x n 1 (S) can be computed as the inventory position at the beginning of the previous period, S, less the total depletion in inventory in the previous period, (D n -l n (S)). Thus, x n 1 (S) = S (D n l n (S)), or l n (S) = D n + x n 1 (S) S. (4.14) An interesting property of x n (S) is that it is a piecewise linear function of S. As we will see, x n (S) has a slope of 0 or 1 in all its linear segments and it also always lies between 0 and S. To obtain these results, we prove the following Lemma. Lemma 3 (a) x n (S) is piecewise linear in S. (b) x n(s) {0, 1} wherever x n ( ) is differentiable. (c) 0 x n S. Proof : We use a proof by induction. Observe that x n 1 (S) = [(x n (S) D n1 ) + + q n (S) D n2 ] + = [S (D n2 + min(x n (S), D n1 ))] + using the identity (a b) + = a min(a, b). It is easy to establish the three statements in the Lemma for x n 1 (S) using the statements for x n (S). Please see Janakiraman and Roundy (2004) for a detailed proof of this result when lead times are integers; the proof here is 15

18 virtually the same. Q.E.D. Define the one period cost function in period n to be v n (S) = d h x n 1 (S) + b l n (S), that is (4.15) v n (S) = (h + b)x n 1 (S) b S + b D n. (4.16) Compare this definition with definition (3.2) in section 3. The definition of the one-period cost function in (4.15) is the same as (3.2) without the expected value operator, except for the following difference. The holding cost for the on-hand inventory at the end of period n (beginning of period n 1) is charged in period n in (4.15) whereas it was charged proportional to the inventory on hand at the beginning of period n in (3.2). We define the cost function v n this way purely for algebraic simplicity. We are interested in proving the convexity of the function N n=1 α N n v n (S). Figure 2 goes here. In the graph above (see Figure 2), we have generated plots of x n (S) for a particular example. It can be observed from the figure that the individual x n ( ) functions are not convex. However, it can be observed that at any point where the slope of x n ( ) drops from 1 to 0 for some n, we can find some n < n such that the slope of x n ( ) increases from 0 to 1. This observation can be formalized and used to prove that m n=0 x n (S) is a piecewise linear function whose slope is increasing. The convexity of the cost function follows directly from this fact. Such a proof is available in Janakiraman and Muckstadt (2001). That proof is long and complicated; but, there is a simpler way of proving the convexity result, which we will present next. To prove the convexity conjecture, we first state Lemma 4: Let {a m } be a non-negative, increasing sequence. For any realization of demands {(D n1, D n2 )} and any n, the function N m=n a m x m (S) is convex. The proof is given in the Appendix. 16

19 The convexity of the discounted cost function is a simple corollary of Lemma 4. Corollary 1 N m=1 α N m v m (S) is convex in S. Proof : Application of Lemma 4 with a m = d α N m shows that N m=1 α N m x m 1 (S) is convex. Consequently, equation (4.16) implies that N m=1 α N m v m (S) is convex. Q.E.D. Thus the discounted cost function is convex in S for every realization of the sequence of random variables {(D n1, D n2 )}. We exploit the convexity of this function in the next section to find the best order-up-to S policy using a bisection search. 5 Computational Investigation In this section, we describe computational experiments used to investigate the performance of heuristics derived directly from bounds developed in section 3.3 and the performance of the best order-up-to policy. Even though we proved our analytic results only for finite horizon models, similar results hold for the infinite horizon discounted cost and infinite horizon average cost models; the proofs for the infinite horizon, average cost case would involve some tedious technical discussions which we have avoided in the interest of space. For our computational investigation, we chose the infinite horizon average cost to compare the performance of policies in our environment due to the following reason. If we use a finite or an infinite horizon discounted cost performance measure, we would have to decide the starting state, that is, the on hand inventory in the first period; we would then be left with the question of how sensitive the comparison of the policies is to the starting state. The infinite horizon average cost model does not have this issue. 17

20 We use two distributions for modeling demands (D n1, D n2 ). In the first model, we assume that the distribution of (D n1, D n2 ) is bivariate normal. In the second model, we assume that D n1 and D n2 are stochastically independent and each of these is a random variable with a gamma distribution. The bivariate normal and gamma distribution were chosen for the following reasons: (a) the lower and upper bounds we have developed use the inverse CDFs of the demand distributions, and these inverse CDFs are readily available for the gamma and the normal distributions in Excel; (b) the bivariate normal is useful to investigate the effect of correlation between demands in the first and second parts of the period; (c) the gamma distribution has a heavier tail than the normal and it is interesting to see how the performance of the heuristics vary depending on which distribution is used. The cost parameters we use are b/h {5, 25, 100}. We use the following demand parameters: (i) Bivariate Normal: D n1 has mean µ 1 and standard deviation σ 1, D n2 has a mean µ 2 and standard deviation σ 2, and the two random variables have a correlation of ρ. The parameter values that we used are: (µ 1, µ 2 ) {(10, 20), (20, 10)}; if (µ 1, µ 2 ) = (10,20), σ 1 {1.75, 3.5, 7}, σ 2 {4, 8, 12} and if (µ 1, µ 2 ) = (20,10), σ 1 {4, 8, 12}, σ 2 {1.75, 3.5, 7}; ρ { 0.5, 0, 0.5}. (ii) Gamma: β = 20, α 1 {0.25, 0.5, 1, 2}, α 2 {0.25, 0.5, 1, 2}. In total, with all the possible cost parameters, we had 162 sets of parameter values with the bivariate normal distribution and 48 sets of parameter values with the gamma distribution. We chose three heuristics for performance evaluation. The first heuristic policy uses the least of the two upper bounds derived in section 3.3 as the order quantity. That is, the amount ordered in any period that starts with x units of inventory on hand equals UB(x), 18

21 where UB(x) = d [min( Φ 1 (U), (Φ Φ 1 ) 1 (U) x)] +. The second heuristic is the analogous policy based on the lower bounds derived earlier. That is, the amount ordered in any period that starts with x units of inventory on hand equals LB(x) where LB(x) = d max{ lb(y) : y x } and lb(x) = d [ max( Φ 1 (L) x, (Φ Φ 1 ) [ 1 L Φ 1 ( Φ 1 (U))Φ 1 (x) ] x) ] +. Here, lb(x) is the maximum of the two lower bounds at x. However, lb(x) is not monotone. We exploit the fact that the optimal policy is a decreasing function of x to tighten this lower bound with our definition of LB(x). The third heuristic is a base-stock or order-up-to policy according to which q S (x) units are ordered if a period starts with x units of inventory where q S (x) = d (S x) + and S is the best order-up-to level. S was determined using a bisection search procedure. For each policy, the evolution of the starting inventory level in a period is a Markov chain with a transition matrix that depends on the policy. Starting with a uniform distribution of the inventory level, we used the transition matrix corresponding to the policy to compute the probability distribution of the starting inventory level in the twentieth period. We verified that this distribution was an accurate proxy for the steady state distribution. This steady-state distribution was used to compute the infinite horizon average cost. For more details on the computational experiments, please see the appendix. It is worth noting that we used the upper and lower bounds to limit our search for the optimal policy. Surprisingly, the optimal policy was significantly close to the upper bound policy in most 19

22 cases. This helped in reducing the computational effort in determining the optimal policy considerably. and 4. The heuristic policies and the optimal policy for two examples are plotted in figures 3 Figures 3 and 4 go here. We present the results for the upper bounding heuristic (UB) and the best order up to policy (S ) in Tables 1-8. We omit the results corresponding to the lower bounding heuristic because its performance is significantly inferior to the other policies. The table heading and the column and row headings specify the parameters. The entries in Tables 1-4 and 6-7 are the average percentage difference in costs between each heuristic (UB/S ) and the optimal policy. Tables 5 and 8 summarize the overall performance of the heuristics with the minimum, maximum and average percentage errors over all the experiments for the normal and gamma distribution cases, respectively. Tables 1-8 go here. The average optimality gaps of the UB heuristic are 0.76% and 2.14% for the normal and gamma distribution cases, respectively. The maximum differences from the optimal costs are 3.38% and 7.21%, respectively. The S policy has average differences of 0.8% and 0.5% and maximum differences of 2.85% and 1.8% from the optimal costs for the normal and gamma distribution cases, respectively. A major reason for the optimality gap being so small when we use these heuristics is the flatness of the cost curves around the optimal solution. To illustrate this idea, we have plotted f n (x, q) as a function of q for two values of x for one of our test examples in Figure 5. We can observe from the plot that though the order quantities suggested by the heuristics are not very close to the optimal quantities, the costs are almost 20

23 identical to the optimal cost. Figure 5 goes here. We observe that the performance of both heuristics improves as the b/h ratio increases. When the demand distribution is bi-variate normal, as ρ increases, the performance of the UB heuristic is unaffected, while the performance of the order-up-to policy deteriorates. There does not seem to be any other strong trend in the performance of either of the heuristics with respect to the problem parameters. To summarize these results, it appears that both the UB heuristic and the best orderup-to policy both perform very well. However, the UB heuristic is easy to implement using spreadsheet functions, whereas it is necessary to use a search procedure, each step of which involves policy evaluation, to determine the best order-up-to level. 6 Conclusions We have examined a single location periodic review inventory control problem with lost sales and fractional lead times. This work was motivated by applications in service part supply chains and retail chains; emergency orders in the former example and product stock-outs in the latter are much better modeled as lost sales rather than backorders. Research directed at inventory control problems with lost sales and lead times is scarce. Among the few papers that consider such problems, Morton (1969) s is a significant one, and our work benefits considerably from his analysis of lost sales problems with integer lead times. We model the optimal inventory control problem as a stochastic dynamic program and analyze properties of the objective function as well as the optimal policy. We present upper and lower bounds on the optimal policy. These bounds are based on the inverse cumulative distribution functions of some demand random variables and are very easy to compute. 21

24 These computations are direct and non-recursive, and, these bounds are easily implementable heuristics in practice. In addition to these heuristics, we also analyze properties of the system when order-up-to S policies are used. The main result we prove in this setting is the convexity of the cost function with respect to the order-up-to parameter S. In our experiments, we use this convexity property to determine the best order-up-to levels for this system using bisection search. Our computational investigation reveals that the upper-bound heuristic and the best order-up-to S policy both perform very well. In fact, on an average, they are only 1.45% and 0.66% more expensive than the optimal policy, which is computationally intensive to determine in large scale problems. 7 Appendix 1: Description of Computational Experiments 1. Truncation/Discretization of demands: In the experiments with the gamma distributions, demands are truncated at 250. All probabilities in this range are scaled up uniformly to sum up to one. In the experiments with the normal distributions, the vector of demands in the two parts of a period is restricted to be in the Cartesian product of {0, 1,..., 100} and {0, 1,..., 100}. The probability mass at some integer pair (u, v) for this discrete distribution is a constant multiplied by the joint density of the bi-variate normal distribution at (u, v), where the constant is chosen such that the masses add up to one. 2. Lower and Upper Bounds: Since LB(x) and UB(x) might not be integers, we used the floor of LB(x) and the ceiling of UB(x) as the lower and upper bounds on the optimal order quantity, respectively. 3. Policy Evaluation: We are interested in finding a close approximation for the long run average cost performance of different policies. Starting from a uniform distribution of states, the actual probability distribution of on hand inventory after 20 periods, 22

25 given any policy, is computed. In our computational experiments, we observe that the probability distribution of on hand inventory changes negligibly after 20 periods. Therefore, we use this distribution as the proxy for the steady state distribution. The long run average cost attained by this policy is then found using its proxy steady state distribution. 4. Finding the Optimal Policy: One way to determine the optimal policy for the average cost performance measure is to find the optimal policies with the infinite horizon discounted cost measure for a sequence of discount factors converging to one. However, this process is computationally very intensive. Instead, we take the following approach. Using a discount factor of one, we compute the optimal policies for finite horizon problems of increasing length until the policies converge. We take the converging policy as an approximation for the optimal policy with respect to the long run average cost performance measure. 5. Finding the Optimal Base-stock Level: We perform a bisection search to find the optimal base-stock level. For a given base-stock level, we compute the performance by finding a proxy for the steady state distribution of on hand inventory levels as described under Policy Evaluation. 23

26 8 Appendix 2: Proofs Proof of Theorem 1: Our proof is by induction. Recall that f 0 (x) = h x. Thus statements (b) and (d) are trivially true for f 0 ; also, we have f 0 (x) = 0 for all x. Now, we assume that the statements (b) and (d) in the theorem are true for functions f 0, f 1,..., f n 1 and that all these functions are convex. Now, let us show that all the statements (a) - (f) are true for period n as well. Next, to simplify notation, let θ(x) = d ((x D n1 ) + + q n (x) D n2), θ 1 (x) = d x D n1 + q n(x) D n2, θ 2 (x) = d q n (x) D n2. That is, θ(x), θ 1 (x) and θ 2 (x) are random variables that depend on D n1 and D n2. We use 1( ) to denote the indicator function of an event ( ). From (3.2) and (3.4), we get f n (x, q) = h x + b E Dn [(D n1 x) + ] + b E Dn [(D n2 ((x D n1 ) + + q)) + ] +α E Dn [f n 1 (((x D n1 ) + + q D n2 ) + )]. Differentiating this expression with respect to q, we get F n (x, q) = α E Dn [f n 1 ((x D n1) + + q D n2 ) 1((x D n1 ) + + q D n2 > 0)] = b b P ((x D n1 ) + + q < D n2 ) (8.17) +α +α x u 1 =0 x u 2 =x u 1 +q x u1 +q u 1 =0 u 2 =0 q u 1 =x u 2 =0 φ 12 (u 1, u 2 ) du 2 du 1 b u 1 =x u 2 =q f n 1(x u 1 + q u 2 )φ 12 (u 1, u 2 ) du 2 du 1 φ 12 (u 1, u 2 ) du 2 du 1 f n 1 (q u 2) φ 12 (u 1, u 2 ) du 2 du 1. (8.18) 24

27 Let us now compute the derivatives of F n (x, q) w.r.t. x and q. F n (x, q) x F n (x, q) q x = b = b + α u 1 =0 x φ 12 (u 1, x u 1 + q)du 1 u 1 =0 (b + αh αb) x + α + α + α + α u 1 =0 x u 1 =0 x x u1 +q u 2 =0 x u 1 =0 x + α f n 1 (0)φ 12(u 1, x u 1 + q)du 1 u 1 =0 f n 1 (x u 1 + q u 2 )φ 12 (u 1, u 2 )du 2 du 1 φ 12 (u 1, x u 1 + q)du 1 x u1 +q u 2 =0 u 1 =0 q u 1 =x u 1 =x φ 12 (u 1, x u 1 + q)du 1 > 0. + b φ 12 (u 1, q)du 1 u 1 =x f n 1(x u 1 + q u 2 )φ 12 (u 1, u 2 )du 2 du 1 f n 1 (0)φ 12(u 1, x u 1 + q)du 1 u 2 =0 f n 1 (q u 2)φ 12 (u 1, u 2 )du 2 du 1 f n 1(0)φ 12 (u 1, q)du 1 > 0. The inequalities above are obtained by using the facts that f n 1 (x) (h b) and f n 1 (x) 0. We have now shown that F n (x, q) is strictly increasing in both x and q. Using statement (b) for n 1, it can also be seen that F n (x, q) is strictly positive in the limit as q approaches for any given x. Since F n (x, q) is strictly increasing in q, q n(x) equals zero if F n (x, 0) 0 and q n (x) is the unique solution to the first order condition F n(x, q) = 0, otherwise (continuity of F n (x, q) ensures the existence of such a solution). As a consequence of these results, we know that x n such that q n(x) > 0 if and only if x < x n. The continuity and monotonicity properties of F n (, ) also let us define x n more precisely as the unique solution to the equation F n (x, 0) = 0. Thus we have shown statements (c) and (e). Before proving the other statements, we derive a useful expression for f n (x) by differentiating both sides of the expression f n(x) = f n (x, qn (x)). We use (3.2)-(3.4) and the fact that [ f n(x, q)/ q] q=q n (x) = F n (x, qn (x)) = 0 25

28 when qn (x) > 0 and ( q n (x)/ x) = 0, otherwise. f n (x) = df n(x, qn (x)) = [ f n (x, q)/ x] q=q dx n (x) + [ f n (x, q)/ q] q=q n (x)( qn (x)/ x) = h b P (x < D n1 ) b P (x > D n1, θ 1 (x) < 0) +α E Dn [f n 1 ((θ(x))+ )1(x > D n1, θ(x) > 0)] (8.19) = h b Φ 1 (x) b + α x u 1 =0 x u 1 =0 x u1 +qn (x) u 2 =0 Evaluating this expression above at x = 0, we get u 2 =x u 1 +q n (x) φ 12 (u 1, u 2 ) du 2 du 1 [ f n 1 (x + q n (x) u 1 u 2 )φ 12 (u 1, u 2 ) du 2 du 1 ]. (8.20) f n (0) = h b. (8.21) Now, we prove statement (g), that is, f n ( x n) = h by using the relations F n ( x n, 0) = 0 (using the first order condition since qn ( x n) = 0 and qn (x) > 0 if x < x n) and (8.19). From (8.17), we get 0 = F n ( x n, 0) = bp ( x n < D n1 + D n2 ) + αe[f n 1( x n D n )1( x n > D n )]. From (8.19), we get f n ( x n) = h bp ( x n < D n1 ) bp ( x n > D n1, x n < D n1 + D n2 ) + αe[f n 1 ( x n D n )1( x n > D n )]. Combining these two equations we can see that f n ( x n) = h. Next, we prove statement (f). To do this, we exploit the fact that F n (x, q n(x)) = 0 x [0, x n ]. Let H(x) = d F n (x, qn dh (x)). Thus = 0, 0 x x dx n. However, ( ) ( ) ( ) dh dx = Fn (x, q) Fn (x, q) q + n (x), when 0 x x n. x q x q=q n(x) q=q n(x) Using expressions developed earlier for the partials of F n, we get ( dh dx = (b + α f n 1(0)) 1 + q n (x) ) x φ 12 (u 1, x u 1 + q x n(x)) du 1 u 1 =0 26

29 ( ) q +(b + α f n 1 (0)) n (x) φ 12 (u 1, qn x (x)) du 1 u 1 =x x x u1 +qn(x) +α [f n 1(x + qn(x) u 1 u 2 )φ 12 (u 1, u 2 ) du 2 du 1 u 1 =0 u 2 =0 (1 + q n (x) x )] ( ) q n (x) q +α f n 1(q n(x) u 2 ) n (x) φ 12 (u 1, u 2 ) du 2 du 1 u 1 =x u 2 =0 x in the region 0 x x n. Note that (b + α f n 1(0)) > 0 (by statement (d) for n-1) and f n 1( ) 0. This implies that 1 < dq n (x) dx < 0, 0 x x n since the equation dh = 0 can now be seen to take the form dx ( ) ( ) 0 = (positive term) 1 + dq n(x) dq + (positive term) n (x). dx dx Thus we have proved statement (f). Next, we prove that f n (x) is strictly positive. First, we differentiate both sides of equation (8.20) we get f n (x) = b φ 1(x) + b x u 1 =0 φ 12 (u 1, x u 1 + q n (x)) du 1 (1 + dq n(x) dx ) b φ 12 (x, u 2 ) du 2 u 2 =qn (x) x [ ] x u1 +qn +α (x) f n 1 (x + q n (x) u 1 u 2 ) (1 + dq n(x) u 1 =0 u 2 =0 dx ) φ 12 (u 1, u 2 ) du 2 du 1 [ ] x + α f n 1 (0)φ 12(u 1, x u 1 + qn (x)) (1 + dq n(x) u 1 =0 dx ) du 1 +α q n (x) u 2 =0 f n 1(q n(x) u 2 ) φ 12 (x, u 2 ) du 2. (8.22) Observe from equation (8.22) and statements (d) and (f) that f n (x) is strictly positive for n 1. In other words, f n (x) is strictly convex, n 1. 27

30 Recall from (8.21) that f n (0) = h b and from statement (g) that f n ( x n) = h. Statements (b) and (d) are direct consequences of these facts and the strict convexity of f n. We have now proved all the statements listed in the theorem. Q.E.D. Proof of Lemma 1 : Before getting into the details, it will be useful to recognize some basic inequalities which are derived directly from Theorem 1. Define q n to be qn(0). As a consequence of (3.7) and (3.12) we have the following: 0 qn (x) q n, and x x + qn (x) x n, 0 x x n. (8.23) x 1 < x 2 if and only if f n (x 1) < f n (x 2). (8.24) Recall that f n ( x n) = h. The two inequalities stated in the theorem are now obtained easily using (8.19). First, we derive the upper bound formula for f n(x). First, we use the fact that f n( ) is an increasing function in (8.19) to get f n(x) h b P (x < D n1 ) b P (x > D n1, θ 1 (x) < 0) + αe[f n 1( x n )1(x > D n1, θ(x) > 0)]. Since f n( x n ) = h, we can rewrite this inequality as f n (x) h b + (b + αh)p (x > D n1, x D n1 + q n (x) > D n2). The right hand side of the above inequality is clearly less than (h b) + (b + αh)φ 1 (x). This is the upper bound for f n(x). Next, we derive the lower bound. To do this, we will be manipulating (8.17) and (8.19). Using the fact that F n (x, q n(x)) = 0 when x x n along with (8.17), we get αe[f n 1(θ(x))1(θ(x) > 0)] = bp (θ(x) < 0). 28

31 Manipulating (8.19) using the above equality, it can be verified that f n (x) = h b P (x < D n1, q n (x) > D n2) α E Dn [f n 1 (q n (x) D n2)1(x < D n1, q n (x) > D n2)]. Since f n 1 ( ) is an increasing function and f n 1 ( x n) = h, we can see that the right hand side is bounded below by (b + αh h) + Φ 1 (x)(b + αh), which is the lower bound stated in the theorem. Q.E.D. Proof of Theorem 2 : First, it is useful to rewrite π n(x) as π n(x) = E Dn E D(n 1)1 [ 1((x Dn1 ) + + q n(x) D n2 D (n 1)1 ) ] = E Dn ( Φ1 ((x D n1 ) + + q n (x) D n2) ). (8.25) Both inequalities will be established by using Lemma 1 and the relation F n (x, qn (x)) = 0 x (0, x n ). The sequence of steps is as follows: (i) consider equation (8.17) and the expression F n (x, qn (x)) = 0, (ii) use Lemma 1 to eliminate f n 1 from this statement, (iii) we would then have an inequality with zero on one side and an expression that depends only on h, b, α and the distribution functions of demand, (iv) substitute some probability expressions with alternate expressions that bound the original probabilities from below or above. First, we derive the upper bound. 0 = F n (x, q n(x)) = bp ((x D n1 ) + + q n (x) < D n2) + αe Dn [f n 1 ((θ(x))+ )1(θ(x) > 0)] = bp (θ(x) < 0) + αe Dn [f n 1(θ(x))1(θ(x) > 0)] [ = b + E Dn (b + α f n 1 (θ(x)))1(θ(x) > 0)] b + E Dn [(b + α [(h b α h) + Φ 1 (θ(x))(b + α h)])1(θ(x) > 0)] (using Lemma 1) = b + E Dn E Dn 1 [(b + α (h b α h) 29

32 + α(b + α h)1(d (n 1)1 < θ(x)))1(θ(x) > 0)] (because Φ 1 (θ(x)) = P (D (n 1)1 < θ(x)) = E Dn 1 [1(D (n 1)1 < θ(x))]) b + E Dn E Dn 1 [(b + α (h b α h) +α(b + α h)1(d (n 1)1 < θ(x)))1(θ(x) > D (n 1)1 )] (because 1(θ(x) > 0) 1(θ(x) > D (n 1)1 )) = b + (b + α h)p (θ(x) > D (n 1)1 ) = b + (b + α h)p ((x D n1 ) + + qn (x) D n2 > D (n 1)1 ) = b + (b + α h)πn(x). Therefore, we have π n(x) b. b+αh To prove the other inequality, we again start with the expression F n (x, qn (x)) = 0. 0 = F n (x, qn (x)) = b + bp ((x D n1 ) + + qn(x) > D n2 ) + αe Dn [f n 1((θ(x)) + )1(θ(x) > 0)] = b + E Dn [(b + αf n 1 ((x D n1) + + qn (x) D n2)) 1((x D n1 ) + + qn(x) > D n2 )] b + E Dn [(b + α(h b) + α(b + αh)φ 1 ((x D n1 ) + + q n (x) D n2)) (using Lemma 1) 1((x D n1 ) + + q n(x) > D n2 )] b + E Dn [b + α(h b) + α(b + αh) Φ 1 ((x D n1 ) + + q n(x) D n2 )] ( since 1((x D n1 ) + + q n (x) > D n2) 1 and (b + α(h b) + α(b + αh) Φ 1 ((x D n1 ) + + q n (x) D n2)) 0) = α(h b) + α(b + αh)π n (x) ( using equation (8.25) ). Therefore, π n(x) b h b + αh. Q.E.D. 30