Inventory Systems with Stochastic Demand and Supply: Properties and Approximations

Transcription

1 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Inventory Systems with Stochastic Demand and Suly: Proerties and Aroximations Amanda J. Schmitt Center for Transortation and Logistics Massachusetts Institute of Technology Cambridge, MA, USA Lawrence V. Snyder Det. of Industrial and Systems Engineering Lehigh University Bethlehem, PA, USA Zuo-Jun Max Shen Det. of Industrial Engineering and Oerations Research University of California Berkeley, CA, USA December 12, 2009 ABSTRACT We model a retailer whose sulier is subject to comlete suly disrutions. We combine discrete-event uncertainty disrutions) and continuous sources of uncertainty stochastic demand or suly yield), which have different imacts on otimal inventory settings. This revents otimal solutions from being found in closed form. We develo a closed-form aroximate solution by focusing on a single stochastic eriod of demand or yield. We show how the familiar newsboy fractile is a critical trade-off in these systems, since the otimal base-stock olicies balance inventory holding costs with the risk of shortage costs generated by a disrution. 1

2 Working Paer, Forthcoming in the Euroean Journal of Oerational Research 1 Introduction Suly disrutions can have drastic imacts on firms who fail to rotect against them. Traditional inventory models focus on demand uncertainty and design the system to best mitigate that risk. However, the effects of suly disrutions can have very different imlications for system design. In the ast five years, there has been an exlosion of research on inventory and suly chain models with suly disrutions. In this aer, we examine a single-stage inventory system with disrutions and introduce an effective aroximation for systems with both disrutions and demand or yield uncertainty. These results constitute a set of tools that will be useful for future research on inventory models with suly disrutions. We examine otimal base-stock inventory olicies using infinite-horizon, eriodic-review models, for a single sulier whose single retailer is subject to stochastic disrutions. Due to the comlexity of mixing discrete and continuous distributions in modeling, it is comlicated to analyze suly disrutions in combination with either demand uncertainty or yield uncertainty. We examine both cases in this aer. Our first model considers stochastic demand. We develo an aroximate technique to set base-stock levels since the model cannot be solved in closed-form. This technique determines aroximately how many eriods worth of demand should be ket in stock based on the exected duration of disrutions and relative weight of enalty costs), and then sets safety stock levels accordingly. It essentially assumes that all other eriods exerience demand equal to their mean, or deterministic demand, so we call this the Single Stochastic Period SSP) aroximation. We comare the SSP erformance to the otimal solution found numerically), and demonstrate that on average it generates a cost increase of 0.17% and outerforms aroximations based on uniform or triangular distributions for demand. We next consider suly disrutions and yield uncertainty together, and aly the SSP aroximation. We show that it erforms very well in this system, with an average cost increase of 0.03%. There are many benefits of having a closed-form aroximate solution, such as the one we develo, for a roblem which would otherwise require numerical otimization. A closed-form solution clearly demonstrates the sensitivity of solutions to inut arameters. It can also be embedded into more comlicated models to add tractability. Closed-form aroximations 2

3 Working Paer, Forthcoming in the Euroean Journal of Oerational Research are also useful tools in ractice, since they are easier to imlement and use on an ongoing basis. The remainder of the aer is outlined as follows: In Section 2 we review relevant literature for the toic. In Section 3 we introduce our aroach for modeling disrutions and review a result that will be used later in the aer. We resent aroximate solutions for roblems with disrutions and stochastic demand in Section 4, and with disrutions and yield uncertainty in Section 5. We summarize our findings in Section 6. 2 Literature Review Suly uncertainty is tyically modeled as comlete disrutions, where suly halts entirely, or as yield uncertainty, where the suly quantity received varies stochastically. One of the first authors to discuss the imact of yield uncertainty was Silver [1976], who analyzes how to modify the EOQ order quantity in order to coe with variance in receit quantities. He considers variance that is either indeendent of order quantity or directly roortional to it. Yano and Lee [1995] rovide an extensive review of the aers on yield uncertainty models. They stress that suly uncertainty is very comlex and since closed-form solutions are often unachievable, valid heuristics must be further studied and develoed. Many aers on suly uncertainty focus on suly becoming comletely unavailable in the case of disrutions. Parlar and Berkin [1991] analyze the EOQ model with disrutions. Berk and Arreola-Risa [1994] ublished a correction to Parlar and Berkin s model, addressing logic errors regarding the occurrence of stock-outs and associated costs. Snyder [2009] introduces a closed-form aroximation for the roblem, and Qi et al. [2007] extend the model to include disrutions at both the sulier and retailer. Parlar and Perry [1995, 1996] extend it to include fixed costs and multile suliers. Some of the other contributors to the suly disrution field who do not focus on the EOQ model include Guta [1996], who considers fixed lead times in variable suly models and evaluates aroximate methodologies for a Q, r) system with lost sales, Parlar [1997], who considers a Q, r) system with backordering, and Song and Zikin [1996], who consider variable lead times and variable order quantities with a dynamic rogramming aroach. Güllü et al. [1997] examine dynamic deterministic demand over finite-horizon 3

4 Working Paer, Forthcoming in the Euroean Journal of Oerational Research and non-stationary disrution robabilities, and relate the otimal base-stock level to the newsboy fractile. Dada et al. [2007] extend the stochastic-demand newsboy model to include multile unreliable suliers. Snyder and Shen [2006] simulate inventory systems with either suly disrutions or demand uncertainty to study how the two sources of uncertainty can cause different inventory level and lacement decisions to be otimal. Schmitt and Singh [2009] also use simulation to test the imact of different tyes of uncertainty, including disrutions, stochastic demand, and discrete jums in demand. They test the advantages of various mitigation strategies for each of these risks. Generally aers focus on either yield uncertainty or suly disrutions. Chora et al. [2007] model both, analyzing the costs involved in bundling the variance from these two distinct sources in a single-eriod setting. They comare comlete disrutions to additive yield uncertainty and stress the imortance of correctly identifying and analyzing the tyes of stochasticity in suly. Schmitt and Snyder [2009] also consider a system with both yield uncertainty and suly disrutions, extending the analysis to an infinite-horizon setting. They demonstrate the imortance of considering the long-term imact of disrutions through multile-eriod analysis. Tomlin [2006] discusses three general strategies for coing with suly disrutions: inventory control, sourcing, and accetance. Inventory control strategies involve ordering and stocking decisions and can be considered mitigating, roactive techniques. Sourcing strategies are contingency lans and can be reactive to an actual shortage or used roactively in lanning for a otential shortage, and involve back-u sulier usage. Accetance means choosing not to roactively mitigate disrutions. Tomlin formulates an infinite-horizon, eriodic-review base-stock system when both an unreliable sulier subject to disrutions) and/or a more exensive, erfectly reliable sulier are available. He roves that single sourcing is otimal when the firm is risk-neutral and demand is either stochastic or deterministic, but that if the firm is risk-averse, dual-sourcing is often otimal. He resents a formula, a secial case of which is resented in Theorem 1 in Section 3 below, for the otimal base-stock level under deterministic demand and random disrutions. Snyder and Tomlin [2008] examine how inventory systems can be develoed to take advantage of advanced warnings of disrutions. They consider a system where the disrution 4

5 Working Paer, Forthcoming in the Euroean Journal of Oerational Research rofile changes over time; advanced warning of disrutions can change their anticiated robability of occurrence. They conclude that a threat advisory system can be extremely beneficial and allows increased cost savings esecially when the disrution robabilities are significantly different in different states, and that tight caacity reduces the benefit of a threat advisory system. Where Snyder and Tomlin consider how changing disrution robabilities should imact mitigation behavior, Golany et al. [2009] consider how mitigation behavior can imact disrution robabilities. They indicate that when disrutions are strategic caused intentionally, e.g. by terrorists), then mitigation at a location may reduce the likelihood of that location being targeted. order to lower the likelihood overall. They advocate distributing mitigation across a network in Contracting and financial investment are other methods of mitigating suly risk. Babich [2008] discusses financial investment in a sulier to revent a suly disrution caused by sulier bankrutcy. He considers rofit-sharing as one method of imroving the sulier s financial stability. Yang et al. [2009] consider a manufacturer who may ay to increase its information on how reliable a sulier may be, in order to better decide whether to use an alternate suly source. Wagner et al. [2009] rovide emirical evidence from the automotive industry that suliers disrution risks are tyically ositively correlated, and Babich et al. [2007] include correlation in their model; they discuss how ricing cometition between multile unreliable suliers can lead to a diversified suly base and less disrution imact risk for a retailer. When suliers disrution correlation is negative, then then can comete based on reliability, whereas when their disrution correlations are ositive they must comete more based on cost. In this aer, we contribute the following tools for analysis of inventory systems subject to suly disrutions: Exact and aroximate exected cost functions when suly is disruted and demand is stochastic A closed-form aroximation for the otimal base-stock level when suly is disruted and demand is stochastic A closed-form aroximation for the otimal base-stock level when suly is disruted and suly yield is stochastic Throughout the aer, we focus on how the familiar newsboy fractile is a critical value 5

6 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Notation S i π i F ) d h Table 1: Disruted-Suly System Notation Definition Base-stock level for the system Index reresenting being in a state with i disrutions in a row Probability of being in state i Cumulative distribution function for the disrution states Deterministic demand at the retailer er eriod Penalty cost er item er eriod Holding cost er item er eriod in systems with suly uncertainty, as it is for demand uncertainty, since the otimal basestock olicies balance the costs of over-stocking with the risk of costs due to suly shortage from disrutions. 3 Suly Disrutions and Deterministic Demand This section serves to introduce our method of modeling suly disrutions and discuss known otimal olicies for coing with them. We consider deterministic demand and suly yield, but relax these assumtions in subsequent sections. A full list of the model arameters and variables are given in Table 1. The order of events in a eriod is as follows: At the beginning of a eriod, the retailer orders u to the base-stock level, S. It then receives material instantaneously in a non-disruted eriod, or receives nothing in a disruted eriod. Demand is satisfied from on-hand inventory, and any unsatisfied demand is backordered. Holding or enalty costs are then assessed for the ositive or negative ending inventory level. The decision variable for the model is the base-stock level. Disrutions are modeled using an infinite-state discrete-time Markov chain DTMC), where the state reresents the number of consecutive disruted eriods. We denote π i as the robability of being in state i and assume that π i are decreasing with i. F i) = i j=0 π j as the cdf of this distribution. We define The following Proosition gives results that are a secial case of the 2-sulier model resented by Tomlin [2006]. 6

7 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Theorem 1 Tomlin, 2006). For a retailer with deterministic demand and suly disrutions with mf π i and cdf F i), a) the exected cost er eriod is given by E[C] = π i [hs i + 1)d) + + i + 1)d S) + ] 1) i=0 b) the exected cost given in 1) is convex. c) S = jd, where j is the smallest integer such that F j 1) Proof: Follows from Tomlin [2006]. +h. Theorem 1 indicates that at otimality, the system will not stock out +h ercent of eriods. These results arallel that of the familiar newsboy model with deterministic suly but stochastic demand, where the otimal order quantity for that model also rovides a solution that causes the system to not stock out in +h ercent of eriods. 4 Suly Disrutions and Stochastic Demand We now consider a retailer subject to both suly disrutions but deterministic yield) and random demand. In Section 4.1 we establish the exact exected cost function. In Section 4.2 we develo an aroximation that assumes that the demand is stochastic in at most one eriod er order cycle and deterministic otherwise. We demonstrate how, by choosing which eriod is treated as stochastic, one also chooses the base-stock level, and we show how to find the otimal choice of the stochastic eriod. We justify this aroach mathematically, using roerties of the demand distribution, as well as numerically, in our comutational study. We call this the Single Stochastic Period SSP) aroximation. In Section 4.6 we establish other aroximations and comare those aroaches to the otimal and to the SSP aroximation. We add stochastic demand to the model with df mx), where X has a mean of µ er eriod and variance of σ 2. For i eriods of demand, we denote the df as m i x i ) with no subscrit imlying i = 1); X i has a mean of iµ and variance of iσ 2. 7

8 Working Paer, Forthcoming in the Euroean Journal of Oerational Research 4.1 Exact Model The exected cost in a eriod with a successful delivery is: h S S x)mx)dx + S x S)mx)dx 2) Define a cycle as the time between successful deliveries. If the suly is disruted for i 1 eriods in a row thus the cycle is in its i th eriod), the costs are: h S S x i )m i x i )dx i + S x i S)m i x i )dx i 3) Thus we have the following exected cost function: cs) = S π i 1 h S x i )m i x i )dx i + S x i S)m i x i )dx i ) 4) Although the summand is similar to the newsboy cost function, the effective demand distribution is different for each term of the sum, and therefore the sum does not collase into a single newsboy function. The cost in 4) cannot be minimized in closed form. For the remainder of the aer, we assume the demand is normally distributed. Our analysis holds for any demand distribution with the aroriate loss function substituted into the results below and other minor changes made. For a single-eriod demand we have X Nµ, σ 2 ), with φ ) and Φ ) denoting the df and cdf of the distribution, resectively. For multile eriods, X i Niµ, iσ 2 ). We can then simlify the cost as given in the following roosition. Proosition 2. The exected cost for a single retailer subject to normally distributed demand and suly disrutions is convex and is equal to: cs) = π i 1 hs iµ) + σ )) S iµ i + h)g σ i where Gr) = v r)φv)dv reresents the standard normal loss function. r Proof: See Aendix, Section A.1. 5) The derivative of 5) is given in the roof of Proosition 2 as d ds cs) = ) S iµ π i 1 + h)φ σ i ) 6) 8

9 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Seeking a minimizer for 5), we set 6) equal to zero and find: ) S iµ π i 1 Φ σ i = + h ) S iµ Since Φ σ aears in 7) and we cannot searate S from an infinite number of i terms, a i closed form minimizer for equation 5) cannot be found directly. Thus we look to aroximate this cost with the aroximation in the following sections. Proosition 3. When the base-stock level S is set otimally for a system with suly disrutions and stochastic demand, the tye-1 service level i.e., the robability that all demands in a given eriod will be met from stock) equals Proof: See Aendix, Section A.2. +h. Thus the familiar otimal service level from the classic newsboy roblem and from Theorem 1 is still otimal when both stochastic demand and suly disrutions are resent. 7) 4.2 Single Stochastic Period Aroximation Let S be fixed and call I the aroximate number of eriods of mean demand stocked in the base-stock level; S = Iµ. We show here that by aroximating the loss function terms for i I, we can aroximate the demand as deterministic for all eriods of a cycle other than the I th where a cycle is defined as the time between successful deliveries). Thus we call this the Single Stochastic Period SSP) aroximation. The loss function term involving I, G S Iµ σ I ), is significant since S = Iµ and Gr) cannot be aroximated easily for small r. However for j I, the term in arenthesis for the loss ) S jµ function, σ, is either relatively large or small. Thus we can develo aroximations j ) for G. S jµ σ j It is well known that Gr) = φr) r1 Φr)) [Zikin, 2000]. Thus if r is a large ositive number, corresonding to j < I, Gr) = 0. If r is a large negative number j > I), Gr) = r. We use this to write out the SSP cost aroximation. Take the sum I 1 j=1 to 9

10 Working Paer, Forthcoming in the Euroean Journal of Oerational Research equal 0 if I = 1. The SSP cost aroximation is: I 1 cs) = π j 1 hs jµ)) + π I 1 hs Iµ) + σ )) S Iµ I + h)g σ + I j=1 π j 1 hs jµ) σ )) S jµ j + h) σ j j=i+1 I 1 = π j 1 hs jµ)) + π I 1 hs Iµ) + σ )) S Iµ I + h)g σ I j=1 π j 1 S jµ)) 8) j=i+1 Note that the demand terms in the summations are deterministic; our aroximation of the loss terms for eriods not equal to I means that, in effect, we assume the demand equals the mean jµ) for j I. Although 8) holds for any I, we show in Section 4.3 that, for given values of the inut arameters, there is a unique value of I that is valid in a sense defined below). Therefore, when differentiating cs) to otimize it, we may treat I as a constant. 4.3 SSP Solution We find the solution for 8) in two arts below. The first is the exact solution when the derivative of 8) yields a well-defined solution, and the second is for when it does not Well-Defined Solution Recall that F ) is the cdf of the disrution distribution, and let F r) = 0 for r < 0. The solution to 8) is given in the following Proosition. Proosition 4. Given the aroximate cost in 8) for a retailer subject to uncertain demand and disruted suly, for fixed I, the S that minimizes cs) is S = Iµ + σ ) F I 2) IΦ 1 +h π I 1 Proof: See Aendix, Section A.3. We establish roerties for I such that 9) is well defined in the following roosition. 9) 10

11 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Proosition 5. There exists at most one I such that both F I 2) < and F I 1) > +h and the argument to Φ 1 in equation 9) is in 0, 1) iff both of these inequalities hold. Proof: See Aendix, Section A.4. +h, We discuss how to find the solution when the argument to Φ 1 in equation 9) is not in 0, 1) in the next section. We had reviously taken I as fixed in terms of the inut arameters, and Proosition 5 confirms this; I is I = j + 1 for the minimum j such that F j) >. Note that this is exactly the solution given in the deterministic demand case +h in Theorem 1, with the excetion that this is a strict inequality and Theorem 1 also holds for F j) = +h Balanced Stock Solution Suose F j) = +h for some j; then by Proosition 5 the solution given in 9) is not well defined for any I; essentially we do not know whether to set I = j + 1 or j + 2. We derive an alternate method to set S for that case in this section. Let I 1 = j + 1. Plugging this into 9) gives: S 1 = j + 1)µ + σ ) F j 1) j + 1Φ 1 +h π j = j + 1)µ + σ ) j + 1Φ 1 πj π j = j + 1)µ + σ j + 1Φ 1 1) 10) Since Φ 1 1) =, the right-hand side of 10) equals j + 1)µ +. Relace with M, denoting a very large number, and we have S 1 = j + 1)µ + M 11) Now let I 2 = j + 2. Plugging this into 9) gives: S 2 = j + 2)µ + σ j + 2Φ F j) ) 1 +h = j + 2)µ + σ j + 2Φ 1 0) 12) π j 1 Since Φ 1 0) =, the right-hand side of 12) equals j + 2)µ. Again relace with M and we have S 2 = j + 2)µ M 13) 11

12 Working Paer, Forthcoming in the Euroean Journal of Oerational Research We now set 11)=13): j + 1)µ + M = j + 2)µ M M = µ 2 14) Plugging this in for M guides us to slit the difference between the I 1 and I 2 solutions. We call this the balanced solution to our aroximatation, since it balances the solutions for I = j + 1 or j + 2. Thus if F j) = +h oints ), we set I = j + 1 and the solution as S = µi ). for some j we will refer to these oints as jum Final Solution We combine the solutions for S given in Sections and above as follows. µ ) I + S 1 2, if there exists I such that F I 1) = = ; +h Iµ + σ ) IΦ 1 F I 2) +h π I 1, for the smallest I such that F I 1) >. 15) +h This solution yields S that can be greater than or less than the otimal S. The following Proosition describes the aroximate solution s behavior based on the inut arameters. Proosition 6. S is increasing with µ, increasing with increasing or decreasing with σ. Proof: See Aendix, Section A.5. +h for fixed I, and can be either We discuss the difference between the otimal and aroximate costs in the following section, and resent numerical results and comarisons for the costs and solutions in Section Difference Between the Exact and SSP Aroximate Cost We want to evaluate the difference between the true cost and the SSP aroximate cost in order to examine the extent to which each inut affects the aroximation s accuracy. The following Proosition comares the aroximate and exact costs. Proosition 7. The difference cs) cs) is always ositive, and is given by: cs) cs) = σ + h) I 1 j=1 ) S jµ π j 1 jg σ + j 12 j=i+1 ) ) jµ S π j 1 jg σ 16) j

13 Working Paer, Forthcoming in the Euroean Journal of Oerational Research This difference aroaches as σ aroaches. Proof: See Aendix, Section A.6. Thus the aroximate cost function always underestimates costs. Unfortunately there is no fixed bound on the aroximation error for a given S, as indicated by the fact that 16) aroaches as σ aroaches. However we show in the following section that while extreme error cases may occur, tyically the SSP aroximation erforms extremely well. 4.5 SSP Numerical Evaluation We examine the aroximation erformance in three arts: first we test the solution over a wide range of random inuts to see its general erformance. Next we see how secific inut arameters affect its erformance. Finally we evaluate the aroximation secifically at the jum oints, where F j) = +h for some j. We secify the DTMC for the disrution states using two arameters: α, the robability of a disruted eriod following a non-disruted eriod failure robability), and β, the robability of a non-disruted eriod following a disruted eriod recovery robability) Varying All Inuts We created 1000 data sets, setting µ = 100 and h = 1 and generating the other inuts randomly. We drew the newsboy fractile,, uniformly from [0.5,0.95], the failure roba- +h bility α from [0.0,0.5], the recovery robability β from [0.1,1], and σ from [0,33.33]. While in reality a failure robability of 50% and recovery robability of only 10% may not be entirely reasonable, we selected these wide ranges in order to thoroughly evaluate the SSP aroximation s erformance. In general, the aroximation erformed extremely well. The average ercent error in the absolute value of the difference between the exact and aroximate base-stock solutions, that is, S S, was 1.1%, and the average cost increase, c S) cs ), was 0.17%. We calculated S cs ) S numerically using Excel Solver.) The worst three cost increases came when either σ was high or both σ and were high, generating cost increases of 11%, 25%, and 62%. In all other cases the cost error was less than 7%, and in 99.1% of the cases it was less than 2%. The error was not clearly related to α and β within the ranges tested, but it increased as 13

14 Working Paer, Forthcoming in the Euroean Journal of Oerational Research % cost increase 70% 60% 50% 40% 30% 20% 10% 0% Newsboy Fractile /+h)) a) % cost increase 70% 60% 50% 40% 30% 20% 10% 0% Demand Standard Deviation Sigma) b) Figure 1: Cost Increase vs. a) +h and b) σ % increase 6% 5% 4% 3% 2% 1% 0% Sigma % cost increase % S increase Figure 2: Aroximate Solution Percent Cost Increase for Increasing Sigma the newsboy fractile and σ increase, as shown in Figure 1. With the excetion of a few high observations when σ or are high, there is no obvious trend in the grahs because the error is consistently close to zero Varying σ We tested the solution for increasing σ values, since this is one of the inuts that increases the difference in the costs, as given in 16). We fixed µ = 100, h = 1, = 20, α = 0.2, and β = 0.4, and found that as σ increases the aroximation error increases. This occurs because larger demand variances causes the aroximations we made for the loss function terms, G ), to be less accurate. A grah of the base-stock solution error and cost increase error of the aroximate solution is given in Figure 2. For this evaluation, the aroximation erforms very well; the average cost increase for this data sets was 0.07% and maximum was 0.33%. However, some extreme error cases may still occur, as exemlified by the single unusual case in Section when the cost increase was 62%. 14

15 Working Paer, Forthcoming in the Euroean Journal of Oerational Research S /+h) S* ~S Figure 3: Base-stock Solutions for Increasing Newsboy Fractiles Varying +h We also tested the aroximation erformance as the newsboy fractile,, increases. From +h Proosition 3, we know that the newsboy fractile is the otimal service level for the system. We fixed σ at 15 and maintained all other inuts as given in Section We noted that the 1000 random inuts tested in Section never required the first solution for S from 15) to be used, where F j) = +h for some j. For this data set, that solution is used twice, for j = 1 and 2. Figure 3 comares the otimal and aroximate S solutions. Clearly the two solutions match very closely, as it is difficult to see the aroximate solution behind the otimal values. The two cases where F j) = +h occur at +h = 2 3 and 4, yielding aroximate S solutions that are 2.8% and 1.1% higher than the otimal solution, 5 resectively. However the cost increases at those oints are only 0.003% and 0.002%. The average increase in cost for all S solutions for this data set was 0.003% Varying the Disrution Parameters We also tested the aroximation erformance as the disrution arameters change. We reset = 20 and ket σ = 15, then set β = 0.4 and increased the failure robability, α, and roduced Figure 4. Next we set α = 0.2 and increased the recovery robability, β, to roduce Figure 5. Both figures show how well the SSP solution matches the otimal solution; the average absolute value of the error in the base-stock solution was 0.5% for Figure 4 maximum absolute value of the error of 3.2%), and 0.6% for Figure 5 maximum of 3.6%). In general, the error in using the SSP aroximation is less sensitive to changes in the disrution arameters than to changes in the demand arameters. This is because the 15

16 Working Paer, Forthcoming in the Euroean Journal of Oerational Research S alha S* ~S Figure 4: Base-stock Solutions for Increasing Failure Probabilities S S* ~S beta Figure 5: Base-stock Solutions for Increasing Recovery Probabilities SSP does not aroximate the distribution for the disrutions at all, but it does aroximate the demand for multile eriods into a single stochastic eriod. The figures also demonstrate the extreme inventory levels that may be necessary if disrutions are very likely or exected to be very long. When β is very low, meaning disrutions are very long in length, significant quantities of inventory must be carried. For examle, when β = 0.1 meaning disrutions average 10 eriods in length), the otimal solution indicates that over 25 extra eriods worth of demand should be ket on stock to rotect against disrutions Results when F j) = +h None of the 1000 original data sets tested in Section generated a solution that had F j) = +h which we refer to as jum oints), requiring the first solution for S from 15) to be used. This demonstrates the low robability of such a case occurring randomly. To more thoroughly exlore the aroximate solution at the jum oints, we generated 1000 new random data set where we forced this to occur. We ket all random data already generated 16

17 Working Paer, Forthcoming in the Euroean Journal of Oerational Research cost ercent increase 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% newsboy fraction, /+h) Figure 6: Cost Increase at Newsboy Jum Point Solutions < 200) with the excetion of. The enalty cost was determined so that F j) =, where the +h aroriate j was determined so that I was between 1 and 10, I being set randomly and uniformly. We found that for many random data sets generated, +h aroached 1 in order to make j as high as was required, and this often made the exact system model unstable the otimal solution aroaches ). Thus we generated an excess of random data sets and chose the first 1000 such that +h for stability. The aroximation error generally increases with. When the fractile is unrestricted, +h we found one instance where the cost error reached 358%. However, when the enalty cost is less than 200 times the holding cost ), < the average cost error is 0.1% and all +h but one cost error is less than 6%. Figure 6 demonstrates this. The average cost increase for all newsboy fractiles was 1.3%. Note that while the cost error for this data set is higher on average and at extremes than for the original data set, the likelihood of these cases occurring is also smaller; while F j) is determined by the characteristics of the suly rocess, +h is determined indeendently by the cost structure and it is not likely that these values would be exactly equal. 4.6 Alternate Aroximations The SSP aroximation is limited to examining just one stochastic demand eriod er order cycle since including more than one does not yield a closed-form solution for S. S aears inside multile Φ ) terms, so an inverse cannot be taken to solve for S. We can include more stochastic terms if we aroximate the normal distribution with another distribution. In this section, we use the uniform and triangular distributions, as suorted by Scherer et al. 17

18 Working Paer, Forthcoming in the Euroean Journal of Oerational Research [2003], and numerically comare their erformance to the SSP aroximation in Section 4.7. We first introduce two simle aroximations where one of the sources of uncertainty is ignored Simle Aroximations Two simle aroximate solutions can be found by either assuming that demand stochasticity can be ignored σ = 0) or that suly disrutions can be ignored α = 0). The first uses the solution given in Theorem 1: S σ=0 = jd, where j is the smallest integer such that F j 1) The second uses the classic newsboy solution for the base-stock level: ) S α=0 = µ + σφ 1 + h + h We comare these solutions with the use of aroximations that account for both sources of uncertainty in our numerical evaluations. 17) 18) Uniform Aroximation If we aroximate the normal cdf terms in 6) with the uniform cdf, then we can include an infinite number of stochastic demand terms since the uniform cdf is linear and therefore more tractable. Scherer et al. [2003] roose using a uniform distribution with mean iµ and range [iµ σ 3i, iµ+σ 3i] to match the first and second moments mean and variance) of a normal distribution with mean iµ and standard deviation σ i. This leads to an aroximation for the cdf of F U x) = x iµ+σ 3i 2σ. Substituting this into the first derivative of the exact cost, 3i 6), leads to the following aroximate solution. Proosition 8. If the demand distribution Niµ, σ i) is aroximated with Uiµ σ 3i, iµ+ σ 3i), then for a retailer subject to uncertain demand and disruted suly, the otimal solution to the aroximation is 1 S U = [ µ π i 1 i + 2σ 3 + h 1 ) ) 19) 2 ] π i i 1 Proof: See Aendix, Section A.7. 18

19 Working Paer, Forthcoming in the Euroean Journal of Oerational Research For the behavior of this solution, we have the following Proosition. Proosition 9. S U is increasing with µ and σ, increasing with with +h if h >,. Proof: See Aendix, Section A.8. +h if > h, and decreasing Triangular Aroximation Suose instead of considering a single stochastic eriod, we also consider one eriod above and below I as well. We can do this if we aroximate the normal cdf with that of the triangular, since the triangular distribution is iecewise linear. The triangular distribution has been shown to aroximate the normal distribution better than the uniform [Scherer et al., 2003]. If we include 3 loss terms instead of just 1 as we did in the SSP cost, 8), the aroximate 3-term cost is: c 3 S) = I 2 π j 1 [hs jµ)] + j=1 π I 2 [hs I 1)µ) + σ )] S I 1)µ I 1 + h)g σ + I 1 π I 1 [hs Iµ) + σ )] S Iµ I + h)g σ + I π I [hs I + 1)µ) + σ )] S I + 1)µ I h)g σ + I + 1 π j 1 [hs jµ) σ )] S jµ j + h) σ j j=i+2 Note that this assumes I 2; if I 1, then the term involving π I 2 and the first summation are zero. The first summation is also zero if I = 2. This yields the following derivative: [ ) d S I 1)µ ds c 3S) = h + )F I 3) + + h) π I 2 Φ σ + I 1 ) )] S Iµ S I + 1)µ π I 1 Φ σ + π I Φ I σ I + 1 This is where an aroximation for the Φ ) terms is needed. To aroximate the normal distribution with the triangular distribution, we use the following triangular cdf as roosed 20) 21) 19

20 Working Paer, Forthcoming in the Euroean Journal of Oerational Research by Scherer et al. [2003]: 0, x < t 1 ; x t 1 ) 2 T x) =, t t 3 t 1 )t 2 t 1 ) 1 x t 2 ; 1 t 3 x) 2, t t 3 t 1 )t 3 t 2 ) 2 < x t 3 ; 1, x > t 3. 22) where t 1, t 2, and t 3 reresent the minimum, mean, and maximum of the ossible values for x. To aroximate the normal cdf with 22), where µ i and σ 2 i are the mode and variance of the normal distribution being aroximated so for i demand eriods, µ i = iµ and σ i = σ i), we use t 2 = µ i, t 1 = µ i σ i 6, and t3 = µ i + σ i 6. Thus for the triangular aroximation, we have: T S) = 0, S < µ i σ i 6); S µ i +σ i 6) 2, µ 12σi 2 i σ i 6) S µi ; 1 µ i+σ i 6 S) 2, µ < S µ 12σi 2 i + σ i 6); 1, S > µ i + σ i 6). An issue arises around the mean; we do not know whether S is less than, greater than, or ) equal to Iµ, so we are unsure how to exactly aroximate the middle term of 21), Φ, with the triangular distribution we do not know how to choose between the 2 nd and 3 rd cases of 23)). We move forward by testing both cases. Also, unlike the SSP aroximation, I cannot be determined by checking the condition in Proosition 5. Thus we assume that the best I to use here is that given by the deterministicdemand solution in Theorem 1, where I is the minimum I such that F I 1) +h. Since the triangular cdf involves the square of the S term, the solution is in terms of the quadratic equation coefficients. The quadratic coefficients for the two cases which solve the aroximation are given in the following Proosition. Proosition 10. The solution for the triangular 3-term aroximation is given as S Iµ σ I 23) S t = b ± b 2 4ac 2a where either, for case 1, µ I σ I 6) S µi, a = + h [ πi 2 12σ 2 I 1 π I 1 π ] I I I ) 25) 20

21 Working Paer, Forthcoming in the Euroean Journal of Oerational Research b = 2 + h) 12σ 2 [π I 2 ) ) )] µ σ 6 π I 1 µ + σ 6 π I µ + σ 6 I 1 I I ) c = + h)f I 3) + π I 1 + π I ) + + h [ π 12σ 2 I 2 I 1)µ 2 2µσ ) 6I 1) + 6σ 2 π I 1 Iµ 2 + 2µσ ) 6I + 6σ 2 π I I + 1)µ 2 + 2µσ )] 6I + 1) + 6σ 2 27) or, for case 2, µ < S µ I + σ I 6), a = + h 12σ 2 b = 2 + h) 12σ 2 [ πi 2 I 1 + π I 1 I [π I 2 π ] I I + 1 ) ) )] µ σ 6 + π I 1 µ σ 6 π I µ + σ 6 I 1 I I ) 29) c = + h)f I 3) + π I ) + + h [ π 12σ 2 I 2 I 1)µ 2 2µσ ) 6I 1) + 6σ 2 + π I 1 Iµ 2 2µσ ) 6I + 6σ 2 π I I + 1)µ 2 + 2µσ )] 6I + 1) + 6σ 2 30) Proof: See Aendix, Section A.9. In alying this aroximation, we solve for all S t values and choose the solution which yields the lower exected exact cost. 4.7 Aroximation Comarisons We evaluated the Uniform and Triangular aroximation techniques using the same set of 1000 random data sets tested in Section A summary of the cost error results aroximation solution cost increase above the otimal) is given in Table 2. No aroximation erformed better than the SSP aroximation. When comared to the otimal solution, the Uniform aroximation had an average absolute value of the ercent error of 34.5% for S and 14.6% for exected cost. The Triangular aroximation had average absolute value of the ercent errors of 12.1% and 8.2% for S and the exected cost, resectively. There were occasional observations where the Uniform or Triangular aroximations outerformed the SSP aroximation, but the averages and ercentiles for the errors make it clear that the SSP is a more reliable aroximation overall. With an average absolute value of error of 1.1% for S and 0.2% for exected cost, the SSP solution clearly outerforms the alternate 21

22 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Table 2: Aroximation Cost Increase Error Results Aroximation SSP Uniform Triangular σ = 0 α = 0 Average Cost Error 0.2 % 14.6 % 8.2 % 44.5 % % Maximum Cost Error 61.8 % % % % % Percent < 1% error 97.6 % 15.2 % 48.7 % 45.0 % 18.1 % Percent < 5% error 99.6 % 33.9 % 70.5 % 55.8 % 39.4 % aroximations; it is better able to cature the stochasticity of the demand with its single stochastic eriod than the Triangular or Uniform aroximations are able to do with three or all stochastic eriods. We also include the cost error for two simle aroximations discussed in Section The first, labeled σ = 0, assumes deterministic demand and uses the solution given in 17). This solution has the worst average and maximum cost errors of all solutions. The second simle aroximation ignores disrutions, labeled α = 0; while not as bad as the σ = 0 solution, the error for this solution is still very high and few solutions have a small error only 18.1% have less than 1% cost error). The cost error tends to be greater for the σ = 0 solution when disrution robabilities are low, since otimal costs are also lower in those cases. Low disrution robabilities mean the otimal base-stock level is relatively close to the mean, but the σ = 0 solution never stocks less than 2 eriods worth of demand and incurs heavy holding costs in those cases. In contrast, the α = 0 solution erforms very oorly when disrution robabilities are higher. Clearly there is a need to address the stochasticity of both demand and suly in setting inventory base-stock levels. Figure 7 shows the ercent cost error for all 1000 observations, in increasing order of error sorted searately for each aroximation), for each of these aroximations. The simle aroximations, σ = 0 and α = 0, have error values that extend above the uer limits of the grah. Desite only being able to incororate a single stochastic demand eriod into its model, the SSP aroximation erforms very well and clearly outerforms the other aroximations. With 97.6% of its solutions for the general random data) roviding a cost error of less than 1%, and 99.6% roving a cost error less than 5%, it is a ractical otion for setting base-stock levels in a system with both suly disrutions and stochastic demand. 22

23 Working Paer, Forthcoming in the Euroean Journal of Oerational Research 250% 200% % cost increase 150% 100% Tri. error Unif. error SSP error sigma=0 error alha=0 error 50% 0% ranked observation Figure 7: Aroximation Cost Error Comarison 5 Suly Disrutions and Stochastic Suly Yield We now consider another combination of discrete and continuous uncertainty in an inventory system: suly disrutions and yield uncertainty. We model demand as deterministic, equal to d er eriod, with id = i times d i eriods of demand). We consider additive yield uncertainty, assuming that the quantity received from the sulier is normally distributed with a mean equal to the quantity ordered and standard deviation of σ y indeendent of the order quantity). Note that this means deliveries could be either greater or less than that ordered, but this could be aroximately adjusted by adjusting the mean. We do not exlicitly include unit costs in our model, so either the sulier or buyer could be held accountable for any excess units delivered. The reader is referred to Yano and Lee [1995] for a review of more comlex models of yield uncertainty. The use of additive yield as oosed to yield that is roortional to the order size) is justified by Chora et al. [2007] as being realistic in the case where contracts are based on roduction batches but the exact yield is stochastic e.g., flu vaccines or semiconductors). This assumtion also allows us to formulate the SSP model and roduce useful insights that would not be achievable with a roortional yield formulation. Schmitt and Snyder [2009] show that the following is the exected cost 23

24 Working Paer, Forthcoming in the Euroean Journal of Oerational Research for this system: ))] id S c y S) = [π i 1 id S) + + h)σ y G σ y 31) While no closed-form otimal exression can be found for 31), we use it to determine the otimal service level in Proosition 11. Proosition 11. When the base-stock level S is set otimally for a system subject to suly disrutions and additive yield, the tye-1 service level i.e., the robability that all demands in a given eriod will be met from stock) equals Proof: See Aendix, Section A.10. +h. We aly the SSP aroximation aroach by considering only one stochastic eriod from 31) and aroximating the rest. We resent this model and numerical results in the following sections. 5.1 Aroximate Cost Formulation ) ) ) Our aroximation is that I S, G jd S jd S id S d σ y 0 for j > I and G σ y σ y for j < I. We formulate our aroximate cost as follows. I 1 ))] jd S c y S) = [π j 1 jd S) + h)σ y + = j=1 )) Id S π I 1 Id S) + + h)σ y G + σ y σ y j=i+1 [π j 1 jd S))] I 1 )) Id S π j 1 hs jd)) + π I 1 Id S) + σ y + h)g j=1 j=i+1 π j 1 S jd)) 32) 5.2 Aroximate Model Solution In solving this system, we again found that I is determined by the inut arameters to be the minimum I such that F I 1) >. Proof of this is omitted since it follows the same +h aroach as the roof of Proosition 5. We take a derivative of 32) to find the exact otimal solution when this inequality holds. For the case when F I 1) = 24 +h σ y for some I, we again

25 Working Paer, Forthcoming in the Euroean Journal of Oerational Research S /+h) S* ~Sy Figure 8: Base-stock Solutions for Increasing Newsboy Fractiles with Uncertain Yield alied the balanced stock argument from Section to determine the base-stock level. This leads to the following solution to the system. Proosition 12. Given the aroximate cost in 32) for a retailer subject to disruted suly and yield uncertainty, the otimal base-stock level for this model is: d ) I + S 1 2, if there exists I such that F I 1) = +h y = ; else Id σ y Φ 1 F I 1) ) +h π I 1, for the smallest I such that F I 1) >. 33) +h Proof: Follows the same aroach as the roofs of Proositions 4 and Numerical Evaluation In order to see how the aroximation erforms for both cases when F j) does and does not equal +h for all j, we again tested data with inuts α = 0.2, β = 0.4, µ = 100, σ = 15, and h = 1 for increasing, which generates F I 1) = +h +h for I = 1 and 2. A grah of the otimal and aroximate S solutions is given in Figure 8. Since the aroximate solution is barely visible behind the otimal solution, clearly the aroximation erforms very well. The highest cost increase from any of the aroximate solutions shown in Figure 8 is %. We also tested 1000 random data sets from Section 4.5.1, making the σ from that data set equal to the σ y for this system. The SSP aroximation erformed very well; the average cost increase was 0.02%, with a maximum occurrence of 5.9% increase. The ercent with a cost error less than 1% was 99.7%. In order to again test the case where F j) =, we also tested the data set from +h Section where we force this case to occur. Recall that this case never occurred in the 25

26 Working Paer, Forthcoming in the Euroean Journal of Oerational Research 1000 original random sets so the likelihood of it occurring naturally is low), but we generated values which made this case occur. The aroximation had an average cost increase of 0.3% for this data, with a maximum increase of 65%. For the data where < 200, the average cost increase is only 0.1% and the maximum observed is 5.1%. Both the average and extreme error cases for the SSP aroximation erformance for this system were better than the system with demand uncertainty. Since the standard deviation of the yield uncertainty is not roortional to I as it is for the demand uncertainty case), this makes the aroximation more accurate for larger I for this system. This hels imrove the accuracy of the SSP erformance for the yield uncertainty system. 6 Conclusions The SSP aroach erforms well for modeling a system subject to suly disrutions. It rovides a closed-form base-stock solution, which is valuable for researchers and ractitioners alike. Researchers may embed it in larger models, or examine the imact of inut arameters. Practitioners can more easily imlement and udate a closed-form solution. The results of this aer also demonstrate that suly disrutions can have significant negative imact on a retailer if it has not roactively rotected itself against them. We have examined three cases with suly disrutions in this aer: 1) deterministic demand and deterministic suly yield, 2) deterministic demand and stochastic, additive suly yield, and 3) deterministic suly yield and stochastic demand. It is interesting to note the imact of disrutions on a retailer in these cases if it does not roactively mitigate them. Since case 1) is entirely deterministic, it would carry no safety stock and disrutions would have the largest imact in this system. In the absence of disrutions and with equal standard deviation either on the demand or the suly yield), cases 2) and 3) would stock the same safety stock quantity. Therefore they would be equally affected by disrutions. Note, however, that safety stock maintained to rotect against regular demand or yield variability is only a fraction of a single eriod of demand. Thus if disrutions are moderate in duration greater than a single eriod), all three cases would suffer shortages of full demand quantities. Clearly disrutions must be rotected against, regardless of the other sources of uncertainty which are already mitigated in the system. 26

27 Working Paer, Forthcoming in the Euroean Journal of Oerational Research Throughout this aer, we have resented several roerties of a system subject to suly disrutions in order to allow retailers to establish best ractices for inventory management in such a setting. We have shown how the otimal base-stock level can be determined by the familiar newsboy fractile. When demand or yield are stochastic, we resented a closed-form Single Stochastic Period aroximate solution that yields very good results. The results from this aer can hel firms roactively and cost effectively rotect against suly disrution risk. 7 Acknowledgements This research was suorted in art by National Science Foundation grants DGE , DMI , and DMI This suort is gratefully acknowledged. We are also thankful for the helful suggestions rovided by anonymous referees. References S. Axsäter. Inventory Control. Kluwer Academic Publishers, Boston, MA, first edition, V. Babich. Indeendence of caacity ordering and financial subsidies to risky suliers. Working aer, Det. of Industrial and Oerations Engineering, University of Michigan, Ann Arbor, MI, V. Babich, A.N. Burnetas, and P. H. Ritchken. Cometition and diversification effects in suly chains with sulier default risk. Manufacturing & Service Oerations Management, 92): , E. Berk and A. Arreola-Risa. Note on Future suly uncertainty in EOQ models. Naval Research Logistics, 41: , S. Chora and P. Meindl. Suly Chain Management. Pearson Prentice Hall, Uer Saddle River, NJ, second edition, S. Chora, G. Reinhardt, and U. Mohan. The imortance of decouling recurrent and disrution risks in a suly chain. Naval Research Logistics, 545): , M. Dada, N. Petruzzi, and L. Schwarz. A newsvendor s rocurement roblem when suliers are unreliable. Manufacturing & Service Oerations Management, 91):9 32, B. Golany, E. H. Kalan, A. Marmur, and U. G. Rothblum. Nature lays with dice - terrorists do not: Allocating resources to counter strategic versus robabilistic risks. Euroean Journal of Oerational Research, 192: ,