Safety Stock or Excess Capacity: Trade-offs under Supply Risk

Safety Stock or Excess Capacity: Trae-offs uner Supply Risk Aahaar Chaturvei Victor Martínez-e-Albéniz IESE Business School, University of Navarra Av. Pearson, 08034 Barcelona, Spain achaturvei@iese.eu valbeniz@iese.eu June, 009 This paper investigates the trae-offs between carrying safety stock an planning for excess capacity when a firm faces uncertain eman an supply. Using well-known results from queueing theory, we characterize a buyer s optimal strategy regaring the amount of capacity that nees to be contracte with a supplier, an the corresponing inventory that it nees to hol. We typically fin that both are higher when eman or supply variability are higher. We use our moel to analyze multi-sourcing strategies: when multiple suppliers are available, we etermine at optimality the size of the supply base, the amount of capacity to be installe, an the inventory levels to be maintaine.. Introuction Matching supply an eman is a ifficult task for many companies. Inee, eman is often unpreictable, an firms must plan for aequate resources to face it. Usually, two ifferent approaches are use to aapt supply to eman. First, a company can carry a sufficiently large amount of inventory or safety stock, that will accommoate eman fluctuations. While this might increase expenses for maintaining this inventory, it also allows the firm to reuce the amount of capacity for prouction. Inee, with higher inventory levels between eman an prouction, the utilization of prouction resources can be increase as inventory ampens any eman shock on prouction scheuling. In other wors, prouction capacity can be use efficiently. An alternative approach is to carry low levels of inventory, an plan for a more flexible capacity, capable of being ajuste with eman changes. This can be achieve by planning a certain amount of excess capacity that will be use in perios of high eman. Of course, in this case, the expenses relate to capacity installation run high, while inventory costs can be lowere. Figure illustrates how a higher capacity level allows inventory to be reuce. The trae-off emerging from these two strategies is a elicate balancing act. Depening on the relative cost of capacity to inventory, firms will fin the appropriate amount of capacity

Inventory Level 3 0 Optimal inventory for capacity=3 0 0 0 30 40 50 60 70 80 90 00 Time Perio Optimal inventory for capacity= Figure : Example of inventory evolution, when capacity is limite exponential eman with average. We observe that, when the installe capacity is lower, the number of instances where inventory levels rop below the safety stock increases, an hence the optimal safety stock require is higher. to be installe an inventory level to be maintaine. This is specially important as some of these ecisions may be irreversible. Inee, increases in supply capacity may either be flexible an short-live, e.g., when temporary employees are hire for seasonal increases of eman, or coul be permanent, e.g., when a new line is installe or a new factory is opene. In such cases, large investments are require, which have a strong impact on the company s fate. Given the impact an irreversibility of such capacity investments, it becomes necessary that aequate cost-benefit analysis supports the ecisions taken. Specifically, one nees to ascertain that the aitional capacity is beneficial in reucing the nee for carrying inventory over a long time horizon. Furthermore, in taking these ecisions, it is important to consier the possibility that installe capacity may fluctuate over time, ue to machine break-owns, supplier isruptions, etc. Inee, in recent times, supply risk management has emerge as a priority for many inustries an may have a strong impact on the capacity ecision. The effect of limite supply capacity on inventory carrying cost is illustrate in apparel manufacturing. As shown by Raman an Kim [7], for apparel manufacturers solving successfully the capacity-inventory trae-off is key. First, they incur a high cost of carrying inventory ue to high working capital for maintaining their inventory. Banks are reluctant to give them loans, for the lack of aequate collateral, since the salvage value of their proucts is very low at the en of selling season. At the same time, they also face a high cost of stock-out since the margins on sales are high. Furthermore, eman is highly uncertain in this inustry an eman shocks can severely eplete the inventory. Finally, the capacity of suppliers is limite an fluctuates, since a supplier of fabric caters to the emans of many other apparel manufacturers, which might be given higher priority. Given the high inventory carrying cost, together with the nee to carry high inventory, it becomes imperative for

such apparel manufacturers to consier investing in a higher capacity from their suppliers, rather than carrying a high inventory. However, given the competitiveness of the inustry, an inappropriate ecision can severely affect the company s future. Also, scarcity of capital resources requires that they optimally ecie on the amount of supply capacity they nee to procure. From an economics stanpoint, an optimal ecision woul be to invest in supply capacity up to the point where the marginal increase in capacity investment is equal to the marginal ecrease in inventory cost. This trae-off between capacity investment an inventory cost gets more involve when capacity is not only limite but also uncertain. A buyer who has investe in the capacity of an uncertain supplier might have too little or too much capacity at its isposal in a given perio, resulting in an uner-utilization of its investment. Given uncertain capacity, the buyer might also consier the alternative of iversifying its supply base. Inee, the buyer can invest in the capacity of more than one supplier. This form of capacity investment has the ae avantage that supply variability is reuce through iversification. However, supply iversification also has costs associate with managing a more complex supply chain. Thus the buyer not only nees to ecie how much to invest in capacity but also how many suppliers to invest in, that is, the buyer nees to fin the optimal iversification strategy, that balances the inventory carrying cost, capacity cost an the iversification cost. In other wors the buyer nees to fin the right balance between inventory levels it maintains an the capacity it installs at each of the potential suppliers. In this paper, we first investigate the trae-off between capacity investment an the cost of carrying inventory. We moel an infinite-horizon perioic-review inventory system, with limite an uncertain capacity. The firm can make a single capacity investment into a single supplier, at the beginning of the moel horizon, which etermines the istribution of capacity available per perio. After taking this ecision, the firm relies on the supplier s realize capacity to serve the eman. We analyze the optimal inventory policy an the optimal capacity level, which minimize the average operating cost of the buyer. Interestingly, to solve the limite capacity inventory problem, we use well-known results from the queueing theory literature on the waiting time in a single-server queue. This allows us to formulate close-form first-orer conitions for the optimal capacity investment an the corresponing optimal inventory levels. We then use our moel to evaluate multi-sourcing strategies. Inee, we provie a framework where a iversification strategy can be optimize, i.e., eciing the optimal size of the supply base, the amount of capacity to be installe, an the inventory levels to be maintaine. Our moel thus contributes to the literature on supply 3

risk management too. The rest of the paper is organize as follows. In we review the relevant literature. In 3 we present our basic moel an fin the optimal inventory level for a given capacity. In 4 we solve the optimal investment ecision in capacity when a single supplier is present. In 5 we fin the optimal capacity investment ecision when sourcing from multiple suppliers is possible. We conclue in 6. All the proofs are containe in the appenix.. Literature Review The literature relevant to this paper can be ivie into three main groups, namely, capacitate inventory moels, capacity investment moels an multi-sourcing moels. Capacitate inventory moels eal with optimal inventory policies when supply is capacitate, an either fixe or ranom. Feergruen an Zipkin [0] show that moifie base-stock policies are optimal uner both average an iscounte cost criterion. In other wors, it is optimal to place orers up to a base-stock level when possible, an otherwise orer full capacity. Ciarallo et al [8] show that an orer-up-to policy is optimal in an infinite-horizon perioic-review setting with uncertain supply capacity an uncertain eman. Tayur [8] further computes the optimal inventory policy for an infinite horizon problem with limite supply capacity, base on the average cost criterion. He uses the concept of shortfall the amount by which the net inventory level falls short of the base-stock level to moel inventory an fins close-form base-stock levels for exponentially istribute eman. Glasserman [] evelops bouns an approximations for setting base-stock levels with fixe limite capacity, uner the average cost criterion. He uses tail probabilities associate with ranom walks to obtain asymptotic approximations as the service level approaches 00%. We closely follow the moeling approach of Tayur an Glasserman an exten it to ranom capacity. The papers above focus on optimizing inventory policies when capacity is given. Much research has been one in jointly consiering inventory an capacity ecisions. Van Mieghem [5] provies a comprehensive review of the topic. We iscuss the two main streams that analyze capacity investment problems. First, newsvenor-type capacity investment moels analyze the capacity-inventory traeoff through a capacitate version of the multi-perio iscounte newsvenor moel. However, since such moels eal with the transient evolution of inventory rather than focusing only on the steay state, it is not possible to fin an optimal capacity level in close-form, in a multi-perio setting. Van Mieghem an Rui [6] stuy newsvenor networks to jointly fin optimal capacity an inventory with multiple proucts uner the lost sales scenario. 4

Interestingly, they use three tools to optimize operating profits, namely capacity an inventory which are ecie before eman is realize an activities that are use to convert inventories into finishe goos by utilizing capacity which are ecie after the eman is realize. However, for the backlog case they only solve the inventory problem for the uncapacitate case. In contrast, we analyze a single-prouct capacity investment ecision, with backlogging an average cost criterion. Angelus an Porteus [] solve the capacity-inventory problem for a manufacturer in the case where both capacity an inventory ecisions can be mae in every perio. That is, capacity can be ae an remove at a certain cost in every perio. They show that it optimal to bring capacity to a certain target interval an characterize these intervals for the lost-sales case. For the backlog case, they fin that capacity an inventory act as economic substitutes, which we observe in our paper as well. However they o not characterize the optimal capacity levels target intervals for this case. Alternatively, queueing-type capacity investment moels stuy manufacturing systems where both eman arrivals an prouction are treate as continuous processes most often as Poisson processes. This results in continuous-review inventory policies where capacity, i.e., the rate of prouction, is a ecision variable. Although such moels can hanle general eman istributions, most times they are limite to exponentially istribute prouction systems, that is, where the time taken to prouce one unit in the prouction facility is exponentially istribute. In contrast, our moel focuses on a perioic-review system where the capacity per perio can be generally istribute. In this stream of literature, Braley an Glynn [4] consier a manufacturer that faces a fixe installation an linear utilization cost structure for capacity. They evelop heavy traffic an Brownian motion approximations for operational costs that yiel insight into the capacity-inventory trae-off. However, they o not evelop any close-form solution for the optimal capacity ecision, because the analysis becomes complex as they allow for correlate eman. Toktay an Wein [9] fin optimal release policies amount of goos release for prouction in a capacitate prouction environment with upate eman forecasts in each perio. De Kok [4] consiers a capacitate packaging facility an iscusses the optimal strategy of meeting service level through either outsourcing extra capacity nees or by postponing the extra capacity nees to the future. Calentey an Wein [6] come closest to our paper in jointly solving the optimal capacity an inventory ecision, with an average inventory cost objective. They consier that back-orer costs are share by both one supplier an one retailer, an solve the steay-state optimal ecision on capacity taken by the supplier an inventory taken by the retailer, for M/M/ queues. However, the focus of their paper is to fin equilibrium transfer payment contracts 5

between the supplier an the retailer to coorinate the supply chain. On a similar game theoretical approach, Cachon an Zhang [5] analyze a buyer-supplier equilibrium in which strategic suppliers select their capacity to minimize operational costs, an the buyer selects its allocation policy to them, in orer to minimize the average elivery lea-time over an infinite horizon. They fin that state-epenent allocation policies, i.e., policies that take into account the present state of workloa on servers, eliver faster lea time compare to state-inepenent allocation policies. Finally, the literature on supply risk management through multi-sourcing is also relate to our work. Tomlin [0] iscusses mitigating supply risk through either sourcing from more than one supplier, compare to carrying safety inventory buffers. Inee, iversification of suppliers is a possible solution to reuce supply isruptions. Feergruen an Yang [9] stuy the planning moels use to etermine procurement quantities, in the presence of stochastic eman an supply efault risk. Babich et al. [3] present a moel for supplier pricing, where the buyer may buy from a single or from many suppliers. Chaturvei an Martíneze-Albéniz [7] consier the esign of optimal capacity auctions when suppliers are unreliable. In this paper, we jointly consier inventory, capacity an multi-sourcing ecisions. 3. The Moel We moel a buyer that faces the eman for a single prouct in every perio. This eman, enote by D n in perio n, is stochastic an exogenously given. It follows a c..f. F D x with mean an is i.i. across perios. The eman is serve from inventory, which is replenishe at the en of every perio at a cost of c per unit, with zero lea-time. If the buyer s inventory falls short of the eman in any perio, then the unmet eman is backlogge to the next perio at a penalty cost of p per unit. On the other han, if all the eman of a particular perio is met, then any leftover units in the inventory are passe on to the next perio at a holing cost of h per unit. The orers place are shippe from one supplier or more than one in 5. This supplier has a limite capacity K n that is available to the buyer in perio n, which may be insufficient to fill the buyer s orer. We moel K n as a ranom variable an follows a c..f. F K x which is i.i.. across perios, with mean k. However, the buyer can control the capacity in place by making an initial one-time investment. As a result, the capacity istribution F K x epens on the buyer s investment, e.g., its mean k is enogenize an can be varie. Since eman is backlogge, the assumption of zero lea-time coul be relaxe easily with no impact on the structure of our results. 6

The buyer thus has two ecisions to take that affect its operating cost: i it must make the necessary capacity investments, an ii it must ecie on an appropriate inventory replenishment policy, that shoul consier the fact that supply is limite by capacity. The objective of the buyer is to minimize its average operating cost over an infinite horizon. This inclues the capacity cost per perio, the inventory carrying cost which inclues holing an back-orering costs, an the variable purchasing cost. Since the average purchasing cost is fixe, equal to c per perio, the problem of the buyer is to fin the right balance between capacity an inventory costs. First, given a capacity level, consier the inventory ecision. At the en of every perio, the buyer observes the capacity of the supplier that is available. The buyer then places the orer which arrives before the eman for the next perio appears. It is well-known that a moifie base-stock policy is optimal for the buyer facing such conitions with infinite horizon. In other wors, if I n is the inventory level at the beginning of perio n after receiving the orer place at the en of perio n, an y is the base-stock level, then it is optimal to orer min{k n, y I n D n }. Thus, the inventory level at the beginning of perio n + is which can also be written as I n+ = { y if y In D n K n I n D n + K n otherwise I n+ = mini n D n + K n, y. Similar to Tayur [8] an Glasserman [] we introuce the shortfall W n = y I n in perio n as the amount by which the inventory falls short of the base-stock level. We can then write Equation as W n+ = maxw n K n + D n, 0 One can observe that the ynamics of the shortfall in Equation are ientical to the ynamics of waiting time in a queueing system with one server. In this queueing system, the shortfall W n represents the time that the n th customer arriving into the system has to wait in the queue. The eman D n represents the time taken by the server to serve the n th customer once it arrives at the server. Finally, the capacity K n represents the interarrival time between the n th an the n + th customer arriving to the queue. Note that the interpretation of the shortfall as a waiting time is not very intuitive, since the shortfall in perio n is interprete as the waiting time of customer n in the queuing system. This is Since capacity ecisions are involve, using an infinite horizon average cost criterion is reasonable, as transient effects shoul be seconary an thus nee not be consiere. 7

very ifferent from manufacturing queues, where time is continuous an queues iscrete. In our case, time is iscrete an queues continuous. As a result, from stanar queuing theory we obtain that when ρ = ED n EK n = k < the istribution of W n converges to W as n such that W = maxw K + D, 0 3 where = inicates equivalence in istribution. For ρ the shortfall increases over time since on average eman is higher than capacity, an hence the istribution of shortfall oes not converge to a steay state. The notation above allows us to formulate the average carrying cost as E D,W pd + W y + + hy W D + Since the expression above is convex in y, we can therefore fin the optimal base-stock level y through a first-orer conition, which yiels PD y W = E W F D y W = p p + h The above result is similar to Tayur [8]. Thus, given a certain capacity istribution, we can optimize the base-stock level accoringly. Hence, we can write the average carrying cost of the buyer as Ca = min y {E D,W pd + W y + + hy W D + } 5 where a represents the vector of investments in capacity, which in turn affects the capacity istribution F K. For example, if the supplier is fully reliable, a woul be the capacity level in place; if the supplier s capacity follows a Gamma istribution of shape ν an scale ξ, then a = ν, ξ. Furthermore, in orer to fin the best balance between capacity investment Ha an inventory cost Ca, the buyer s problem can be written as 4 min a {Ca + Ha} 6 We are now in a position to fin optimal capacity investment that woul solve Equation 6. For this purpose, we follow three steps:. We first characterize the istribution of waiting time, with p..f., F W, in Equation 3 for ifferent istributions of eman an capacity.. We evaluate the average carrying cost in Equation 5 for this particular istribution of waiting time. 8

3. We finally establish optimality conitions for the optimal capacity investment by characterizing buyer s problem in Equation 6. 4. Single Supplier In this section, we stuy the optimal balance between capacity an inventory when there is a single supplier. For this purpose, we analyze the waiting time istribution of a G/G/ queue. Due to the complexity of this task, we first focus on a G/M/ system, i.e., when eman is exponentially istribute, an erive close-form solutions. We then consier the general case where we iscuss analytical an numerical solutions. 4. Exponential Deman Consier first that the eman is exponentially istribute with mean. For a general istribution of supplier capacity, the istribution of shortfall in Equation is given by the istribution of the waiting time in a G/M/ queue. Using results from queueing theory, see Kleinrock [3], we can characterize the istribution of shortfall through its probability ensity function f W w. This p..f. is characterize by a parameter σ, 3 an is given by [ ] σ f W w = σδw + σ e σ w 7 where δw is the elta-irac function. In other wors, the istribution of the shortfall is σ 0 with probability σ an an exponential of rate with probability σ. σ can be foun from the functional equation σ = x=0 e σ x F K x 8 σ or equivalently as σ = L K, where L K represents the Laplace transform of the supply capacity istribution F K. In stanar queueing theory, it has been shown that there exists a unique real solution for σ in the range 0 < σ <, when ρ = /k <. The probability istribution function for the shortfall enables us to characterize the basestock level in Equation 4 as yσ = h ln 0. 9 σ p + h 3 In a G/M/ queue, the steay-state istribution of customers in the system, at arrival instants, is geometric. σ is the parameter that etermines this geometric istribution, i.e., it is the ratio of steay-state probability of fining n + customers in the system, to the probability of fining n customers in the system. However, there is no irect interpretation of σ as a probability ratio in our moel. 9

To characterize the carrying cost in Equation 5 we introuce the ranom variable Z which is the sum of shortfall an eman. Then the istribution of Z can be foun from the convolution of shortfall an eman. Specifically, for exponential eman, the istribution of Z is i.e., Z is exponentially istribute with rate as PZ x = e σ x, 0 σ. We can thus write the carrying cost Ca = E Z pz y + + hy Z + Both the optimal inventory y an the istribution of Z can be uniquely ientifie by σ. Furthermore, σ is etermine by the mean of eman an the parameters of the capacity istribution, i.e., the vector of investments in capacity a. The following lemma provies some structural properties of the carrying cost. Lemma For an exponentially istribute eman, the cost of carrying inventory Ca is convex in a if σ is convex in a. Furthermore, if the capacity investment cost H is also convex in the vector of investments a, then the buyer s problem in Equation 6 is convex an hence the optimal capacity investment is characterize by first-orer conitions. Theorem For an exponentially istribute eman, if Ha an σa are convex in a, then the optimal vector of investment in capacity is given by jointly solving the system of equations H = C = a i a i h h σ σ ln p + h a i The above theorem states that, at optimality, the marginal benefit from buying capacity, achieve as a reuction of carrying cost, shoul be equal to the marginal cost of buying that capacity. From Equation 9, we fin that optimal inventory levels are increasing in σ. Also, if σ is ecreasing in a i then from Equation, we fin that a higher investment cost for a i woul result in a lower a i at optimality. This implies that the optimal inventory level y an the capacity investment a i act as economic substitutes, as observe by Angelus an Porteus []. In the remainer of this section we characterize Ha an σa an show their convexity in a, for ifferent istributions of supply capacity. We first focus on an exponential istribution of supply capacity, an then on general istributions of supply capacity that are commonly foun in the literature: i fixe capacity, ii Bernoulli istribute capacity an iii Gamma istribute capacity. 0

4.. Exponential Capacity For exponentially istribute capacity with mean k, the istribution of shortfall can be characterize by the istribution of the waiting time of an M/M/ queue. In that case, F K x = e x k, an hence the Laplace transform is LK s =. As a result, Equation + ks 8 becomes σ =, which can be written as σ kσ = 0. Since we know + k σ that 0 < σ <, hence the only feasible solution of σ can be σ = k = ρ, the utilization. Proposition For exponentially istribute capacity with mean k, σk = k an convex in k. is ecreasing Since exponential istribution is completely characterize by its mean k, we can efine the capacity investment vector as a = k. Furthermore, consier that Hk = C cap k is linear, with a capacity cost equal to C cap per unit. Since σ is ecreasing an convex in k, with Theorem we can fully characterize the optimality conitions for the capacity investment: k = + h h ln. C cap p + h In other wors, at optimality, the utilization of capacity ρ = k is increasing with the ratio of capacity to inventory cost C cap, an ecreasing in the ratio of back-orering to inventory h cost p h. 4.. Non-Exponential Capacity In general one cannot fin an explicit expression for σ from the functional equation 8, when supply capacity is not exponentially istribute. Hence, to show convexity of σ in a one has to implicitly ifferentiate the functional equation. For a capacity that is fully reliable an fixe at q, we can show σ to be convex in q. Proposition For a fixe capacity q, σ = e σ q, an σq is ecreasing an convex in q. Here we efine capacity investment vector a = q. Letting, Hq = C cap q, we can characterize the optimal capacity ecision from Equation. Note however that σ is implicitly efine, an therefore one nees to numerically fin the optimal capacity investment. Next we consier the case where supply capacity is Bernoulli istribute, i.e., we consier supply capacity to be m with a probability α an 0 with a probability α. Note that when α =, we go back to the previous case, where capacity is fixe. When we efine the

3.5 0 Optimal Mean Capacity 3.5 9 8 7 α=0.5 α=0.9 α= Optimal Inventory 0. 0.4 0.6 0.8 C cap 6 5 4 3 Average Cost 6 5 4 3 = h= p=0 Hm,α=5mα 0. 0.4 0.6 0.8 C cap 0..4.6.8 Mean Capacity Figure : The supply capacity is Bernoulli istribute with parameters m, α an Hm, α = C cap mα. We consier the reliability α to be fixe, an consier the impact of varying m on the right-han sie figure. On the left-han sie figure, we illustrate the sensitivity to the cost of capacity C cap. capacity investment vector as a = m, α, the buyer has two levers to invest in capacity: it can increase the reliability α, or increase the available capacity m when the supplier is reliable. Proposition 3 For a Bernoulli istribute capacity with parameters m an α, σ = αe σm + α an σm, α is ecreasing in m an α, an jointly convex in m, α. Assuming that Hm, α is increasing an convex in m, α, Proposition 3 allows us to fin the optimal capacity ecision from Equation. When we efine the cost of capacity as Hm, α = C cap mα, although the capacity investment is not jointly convex in m, α, we fin numerically that the optimal capacity ecision is still uniquely efine by the first-orer conitions. To illustrate the proposition, Figure presents, for α fixe, the optimal capacity investment m. We can observe in the figure that optimal inventory an mean capacity are economic substitutes. Finally, we consier the case where supply capacity is Gamma istribute with mean φ = νξ, where ν is the shape parameter an ξ is the scale parameter. This is a useful istribution that is unimoal, an usually occurs, for example, when the effective supply capacity is the sum of exponential variables. We use this property to moel the effect of

multi-sourcing on capacity investment in 5. If a = ν, ξ, the buyer has again two levers to invest in capacity, that is, it can either scale up the capacity by investing in ξ or it can invest in making the supply capacity more reliable by investing in ν. 4 We fin that σ is generally convex in ν, ξ, however it is ifficult to prove it. Instea we show in the following proposition the convexity of average carrying cost C in ν, ξ. Proposition 4 For a Gamma istribute capacity with shape parameter ν an scale parameter ξ, the average carrying cost C is ecreasing in ν an ξ, an jointly convex in ν, ξ. When Hν, ξ is increasing an jointly convex in ν, ξ Proposition 4 allows us to fin the optimal capacity ecision from Equation. Usually, the buyer in fact controls the average supplier capacity φ = νξ, as the cost is often expresse as C cap φ. If a = φ, i.e., the buyer can only invest in the mean of supplier capacity, or if a = φ, ν, i.e., the buyer can invest in both mean an the shape parameter, then σ is generally not convex in φ, ν. However, we show in the following proposition convexity of average carrying cost with respect to the mean an the shape parameter of supply capacity. Proposition 5 For a Gamma istribute capacity with mean φ an shape parameter ν, the average carrying cost Cφ, ν is ecreasing in φ an ν, an jointly convex in φ, ν. This proposition is quite relevant since it provies a well-behave strucutre for our moel with multiple sourcing in 5. Similarly as before, when the capacity cost Hφ is increasing an convex in the mean of capacity, the optimal capacity investment is unique an can be foun from Equation. Finally, taking ν given, an varying φ, Figure 3 illustrates the trae-off between capacity an inventory. 4. Non-Exponential Deman When eman is non-exponential then the shortfall is istribute as the waiting time in a G/G/ queue. No close-form formulas exist for this case. In certain cases some results can be obtaine, in particular when the supply capacity is exponential. 4 Note that for k = νξ fixe, when ν we obtain σ e σ k, which is equivalent to having a fixe supply capacity of k. 3

{Optimal Mean Capacity 5 4 3 0 0. 0.4 0.6 0.8 C cap 5 Average Cost 0 9 8 7 6 5 4 ν=0.5 ν= ν=5 ν=00 {Optimal Inventory 4 3 0 0. 0.4 0.6 0.8 C cap 3 = h= p=0 Hν,ξ=5νξ 0..4.6.8 Mean Capacity Figure 3: The supply capacity is Gamma istribute with mean φ an shape parameter ν an Hφ, ν = C cap φ. We consier the shape ν to be fixe, an consier the impact of varying the mean capacity φ on the right-han sie figure. On the left-han sie figure, we illustrate the sensitivity to the cost of capacity C cap. 4.. Exponential Capacity When supply capacity is exponential with mean k, the Pollaczek-Khinchine transform formulas characterize the Laplace transform of the waiting time in the steay-state, L W s = k s L D s + ks 3 where L W s is the Laplace transform of the waiting time istribution, L D s is the Laplace transform of the eman istribution, an the average eman. Inverting the Laplace transform in Equation 3 can provie the istribution of the shortfall, an provies a numerical approach for optimizing the value of the capacity a = k to be installe. Recalling that Z = W + D, Equation 3 yiels that L Z s = L D sl W s = L Dsk ρs L D s + ks 4 an hence the inventory carrying cost can be expresse as Ck = pez py+p+hey Z +. The Pollaczec-Khinchine formula provies the expecte waiting time as ρ + CV EW = D ρ 4

Optimal Mean of Capacity.5.5 3 4 5 C cap 3.5 Average Cost 40 35 30 5 GammaCV D = ExpoentialCV D = GammaCV D =/ UniformCV D =/3 FixeCV D =0 = h= p=0 C cap =4 Optimal Inventory 3.5 0 5.5 3 4 5 C cap 0..4.6.8..4 Mean of Capacity Figure 4: On the left-han sie, optimal mean capacity an inventory level, as a function of the cost of capacity C cap. On the right-han sie, total average cost as a function of mean capacity. where CVD = V ard. Hence, ED [ ] ρ + CV EZ = EW + = + D ρ Unfortunately, Ey Z + cannot be foun in close-form. We use a Fourier-series metho to numerically invert the Laplace transform of Z, as escribe by Abate an Witt []. We procee to show numerical results for the optimal capacity investment for ifferent eman istributions, all with average : a fixe eman of, b uniform eman in [0,], c Gamma istribute eman with mean an ifferent coefficients of variation, of, i.e., an exponential of rate an. We compare the expecte capacity at optimality, together with the inventory level. The right-han sie of Figure 4 shows that, given a certain level of mean capacity k, the higher the variability of the eman, the higher the inventory operating cost. Similarly the left-han sie of the figure shows that, given a certain cost of capacity, the higher the eman variability, the higher the mean capacity that the buyer installs an the higher is the corresponing safety stock y that it keeps. As a result, if Hk = C cap k, the optimal investment in capacity is increasing as eman becomes more variable. This results in a 5

lower utilization, as one woul expect. This implies that, when eman variability increases, the buyer reacts by both lowering the capacity utilization an increasing the safety stock. 4.. Non-Exponential Capacity When the capacity is not exponentially istribute, then we must resort to numerical simulation to fin the istribution of the shortfall. We present below our numerical results for ifferent istributions of supply capacity, an ifferent istributions of eman. We consier i fixe capacity, ii Bernoulli istribute capacity, with α = 0.8, an iii Gamma istribute capacity with ifferent coefficients of variation, of, i.e., an exponential of rate an ; an a fixe eman of, b uniform eman in [0,], c Gamma istribute eman with mean an ifferent coefficients of variation,, i.e., an ex- ponential of rate an. Interestingly, the carrying cost is foun to be convex in the average capacity, which guarantees that the optimization over capacity is well-behave. We show the results in Table after the appenix. As in 4.., we fin that operating cost, inventory levels an capacity investment typically increase as eman or supply becomes more variable. We fin that this tren to be consistent within a istribution, that is, for a given istribution, capacity an inventory levels increase with the coefficient of variation. However, this tren might not hol across istributions since the tail effects of ifferent istributions coul consierably affect the shortfall istribution an consequently the inventory carrying cost. For example, cost an inventory are higher for a Bernoulli istribute capacity coefficient of variation compare to Gamma istribute capacity coefficient of variation, higher. 5. Multiple Suppliers So far, we have consiere the trae-off between capacity an inventory for a buyer that is serve by a single supplier. In this section, we analyze the impact of multi-sourcing on the capacity-inventory relationship. Of course, when the capacity at a supplier is fully reliable, iversification will not help the buyer. In contrast, when capacity is unreliable, there exists an opportunity to iversify the supply base, which reuces the uncertainty of supply its coefficient of variation. This allows the buyer to utilize capacity better, an reuce operating cost. At the same time, the supply chain will become more complex, as more suppliers are present, an this may create aitional costs. We moel this as a cost of iversification C iv to be pai for each supplier inclue in the supply base. When C iv = 0, it is clear that 6

the buyer woul completely iversify its supply base an work with as many suppliers as possible, for a given total capacity. However, when C iv > 0, there is an optimal number of suppliers in the supply base, that shoul not be too large because then the sourcing cost woul be too high nor too low because then the variability of capacity may force the buyer to install a higher level of capacity. We investigate this trae-off an look at the optimal iversification strategy, that is, how many suppliers to be inclue in the supply base, how much capacity to be installe, an what inventory level to keep. For simplicity, we focus our analysis on the case where all suppliers are ientical, i.e., their capacity investment cost, their purchasing cost an their iniviual capacity istribution are the same. In aition, in orer to use the results of 4, we assume that all the suppliers capacity is Gamma istribute with the same shape parameter, e.g., when it is exponential. We also assume, as in section 4., that eman is exponentially istribute, with mean. Essentially the buyer s problem remains the same as in Equation 6, with the exception of the aition of an extra parameter, namely the number of suppliers s. We consier that each supplier has a Gamma istribute i.i.. capacity with shape ν i = ν an scale ξ i = ξ. 5 As a result, the total available capacity to the buyer is Gamma istribute with shape sν an scale ξ. Thus, the average total capacity is φ = sνξ. From Proposition 5, we know that the inventory carrying cost is ecreasing an jointly convex in sν, φ an also in s, φ. Hence, we can etermine the optimal inventory level an the optimal iversification strategy from Equation 9 an Equation respectively. As before, we can use the capacity investment vector to efine the cost structure for iversification. We take a = s, φ. Assume first that Hs, φ is increasing an convex. That is, we assume that marginal cost of contracting with each aitional supplier cost of iversification is increasing an the marginal cost of buying higher mean capacity is increasing too. We can therefore characterize the optimal iversification strategy from Equation. 6 From that equation, we see that the buyer woul iversify less when the cost of iversification H is higher. Similarly, the buyer woul buy higher mean capacity from suppliers s when the cost of capacity H is lower. φ It is worth analyzing the case where H is linear in s, φ: we can efine Hs, φ = C iv s + C cap φ, where C iv is the cost of iversification an C cap is the cost of capacity. Using the optimality equations of Theorem, the optimal supply base an the optimal mean 5 Clearly, when choosing s ientical suppliers, at optimality the buyer woul install the same capacity in all of them. 6 Although s takes integer values, min φ {Hs, φ + Cs, φ} is convex in s, an hence can be optimize by simple search. 7

capacity are foun by solving the equations σ = σφ + νs C iv = h νs h σφ, s σ ln p + h s h σφ, s p + h φ C cap = h σ ln σφ, s σφ, s where an can be foun by ifferentiating implicitly the first equation. s φ These equations provie a value of s that might not be integer. Since the cost function is jointly convex in s, φ, the optimal size of the supply base s is s or s. Finally, φ is given by solving σ = σφ νs + C cap = h σ ln νs h σφ, s p + h φ Once again, from Equation we see that the optimal number of suppliers s is ecreasing in C iv, that is, the buyer woul iversify less when the cost of iversification is high. Similarly the total capacity is ecreasing an the inventory is increasing in C cap. We show this optimal strategy in Figures 5 an 6. In the first figure, we see how the iversification strategy shoul vary as a function of the iversification cost. Interestingly, when the supply base is reuce, the average total capacity to be installe increases, as well as inventory. In the secon figure, we observe the impact of the capacity cost. Here, we see that the number of suppliers tens to increase with C cap. This is intuitive: when the capacity becomes more expensive, it becomes more cost-efficient to reuce its uncertainty, which can be achieve by increasing the number of suppliers. In summary, in a context of supply risk, our moel can be use to jointly evaluate how many suppliers to use, an how much capacity an inventory are necessary. 6. Conclusion The moel evelope in this paper solves the capacity-inventory trae-off of a buyer, in an infinite-horizon perioic-review setting, for the average cost criterion. We consier the general case where both eman an available capacity are stochastic. Using that a moifie base-stock policy is optimal, we show that the shortfall, efine as the ifference between base-stock level an current inventory, is istribute as the waiting time in a queuing system with one server. 5 6 Thus, when eman is exponentially istribute we are able to obtain 8

No. Of Suppliers Total Mean Capacity Optimal Inventory 0 5 0.0 0.04 0.06 0.08 0. 0. 0.4 C iv.3.. 0.0 0.04 0.06 0.08 0. 0. 0.4 C iv 3.4 3. = h= p=0 C cap = 3 0.0 0.04 0.06 0.08 0. 0. 0.4 C iv Figure 5: Optimal iversification strategy as a function of C iv, the cost to be pai per supplier. No. of Suppliers Total Mean Capacity Optimal Inventory 6 4 = h= p=0 C iv =0. 0 0 3 4 5 6 7 8 9 0 C cap 4 3 0 3 4 5 6 7 8 9 0 C cap 6 4 0 3 4 5 6 7 8 9 0 C cap Figure 6: Optimal iversification strategy as a function of C cap, the capacity cost. 9

the first-orer conition for the capacity ecision. We provie close-form solutions for several cases commonly use in the literature to moel supply risk: when capacity is fixe, when it is Bernoulli istribute, an when it is Gamma istribute. When eman is not exponentially istribute, some analytical results can be obtaine when capacity is exponential, an otherwise we iscuss the optimal capacity ecisions base on numerical experiments. We then apply the moel to a situation where the buyer can choose to install capacity at multiple suppliers simultaneously, in orer to iversify supply risk. We fin that, as the cost of iversification increases, the supply base has fewer suppliers, an both the total installe capacity an the safety stock increase. As the cost of capacity increases, the supply base has more suppliers, an the total installe capacity ecreases an the safety stock increases. Compare to the previous literature, the moel provies a useful framework to evaluate supply risk management strategies. In particular, the application with multiple suppliers can be use to analyze more general supply base esign problems, which are receiving increasing attention. This opens irections for future research. For example, while our moel consiers multiple suppliers, they are assume to be ientical. When they are asymmetric, the capacity ecision must take into account that they might offer ifferent supply reliability an ifferent cost. As a result, suppliers must be prioritize, base on cost an reliability. References [] Abate J. an W. Witt 995. Numerical Inversion of Laplace Transforms of Probability Distributions. ORSA Journal on Computing, 7. [] Angelus A. an E. L. Porteus 00. Simultaneous Capacity an Prouction Management of Short-Life-Cycle, Prouce-to-Stock Goos Uner Stochastic Deman. Management Science, 483, pp. 399-43. [3] Babich V., A. N. Burnetas an P. H. Ritchken 007. Competition an Diversification Effects in Supply Chains with Supplier Default Risk. Manufacturing & Service Operations Management, 9, pp. 3-46. [4] Braley J. R. an P. W. Glynn 00. Managing Capacity an Inventory Jointly in Manufacturing Systems. Management Science, 48, pp. 73-98. [5] Cachon G. P. an F. Zhang 007. Obtaining Fast Service in a Queueing System via Performance-Base Allocation of Deman. Management Science, 533, pp. 408-40. 0

[6] Calentey R. an L. M. Wein 003. Analysis of a Decentralize Prouction-Inventory System. Manufacturing & Service Operations Management, 5, pp. -7. [7] Chaturvei A. an V. Martínez-e-Albéniz 008. Optimal Procurement Design in the Presence of Supply Risk. Working paper, IESE Business School. [8] Ciarallo F. W., R. Akella an T. E. Morton 994. A Perioic Review, Prouction Planning Moel with Uncertain Capacity an Uncertain Deman-Optimality of Extene Myopic Policies. Management Science 403, pp. 30-33. [9] Feergruen A. an N. Yang 008. Selecting an Portfolio of Suppliers Uner Deman an Supply Risks. Operations Research, 564, pp. 96-936. [0] Feergruen A. an P. Zipkin 986a. An Inventory Moel with Limite Prouction Capacity an Uncertain Demans, I: The Average Cost Criterion. Mathematics of Operations Research,, pp. 93-07. [] Feergruen A. an P. Zipkin 986b. An Inventory Moel with Limite Prouction Capacity an Uncertain Demans, II: The Discounte Cost Criterion. Mathematics of Operations Research,, pp. 08-5. [] Glasserman P. 997. Bouns an Asymptotics For Planning Critical Safety Stocks. Operations Research, 45, 44-57. [3] Kleinrock L. 975. Queueing Systems. Wiley, 975. [4] e Kok T. G. 000. Capacity Allocation an Outsourcing in a Process Inustry. International Journal of Prouction Economics, 68, pp. 9-39. [5] van Mieghem J. A. 003. Capacity Management, Investment an Heging: Review an Recent Developments. Manufacturing & Service Operations Management, 54, pp. 69-30. [6] van Mieghem J. A. an N. Rui 00. Newsvenor Networks: Inventory Management an Capacity Investment with Discretionary Activities. Manufacturing & Service Operations Management, 44, pp. 33-335. [7] Raman A. an B. Kim 00. Quantifying the Impact of Inventory Holing Cost an Reactive Capacity on an Apparel Manufacturer s Profitability. Prouction an Operations Management, 3, pp. 358-373.

[8] Tayur S. R. 993. Computing The Optimal Policy For Capacitate Inventory Moels. Comun. Statist., 94, pp. 585-598. [9] Toktay L. B. an L. M. Wein 00. Analysis of a Forecasting-Prouction-Inventory System with Stationary Deman. Management Science, 479, pp. 68-8. [0] Tomlin B. 006. On the Value of Mitigation an Contingency Strategies for Managing Supply Chain Disruption Risks. Management Science, 55, pp. 639-657.

Appenix Proof of Lemma Proof. Equation can be re-written as Ca = h y z=0 y zgzz + p z=y z ygzz where gz is the istribution function of Z = W + D. We then obtain h Ca = hy σa ln Substituting the value of y = h p+h p + h σay + σa e in the above equation, we get Ca = hyσa. σa Hence the graient of Ca can be written as a Ca = C σ aσa where a X enotes the vector of partial erivatives of X with respect to each element of the vector a an C h h ln σ = p+h 0. The Hessian for Cya, a can thus be written σ as C σ M + C σ H σ 7 where H σ is the Hessian matrix of σ an M is a symmetric matrix such that M ij = σ a i σ a j. By construction M is positive semi-efinite. Also C σ 0 an C 0. Hence if σ is jointly σ convex in a then the Hessian of Cya, a is the sum of two positive semi-efinite matrices an therefore C jointly convex in a. Proof of Theorem Proof. From Lemma we can write the first-orer conitions for Equation 6 using h C h ln σ = p+h. σ Proof of Proposition Proof. For fixe capacity, Equation 8 can be written as σ = e σq or in other wors q = lnσ σ. 3

q is monotonically ecreasing in σ in the interval 0,. Inee, taking the first erivative with respect to σ we obtain q σ = σ σ lnσ σ < 0 because for all σ 0,, lnσ >. Taking the the secon erivative yiels σ q σ = 4σ + 3σ σ lnσ σ σ 3 Convexity of q in σ implies that σ is convex in q. Proof of Proposition 3 Proof. From Equation 8 we get σ σ σ 3 0 σ = αe σm + α. 8 We first show that σ is ecreasing in m an α, then we show that σ 0 an finally we m show that the eterminant of Hessian of σ is non-negative. From Equation 8 we can write m explicitly as a function of σ: m = σ + α σ ln. α m < 0, as in the previous proof. Similarly, from Equation 8 we can write α as an explicit σ function of σ, σ α = 9 e σm Again α < 0 in the unit interval. σ Differentiating α with respect to σ in Equation 9 yiels α e σm σ = e σm m σ e σm Differentiating again, = α σ m σ e σm e σm e σm 3 e σm m σ e σm σm me 4

m σ Letting x =, the numerator of α σ is m [ e x x e x e x + xe x] = m [ e x x + x + e x] 0 because x + x + e x 0 for all x 0 it is convex in x, takes value zero at x = 0 an has slope zero at x = 0. Therefore α 0. This implies that σ is convex in α. σ To show joint convexity of σ in α an m, we nee to show that the Hessian matrix of σ is positive semi-efinite. For this purpose, we only nee to show that the eterminant of the matrix is non-negative since we have alreay shown that σ is convex in α. We fin the first erivatives as which implies that mσ + α σ α = σ 0 α mσ+α. Also, σ σ m = σ + α 0 mσ+α which implies that σ + α. We now evaluate the Hessian of σ: σ m σ α m σ m α σ α = mσ + α σ+α σ σ+α σ mσ+α σσ+α + σ+α σ α σ σσ+α + σ+α α m σα+σ + m σ mσ+α α We nee to show that the eterminant of the above matrix is positive. In oing so we ignore the common term outsie the matrix, after which we nee to show that mα + σ σ + α 0. mα + σ This is true if σ α =, we nee to show e σm 0 since σ + α. m σ + e σm e σm 5. 0 Substituting for the value of

m σ Taking x =, the above inequality is true if x + x + e x 0 which, as seen above, is true for x 0. Thus the Hessian of σ is positive an σ is jointly convex in α an m. Proof of Proposition 4 Proof. As in the previous proof, we first show that σ is ecreasing in ν an ξ which woul imply that average carrying cost too is ecreasing in ν an ξ. We then show that σ ν 0 an finally we show that the eterminant of the Hessian of average carrying cost is non-negative. For Gamma istribute capacity, σ can be foun from Equation 8 as ν σ = σξ + We can write the above equation as an explicit equation of ν, that is lnσ ν =. ln σξ + Taking the erivative of the Taylor expansion of ν with respect to σ as σ yiels ν lim σ σ = /ξ < 0. Since there is a unique solution to Equation with 0 < σ <, an ν ν σ < 0 for all σ 0,. Taking the first erivative of Equation with respect to σ ν ν σ = lnσ σ ξν σξ + Taking the erivative of Equation 3 with respect to σ we get Letting i = ν ν + σ ν σξ +, ν + lnσ = + σ [σ lnσ] ξ/ + ln [ σξ + ln σ σ < 0, we get. 3 σξ + ]. σξ + ν ν σ = + lnσi lni + ξ/ + lniσ lnσ + lni i + ξσν/ ξσνi/ iσ lnσ lni an substituting the value of ν from Equation gives ν ν σ = lnσi lni + ξσ lnσ + lni/ + ξσi lni lnσ/ iσ lnσ lni 6

The numerator of the above expression can be written as ξσ lnσ [ i lni ξσ lnσ/] / lnσ [ i lni ξσ lni lnσ/ ] lnσ [ iξσ lni/ ξ σ lnσ/ ] > 0 because each term within the square brackets is non-negative since ν = lnσ lni > ξ in steay-state the utilization ρ = ξk <. Which implies that ν 0 an hence σ is convex σ in ν. To show that σ is ecreasing in ξ, we ifferentiate Equation with respect to ξ: σ ξ = ν σσ σξ + νξσ The fact that σ is ecreasing in ν implies in Equation 3 that σξ + νξσ 0. Therefore σ is also ecreasing in ξ. To show that the average carrying cost C is jointly convex in ν an ξ we use the Hessian matrix, see Equation 7. Without loss of generality we assume = showing convexity in ξ or ξ is equivalent. The Hessian can be written as h ln = h p+h σ C ν C ξ ν σ + ν σ C ν ξ C ξ σ + σ σ ξ ν σ ν ξ σ σ + σ σ ν ν ξ σ ν ξ σ + ξ σ For ease of notation we enote partial erivatives with subscripts, e.g., σ ν = σ ν an σ ν = σ νν. We have alreay shown that σ is convex in ν, therefore to show that the average cost is jointly convex in ν an ξ, we nee to show that the eterminant of the Hessian matrix efine above is positive. In other wors we show below that σ νν σ ξξ σ νξ + σ σ νσ ξ σ ξ σ νν σ ξ σ ν + σ ξξ σ ν σ ξ σ νξ 0 4 We evaluate the expression in Equation 4 term by term. The first term on the left-han sie of the equation is the eterminant of the Hessian matrix of σ, that is et σ ν σ ξ ν 7 σ ν ξ σ ξ

Taking i = σξ + the Hessian of σ can be written as i νξσ ξiσ lni + σilni i + νσξ i νσξ σi νσ lni + σ + σi νσ lni + σ + σν lnii+σξ i νσξ σν lnii+σξ σνiν+ σ νσ σ i νσξ i νσµξ The expression for the eterminant of the above matrix is groupe into terms having common enominators, that is, terms having enominators i νσξ, i νσξ an. The eterminant can then be written as σ i i νξσ 4 which is equivalent to ν σ lniνσ ξ lni + σi + 4νσ σξ lni +νi σ lni iνσ lni + σ σ i i νξσ 4 ν σ lni + lniνσ ξ + i σ iνσ lni + σ Now we evaluate the secon term of Equation 4. σ ξ σ ν σ ξν σ + σν σ i νξσ σ νν + σ ξξ = i + νσξ lni σ ν σ ξ i νξσ Also σ νξ = = Aing the two up, we get σ νν σ ξ σ ν + σ ξξ σ ν σ ξ σ νξ = σi ν+ σ lni i νξσ σ σ lni νi + νσξ + i i νξσ i νξσ σ νi lni + σξ σ νı lni + σ σi i νξσ Multiplying the above expression with σ σ σ ξ σ ν νσ ξ σ νν + σ ξξ σ νξ σ ν σ ξ σ σi νξσ + i νξσ σ σ νσ ξ = σ νi lni σ σ lni νi + νσξi i νξσ σ σ lni i νσξ i νξσ i νξσ νσ i lni yiels = 4 σνσ3 i lni i νξσ 4 i νξσ + lni We can now write Equation 4 as σ νν σ ξξ σνξ + σ σ σ ξ σ ν νσ ξ σ νν + σ ξξ σ νξ σ ν σ ξ σ i σνσi νξσ lni + lni = i νξσ 4 +ν σ lni + lniνσ ξ + i σ iνσ lni + σ 8 5

Removing the common term, we nee to show that σνσi νξσ lni+lni +ν σ lni+lniνσ ξ+i σ iνσ lni+ σ 0, that is, ν σ lni + lni + σ νσ lni + σ or in other wors ν σ lni + ν ν + σ lni σ Since ν lni = lnσ an lnσ σ for σ, we can boun lni with σ. It is thus sufficient to show that σ + ν ν + σ lni 0 The above inequality is true for σ ν +. For σ, we nee to show that ν + σ ν ν + σ lni Using the logarithmic inequality lnx x x that lni σ. We then nee to show that νσ ν + σ νσ an applying it to lnσ σ, we get σ which is inee true since σ. Hence we have shown that the average carrying cost C is jointly convex in ν, ξ. Proof of Proposition 5 Proof. For Gamma istribute capacity we can write σ φ, ν = ν 6 σ φ,νφ + ν Where φ is the mean an ν the shape parameter of the capacity. Note that σ φ, ν has been characterize similar to σξ, ν in Equation. Inee, for a Gamma istribution, its mean is φ = νξ where ξ is the scale parameter. Without loss of generality we assume that = since the analysis will remain same with φ as with φ. This implies that in orer to have a steay state istribution of the shortfall we must have φ > =. We fin that σ is not always convex in φ, ν, therefore 9

to show convexity of the average carrying cost, we irectly use Equation 7. The Hessian matrix for the average cost can be written as C φ C ν φ h ln = C φ ν C ν h p+h σ σ φ + σ σ σ φ ν σ σ σ σ φ φ ν + σ φ ν σ σ ν + σ + σ φ ν σ ν We nee to show that this Hessian matrix is positive semi-efinite. For this purpose we formulate the matrix as a function of ν an the scale parameter is ξ = φ. Inee, we can ν write σ as σ ξφ, ν, ν = σ ξφ, ν + ν We can then write the partial erivatives of σ in terms of partial erivatives of σξ, ν so that we can use the results of the analysis of the proof of Proposition 4. For ease of notation we enote partial erivatives with subscripts, e.g., σ φ = σ φ an σ φ = σ φφ. σ φ = ν σ ξ σ ν = φ ν σ ξ + σ ν σ φφ = ν σ ξξ σ νν = φ ν 3 σ ξ φ ν σ νξ + φ ν 4 σ ξξ + σ νν σ φν = φ ν 3 σ ξξ ν σ ξ + ν σ ξν To show convexity of C in φ, ν, we nee to show that the Hessian matrix is positive semi-efinite for all values of ξ an ν. Note that we no longer consier ξ as a function of φ, ν since we will show that the Hessian is positive semi-efinite in the entire space of ξ, ν for ξ 0,ν 0 an νξ >. From here on, we prove convexity of the average carrying cost C in steps. The sketch of the proof is the following. In step we show that the average carrying cost is ecreasing in φ an ν. In step we show that for C to be convex it is sufficient that a function Rν, ξ, σν, ξ is non-negative. Interestingly, this function is inepenent of φ. In step 3 we show that R 0 for all ξ 0 an for all ν. In the consequent steps 4, 6 an 5 we prove that R 0 for all ξ 0 an for all ν < too. Step The average carrying cost Cφ, ν is ecreasing in φ an ν. 30

We have alreay shown in the proof of Proposition 4 that σ ξ 0, which implies that σ φ 0. We have also seen in the proof of Lemma that C C 0 an hence 0. To show that σ φ σ ν 0, we nee to show that σ ν φ ν σ ξ. Letting i = σξ +, this is equivalent to showing that ln i i, which is inee true for all values of i. i Step The average carrying cost Cφ, ν is convex in φ, ν if for all ν, ξ 0 Rν, ξ = i ν σ [ ν + σ ]ln σ + σξ[ ν + σ] ln σ νσξ σ i where σ = σν, ξ an i = σξ +. We saw in the proof of Proposition 4 that σ ξξ 0, which implies that σ φφ 0. Hence C 0. As a result, to show convexity of C in φ, ν we nee to show that the eterminant φ of the Hessian matrix of C is non-negative, that is, we nee to show that σ φφ + σ σ φ σ νν + σ σ ν σ σ φ σ ν + σ φν which reuces to showing σ φφ σ νν + σ σ ν σ φφ + σ σ φ σ νν σ 4 φν + σ σ φν σ φ σ ν After substituting the erivatives of σ with erivatives of σ in the above inequality σ σ ξξ σ νν + σ νσ ξξ + σξσ νν σ σ σ ξν + σ ν ξ 0, σ σ ξ σ ξν ν ν σ ξσ ν + σ ξν σ ξ σ ν Interestingly, this inequality is inepenent of φ since σ can be efine as σξ, ν. therefore nee to show that σ σ σ ξξ σ νν σ ξν + σνσ ξξ + σξσ νν σ ξ σ ν ξ We σ σ ξν ν ν σ ξσ ν + σ ξν σ ν 7 To show the above inequality we first evaluate the iniviual terms. The first term on the left-han sie of the inequality in Equation 7 is similar to what we ha evaluate in Proposition 4 to prove convexity of σ in ν, ξ, that is, σ σ ξξ σ νν σ ξν = σ σ νi lni + lni ξνσ + σi i σν lni + i ξσν 4 σ σ 0 3

The secon term on the left-han sie of Equation 7 is σνσ ξξ + σξσ νν σi lni σνiν + σ = νσ σ i ξσν 4 i σξν + σ νσ ξσ i lni + σilni i σξν 4 i σξν i + νξσ σνi 3 ν + lni νσi lni = σ σ i σξν σ i ξσν 4 ξνσ σiν lni i lni + i σξν i + νξσ = σ σ i iσνlni + νi + iξσν νσi lni i ξσν 4 i σξν σ ξνσ i lni The term on the right-han sie of the inequality in Equation 7 is σ σ σ ξ σ ν ξ σ ξν ν ν σ ξσ ν + σ ξν σ ν σνσ σ σ = νi σξν σσi σ νσ lni + σ + ν lnii + ξσ νi σξν i σξν i σξν σσ i lni σ i lni σν lnii + ξσ νσ ln i + σ + i σξν i σξν 3 i σξν σi σξν σ = σ σ + ii σξν νσ lni + σ + ν lnii + ξσ i σξν i σξν 4 +σνi lnii σξν + σνi lni σ νσ lni + σ + Since i + ξσ = ξ + an σξ + = i σ, the above expression is σ σ i σξν 4 = σ σ i σξν 4 = σ σ i σξν 4 σν lnii + ξσ i σξν σi σξν + σνii σξν lni σii σξν + σνiξ + lni σνi lni σi ν lni + σνi lni ξ + σ i σνξ i σξν σi σξν i + νii σ lni σνi lni σν ξi lni σνi lni σ + σνi lni ξ + i σνξ σi σξν + νi σ lni νσi lni σν ξi lni σνi lni + σνi lni ξ + σ i σνξ 3

We now evaluate the term σ σνσ ξξ + σξσ σ νν σ ξ σ ν ξ σ ξν ν ν σ ξσ ν + σ ξν σ ν lni i σν = σ σ i + νi + ξσν νξ + + i σξν 4 i σξν i σ i σνξ νi lnii σ σ + σ νξ [using that νi = ν + νξ νσξ] lni i σν i + ξσν νξσ + i σξν i = σ σ i σξν 4 = σ σ i σξν 4 = σ σ i σξν 4 = σ σ i σξν 4 σ i σνξ νi lnii σ σi σνξ σ lni iσν i σξν i νσξ σνilni + i νξσ + i σνξ νi lnii σ σi σνξ σ i + σνξ νi lnii σ σi σνξ σ i + σνξ i νσξ σνilni + νσi lni + νii σ lni Finally to show inequality in Equation 7 we nee to show that σ σ σ ξξ σ νν σ ξν + σνσ ξξ + σξσ σ νν σ ξ σ ν ξ σ ξν ν ν σ ξσ ν + σ ξν σ ν = νi lni + lni ξνσ σi i + σν lni + σ σ σ σ i σξν 4 +i νσξ σνilni σ + νσi lni + i + σνξ 0 νii σ lni Removing the common term σ σ ν lni + lni ξνσ + i σ σν lni + σ + νi σ lni an i, we nee to show that i σξν 4 σi + i νσξ σνlni + νσ lni + σ i + σνξ i Replacing lni = lnσ/ν, an iviing both sies by σ, we nee to show that lnσ lnσ ξνσ ν σ + i σlnσ + i νσξ σν σ lnσ σ + + σξν i i σ lnσ σ + i σ σ lnσ Which is equivalent to showing the inequality νξσ lnσ + ilnσ σ + σilnσ ν ν νξσ σ iσ lnσ iσ lnσ + σ lnσ i σ 33

The above inequality can also be written as i + σ lnσ σ i ν σ lnσ + σξ lnσ ν + σ ξ lnσ νσξ σ i or in other wors R = i ν + σ ln σ + σξ ν + σ ln σ νσξ σ ν σ i Step 3 Rν, ξ 0 for ν an for all ξ. 0 0 8 After replacing ν with lnσ in Equation 8, we nee to show that lni i lnσ + σ lni σξ + σ ξ + lnσ σ i σ + σ ξ lni σξ σ 0 ilni Removing the common lnσ > 0 term an writing the secon term on the left han sie [i lni ξ σ]σ lnσ of the above inequality as, we woul nee to show that i σlni i + σ lni σξ + σ ξ + [i lni ξ σ] σ lnσ 0 i σlni Using the logarithmic inequality lnx x for x 0, we can substitute lnσ using x lnσ σ for σ in the above inequality an thus we woul nee to show that σ which is similar to showing that i + σ lni σξ + σ ξ [i lni ξ σ] σ ilni 0 i + σ lni σξ + σ ξ [i ξ σ lni ] σ i Using the inequality lni i σξ, we can show that i 0 since i = σξ +. i lni Also using the logarithmic inequality ln + x x x 0, we get that lni σξ. Using these inequalities it is sufficient to show that which is similar to showing i + σ lni σξ + σ ξ i σ i i + σ lni σξ + σ ξ σ ξ σ i 0 0 0 34

After substituting for lni in the above inequality with lni show that + σ σ σξσ i 0 σξ i we woul nee to or simply i σξ. Since σ 0, therefore i νξσ, hence the above inequality is efinitely ξ true for ν. Therefore Rν, ξ 0 when ν. Step 4 lim Rν, ξ = 0, lim ν ν ξ ξ all values of ξ. Rν, ξ ν = 0 an lim ν ξ Rν, ξ ν = ξ + + 3 ν + ν > 0, for From Equation we see that when =, ν σ =. Due to uniqueness of solution of ξ 0 < σ <, we can also say that σ as ν ξ. Similarly i = σξ + as ν ξ. Using the Taylor approximation for ln σ σ = σ, we get lim Rν, ξ = lim σ + σ σ ν σ ξ The first erivative of R with respect to ν can be written as R ν = ξln σ ν + σ σ ν σ ν + ν + i ln σ σ νσ σ iν + σln σ ν σ +ξln σ ν + σ σ ν νσ ξ σ σν ξ σ i i = 0 σ ν iσ ln σ ν σ σ ν iln σ ν + σ + iln σ ν + σ σ ν σ ν σ ν + ξ ν + σ σ ν σ ξ ln σ ν + σξln σ σ ν σ ν + νσξ i σ ν νσξ σξ i Taking Taylor expansion of the logarithm in Equation 3 we fin that lim ν ξ σ ν = lim i i ln i i = σ ν an hence lim ν ξ = lim ν ξ Rν, ξ, σν, ξ ν i ν + σ νσ i ν + σ ν ξ ν + σ νσξ i = + + = 0. From Equation 5, we can fin the secon erivative of σ with respect to ν. Again, taking the Taylor expansion of the logarithm, we fin lim ν ξ σ ν = lim i σi i + νσξ = 0. 35

The secon erivative of R with respect to ν gives a long expression, which in the limits can be written as lim ν ξ = lim ν ξ Rν, ξ, σν, ξ ν ξln σ ν + σ σ ν σ ν + ν + i ln σ σ νσ σ iν + σln σ ν σ ν +ξln σ ν + σ σ ν νσ ξ σ { i ξ = lim + ξ + + + + + + ξ ν ν ν ν ν ξ ξ + + 3ν + ν = lim ν ξ > 0 for all ξ. Step 5 The equation ν+σ ν σ ν iσ ln σ ν σ σ ν iln σ ν + σ + iln σ ν + σ σ ν σ ν σ ν + ξ ν + σ σ ν σ ξ ln σ ν + σξln σ σ ν σν ξ σ σ i ν + νσξ σ i ν νσξ σξ σ i ν + ν + + ξ + ξ ξ + + ξ + ξ z + can have at most one real negative root. ν+σ ν z = 0, where z = i ln σ σνσξ 0 } After some manipulations, Rν, ξ can also be written as i ln σ ν + σ Rν, ξ = σνσξ ν i ln σ + σνσξ ν + σ ν Letting z = i ln σ 0 in the above equation results in a quaratic equation σνσξ ν + σ ν + σ z + z ν ν The roots of this equation are ν+σ ν ± ν+σ + ν+σ ν ν+σ ν The following scenarios cover the entire space of 0 < σ < an ν, that is,. σ ν +. σ ν + 3. σ ν + an σ ν + 36 ν

The first case gives both positive roots, which are inee not vali roots since z 0. The secon case gives one positive an one negative root an the thir case also gives one positive an one negative root. Step 6 z = i ln σ σνσξ 0 is monotonic in ν for all values of ξ. Substituting ln σ = ν ln i an i = σξ + into z, we get i ln i = i lni 0. Also i i σi ν i ln i in ν. Therefore σi ecreasing in ν for all values of ξ 0. z = i ln i σi = ξ σ ν 0, for all ξ 0, since σ is ecreasing is increasing in ν for all values of ξ 0. Which implies that z is En of the proof. From step we get the sufficient conition Rν, ξ 0 for all ξ an ν. From step 3 we know that this conition is satisfie for ν an for all values of ξ. From step 4 we know that R is positive an increasing in the limit ν. Steps 5 an 6 ξ together imply that equation Rν, ξ = 0 can have at most one real root along ν 0 for all values of ξ 0. However if R were to be less that 0 then it woul have to have at least two real roots between ν follows from steps 3 an 4. Hence R 0 when ν too ξ an for all ξ. This completes the proof. 37

Table : G/G/ Results with Ccap = an unit mean eman. Deman Fixe CV=0 Uniform CV= 3 Capacity Gamma CV= Exponential CV= Gamma CV= Cost= Cost=.47 Cost=3.73 Cost=5.7 Cost=7.77 Fixe CV=0 Cap= Cap=.43 Cap=.78 Cap=.7 Cap=3 Inven= Inven=.94 Inven=.4 Inven=.75 Inven=3.5 Cost=3. Cost=3.79 Cost=4.53 Cost=5.94 Cost=8.39 Bernoulli CV= Cap=.6 Cap=.66 Cap=.89 Cap=.35 Cap=3 Inven= Inven=.89 Inven=.9 Inven=3.3 Inven=3.95 Cost=.85 Cost=3.66 Cost=4.46 Cost=5.88 Cost=8.3 Gamma CV= Cap=.78 Cap=.93 Cap=.9 Cap=.7 Cap=.86 Inven=.8 Inven=.39 Inven=.94 Inven=3.6 Inven=4.8 Cost=3.83 Cost=4.5 Cost=5.8 Cost=6.44 Cost=8.88 Gamma CV= Cap=.03 Cap=. Cap=.05 Cap=.3 Cap=.89 Inven=.43 Inven=3.0 Inven=3.5 Inven=4.6 Inven=4.9 Cost=5.7 Cost=5.8 Cost=6.3 Cost=7.47 Cost=9.8 Gamma CV= Cap=.46 Cap=.46 Cap=.4 Cap=.47 Cap=.94 Inven=3.4 Inven=4.05 Inven=4.4 Inven=5.5 Inven=6.5 38