Cycle Probabilistic Stockout in Supply Chain Management

Transcription

1 Cycle Probabilistic Stockout in Supply Chain Management Miguel A. Lejeune Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, The present study is motivated by a real-life problem faced by a major North American chemical supply chain that seeks to enforce a cycle service level holding over the whole planning horizon composed of a finite number of interdependent timeperiods. We consider a supply chain using maritime transportation, and provides an integrated approach for stochastic inventory-production-distribution planning. The novelty of this research is that it is, to the best of our knowledge, the first one to propose models and develop solution methodologies for the construction of multi-period replenishment plans for multi-stage supply chains enforcing cycle stockout service level requirements. The proposed solution methodologies are based on concepts and new developments in probabilistic programing, and are decomposed into two major steps. The first one identifies demand trajectories such that, if all their requirements are met, the prescribed cycle service level is attained. The second step determines the optimal decisions to be taken in order to satisfy the identified demand trajectories at the lowest possible cost. We evaluate the computational tractability of the models and solution methodologies, highlight the impact on the distribution schedule and on the savings, and derive managerial insights. Another interesting feature of this study is the wide applicability of its approach which can be used to handle most types of random variables, and many scheduling and planning problems arising in multi-tiered supply chains. Key words: cycle service level, supply chain management, marine distribution scheduling, joint probabilistic constraint, static stochastic programming 1. Introduction In this paper, we build probabilistic inventory-production-distribution plans for a multi-stage supply chain subject to cycle non-stockout service levels. We propose models and develop solution methodologies that allow for the integrated management of the production, inventory and distribution functions, and for the satisfaction of requirements of cycle non-stockout service levels. The original motivation for this study came from a problem faced by a North American chemical company which is one of the top five worldwide producers of soda ash and calcium chloryde. The supply chain comprises one supplier in Michigan, two manufacturers in Michigan and Ontario, and about 15 main distributors in harbour cities such as Montreal, Quebec, Cleveland, Oswego, etc. The company has about $300 million revenue, about $100 of which stemming from the calcium chloryde market. The raw material is brine and the end-products are several forms (liquid, flake, pellet) of calcium chloryde. The demand for calcium chloryde is non-stationary and very seasonal - the product is used, for example, for motorways maintenance (de-icing roads, dust control, etc.). Demand shortages result in customer dissatisfaction and loss of goodwill that could be disastrous for the supply chain; therefore, shortage management is clearly a 1

2 2 key supply chain performance driver. Very large and heterogeneous (i.e., differing in loading capacity and speed) tank ships, vessels or barges are used to transport products over the Great Lakes between supply chain entities. Large vessels are very expensive to operate; transportation costs represent about 50% of the total product costs. Contracting out transportation capacity with external logistics service providers is also an option. Transportation scheduling is critical for the supply chain, and the lack of proper planning tools leads to both unpredictability and limited responsiveness. The construction of multi-function replenishment plans is known to be a very complex endeavor [7, 29] due to the combinatorial elements and symmetry involved, and, in particular, to the combined presence in inventory routing problems of large continuous (amounts supplied, inventory levels) and discrete (choice of carriers, routes) variables. The complexity of constructing a replenishment plan is here exacerbated by the stochastic nature of the problem. In this paper, we consider the demand as the source of uncertainty. Presented next is a brief review of papers studying the construction of integrated replenishment plans where the demand is stochastic. Graves and Willems [14] propose a model enforcing a base stock inventory policy in a multi-echelon supply chain with deterministic production and transportation times, and stochastic demand. The authors handle the stochasticity of the demand by introducing the concept of bounded demand defined by a meaningful upper bound on demand over varying horizons [14]. The safety stock is determined in such a way that it covers any demand realization falling within that upper bound. The deterministic optimization problem is solved by using the spanning tree concept, and provides a minimal service time between every successive supply chain stage. Decisions related to the choice of carriers, their routing or scheduling are not considered. Bitran and Yanasse [3] construct a production and sourcing plan over a multi-period horizon by solving a dynamic programming problem in which the demand is random and distribution decisions are not considered. The model contains individual probabilistic constraints which guarantee that the likelihood of not having shortage at a single period is below a certain probability. Equivalent formulations of the dynamic problem are derived, as well as error bounds on the exact solution. Next, Yildirim et al. [31] also build a production plan by solving a stochastic dynamic programming problem. The authors adopt a rolling horizon approach and recursively update a base stock policy. Further on, assuming that the unmet demand is backlogged, Bertsimas and Paschalidis [2] design production policies that minimize the inventory costs and keep the probability of having a shortout below a predefined threshold. These policies can be applied in case of autocorrelated demands and service processes. In all the papers above, the service level is a stagewise one, while, as a contrast, we consider in this study cycle service levels. These impose requirements that must hold over the entire planning horizon, and are therefore very demanding. A cycle service level requires the probability (magnitude) of a shortage

3 3 happening at any time during the planning horizon composed of a number of interdependent time-periods, to be lower than a small prescribed probability (quantity). It contrasts with a stagewise service level p t enforcing stockout requirements at each period considered independently of each other, and, therefore, ensuring that, on average, 100p t % of customers are satisfied all the time. Stated differently, while a cycle service level provides a representative measure of the full responsiveness of the supply chain [1, 17], a stagewise service level provides an expected value measure that reflects the steady-state nature of the supply chain. Indeed, even if the prescribed stagewise service level is high at each period, it does not offer any guarantee about the corresponding cycle service level [25]. A high probability of not having a shortage at a given period t does not offer guarantee about the probability of not having a shortage over the planning horizon composed of a collection of periods, and results in an unknown, possibly very low, cycle service level, especially if the number of periods of entities is large. Cycle service levels are those the considered supply chain motivating wants to enforce. The need to enforce such demanding service levels does not uniquely pertain to this supply chain but can be extended to supply chains operating in highly competitive, malevolent (military) environments [1, 17, 20], or those in which a supply shortage could trigger huge costs to set up the production process and machines all over again [9]. An example of this is provided by the calcium carbonate producer Omya Hustadmarmor [9] for which a stockout would cause the stopping of the production, and this would be extremely harmful due to the astronomical cost of operating the paper machines at the mills. The recourse to cycle service levels in inventory-production planning and scheduling is also emphasized in [5, 30]. Increased globalization and competition further reinforce the need to ensure very strict service levels such as the cycle service levels. A major contribution of this paper is that it is, to the best of our knowledge, the first study that provides model formulation and tractable solution methodologies for the construction of replenishment plans subject to the attainment of cycle service levels. We shall consider three cycle stockout service levels. The first one limits the probability of having a shortage at any point of the planning horizon, while the two other service levels constrain the magnitude of stockout. The modeling approach relies on the static stochastic programming framework. The rationale for developing static stochastic programming models is that, in the problem at hand, and in many other contexts, decisions and arrangements must be taken ahead of time, prior to the observation of the source of uncertainty. Although, in some circumstances, it is possible to take corrective actions after the unfolding of the uncertainty, contracting and scheduling large vessel shipments (plus the domino effect that these decisions have on production and inventory at multiple sites) require a longer planning horizon; these decisions cannot be adjusted at each time period without incurring large additional costs (i.e., contracting out external transportation capacity on a very short notice). We shall show that the constructed replenishment plans using this approach have very appealing features from a managerial

4 4 perspective. Indeed, they are strategic plans that hedge against major risks and uncertainties, and they can also be updated in order to take advantage of favorable realizations of the random variables. For each service level, we develop a two-step solution method. The first step consists of determining a service level demand trajectory such that, if all the requirements imposed by this demand trajectory are satisfied, then the prescribed cycle non-stockout service level is attained. The second step determines the optimal decisions to be taken so that the requirements of the service level demand trajectory identified in Step 1 are met at the lowest cost. The present study also contributes to fill the gap in the literature related to maritime transportation scheduling and planning. In their thorough review of the literature, Christiansen et al. [8] report that only a few models (and even more so in a stochastic context) on ship scheduling and planning have been proposed, while the demand for maritime transportation services has been rising significantly. During the last decade much attention has been directed towards inventory-distribution problems where trucks are used for transportation. In such problems, similar to the vehicle routing problem, a set of orders of known sizes has to be assigned to a given fleet of trucks for delivery at minimal cost. However, larger scale distribution problems (transportation of bulk and semi-bulk products), such as the maritime ones, differ from the distribution problems where other transportation modes are used, since ships operate under different conditions [8]: the fleet size of shipping companies may change across time, the fleet may contain ships of different types, sizes or cost structures, ships pay port fees, the draft of a ship depends on the weight of the load, ships operate mostly in international trade, ship trips span days or weeks, etc. [27, 8]. An example of a real-life maritime inventory-distribution scheduling problem is addressed by one the 2006 Franz Edelman Award finalists [9] who consider a problem faced by a Norwegian company delivering calcium carbonate to European paper manufacturers. The objective is to construct an integrated inventoryproduction-distribution plan such that the operational costs are minimized. Products are various types of calcium also transported from a single processing plant using tank ships of various sizes and with fixed costs of different magnitude. The problem is formulated as a mixed-integer problem, in which the approach to hedge against uncertainty consists of relying upon buffering mechanisms, i.e., large safety stocks. We refer the reader to the Transportation Science special issue devoted to maritime transportation [24] and to [8] for a more extensive review of the literature. It is clearly stated there that, although ship scheduling carries much uncertainty, very few ship scheduling models take this issue explicitly into account. Hence, to the best of our knowledge, our paper is the first one to propose a stochastic, integrated and multi-period planning model in a marine context, and to develop a solution method. The paper is organized as follows. In Section 2, we describe the general probabilistic inventoryproduction-distribution model permitting the construction of sustainable replenishment plans that meet the

5 5 requirements of cycle service levels. In Section 3, we analyze the specifics of the service level that bounds the probability of having a stockout at any time over the entire planning horizon, and propose two modeling and solution techniques. In Section 4, we discuss two service level variants that constrain the magnitude of stockout. We present the modeling approaches, characterize the associated service level constraints, and propose solution methodologies. Section 5 evaluates the proposed methods with respect to the computational tractability, the resulting savings for the company, the impact on the production-distribution scheme, the degree of conservativeness of the various model formulations for a given service level, the degree of stockout risk associated with each service level, and the suitability of the service levels for different degrees of risk aversion. Section 6 offers concluding remarks and managerial insights. 2. Probabilistic Replenishment Planning Model 2.1. Model description and main assumptions We develop here a model that allows the taking of integrated inventory, production and distribution decisions in a three-stage supply chain facing cycle service level requirements. We consider a set I of suppliers transforming raw materials into semi-finished products and delivering them to a set K of manufacturers. These latter provide the end-products and transport them to a set J of distributors in order to satisfy their random demand. Each supplier and manufacturer has a limited, non-stationary production capacity. Each supply chain entity has a limited and stationary storage capacity for raw materials, semi-finished products and/or end-products, and starts, at the beginning of the first period, with a given inventory level. Demand that cannot be satisfied from on-hand inventory is lost. Only direct shipments are allowed and they are carried out using a fleet V of heterogeneous vessels and barges. The lead times depend on the transportation, product loading and product unloading times. There is no transshipment between distributors. The supply chain must address the question of how many raw materials and products it must produce and store at each supplier and manufacturer location. Since shipments are not initiated by orders from the receiving entities, the supply chain must control and plan the inventories, the production and the shipments in order to minimize the risk of the receiving locations running out of stock. Distribution decisions must be made not only with regards to the schedule of the shipments but also with regards to when to call on each delivery entity, how much to supply, and which ship to use. Additional decisions relate to the eventual use of external logistics service providers and to the maintenance schedule of the vessels. These decisions must be taken in a manner that minimizes the total costs of the supply chain and without violating any constraints (storage, production, transportation limitations, etc.). In particular, the plan must be such that a cycle stockout-related service level is satisfied. Some assumptions warrant further explanation. The first one pertains to the enforcement of a cycle service level that holds throughout the whole planning horizon. We discussed previously the increased importance

6 6 and distinctive features of cycle service levels. A second assumption is that backlogging the demand that is not satisfied from the available stock is not allowed. The reason for this is that, very often, the lagged satisfaction of the demand is not accepted by highly demanding customers, those having a greater negotiating power or those suffering from high set up costs when there is a shortage [9]. Another reason for not allowing backorders is the length [26] of each period that can possibly be very long (month, semester, etc.). Third, we assume that the transportation plan is made of direct shipments, i.e. a single loading and discharging location, due to the fact that the demand is much larger than the carriers maximal loading capacity, and to the very high cost of operating a carrier. Preliminary tests also showed that attempting to construct a distribution scheme with carriers delivering diverse destinations within a single trip is computationally intractable. For an in-depth discussion of the advantages to resorting to direct shipments, we refer the reader to [12]. Fourth, the sustainability assumption intends to avoid errors known as end-effects [11], i.e. what is optimal over the short horizon may be suboptimal over the long run. The objective is to avoid draining the inventories and to construct a plan that is reproducible over the next planning horizon. Fifth, we assume that the fleet of carriers is very heterogeneous: ships differ with respect to their loading capacity, cost, speed and ownership (i.e., some belong to the supply chain entities, other can be rented form external logistics sources). This is typical in marine transportation and contributes to the complexity of the problem [27]. Finally, distribution time-windows constraints are introduced, since it is impossible to deliver to certain locations at some periods due to weather conditions, the closing of facilities, etc. For example, no transportation to Cleveland is allowed in the winter months, since the chemical product is then likely to freeze Problem formulation We present here the formulation of the model discussed above. It takes the form of a stochastic problem whose objective function (1) minimizes the sum of the distribution, production and holding costs over the entire planning horizon. The production costs are computed by multiplying the production levels p[i,t] and b[i,t] by the unit production cost P[i,t]. The inventory costs are proportional to the stock levels of products and raw material at each supply chain entity; the unit holding cost at entity i and time t is denoted by h[i,t]. The distribution costs are obtained by multiplying the unit transportation cost c[i, k, v] (from entity i to k with vessel v) by the lead times for each shipment x[i,k,v,t] (between i and k at t with v). The lead times are the sum of the transportation time 2r[i,k,v] which depends on the distance separating the origin i and destination k entity and on the vessel v used, the loading l[i,v], and unloading u[k,v] times. The existing literature shows that the maximization of the profits is very much impacted by the unit selling price, and is likely to result in a very low service level. Moreover, this requires the inclusion in the objective function of a penalty cost for the unmet demand, and it is well known that the quantification of such a penalty cost is very difficult (e.g., opportunity costs, loss of goodwill). These elements provide the

7 7 rationale for defining the objective function as a cost minimization function (1). Hence, the model does not attempt to quantify the cost of losing customers, but instead imposes, through the use of stochastic service level constraints, the minimization of the costs subject to the attainment of a prescribed service level, thus effectively limiting the risk of loss of customers and of market share. min ( ) (p[i,t]+b[i,t]) P[i,t] + ( h[i,t] ( p[i,t] +n[i,t] )) + ( ) h[j,t] z[j,t] 2 t T i I t T i I t T j J }{{}}{{} Production cost Inventory cost ( (l[i,v]+2 ) ) r[i,k,v]+u[v,k] c[i,k,v] x[i,k,v,t] (1) t T i I k K,k i v V } {{ } Distribution cost subject to 0 b[i,t] b max [i,t],i I, t T (2) 0 p[i,t] p max [i], i I,t T (3) 0 q[i,k,v,t] = C[v,t] x[i,k,v,t], i I,k K,v V,t T (4) z max [i] z[j,t] = z[j,t 1]+m[j,t] d[j,t] 0, j J,t T (5) w max [i] w[i,t] = w[i,t 1]+b[i,t] q[i,i,v,t] 0,i I,t T (6) v V i I,i i s max [i] s[i,t] = s[i,t 1]+p[i,t] q[i,j,v,t] 0, i I,t T j J v V (7) z[j,τ] z[j,0], j J (8) i I k K s[i,τ] s[i,0], i I (9) w[i,τ] w[i,0], i I (10) (l[i,v]+2 r[i,k,v]+u[k,v]) x[i,k,v,t] a[v,t], v V,t T (11) x[i,j,v,t] M δ[v,t], v V,t T (12) T δ[v,t] = T 1, v V,δ[v,t] (13) t=1 x[i,j,v,t ] = 0, i I,v V, x[i,j,v,t] = 0, v V,t T (14) Tω ξ (15) x[i,j,v,t] Z +, i I,j J,v V,t T (16) z[j,t],p[i,t],s[i,t],w[i,t],b[i,t],q[i,k,v,t],ω[j,t] 0,i I,j J,t T,v V,k K (17) Constraints (2) and (3) impose upper limits on the manufacturing of semi-finished products b[i, t] and products p[i, t], respectively, since production facilities are capacitated. The set of constraints (4) ensures that the maximum capacity C[v,t] of a transportation carrier c at t is not exceeded, and imposes full-truck load shipments. Indeed, the quantity q[i,k,v,t] delivered from i to k at t with carrier v is set equal to the number of shipments x[i, k, v, t] multiplied by the maximal loading capacity. Denoting, respectively, by m[j,t] and by d[j,t], the supply received by and the demand of j at t, constraints (5) represent the flow balance constraints of end-products at distributors. The ending inventory level at a period t is equal to the

8 8 ending inventory level at t 1 plus the supply received during period t minus the customers demands satisfied by the end of t. Combined with the bound z max, (5) assure that the inventory at the distributors does not exceed the maximum capacity and does not become negative. For ease of exposition, we omit the conversion factors for the transformation of raw materials into semi-finished products and of these latter into end-products. Constraints (6) and (7) are the raw materials and semi-finished products flow balance constraints, and they ensure that the maximum inventory levels w max and s max of raw materials and semifinished products are not exceeded. Constraints (8), (9) and (10) make the plan reproducible over the next planning horizon by ensuring that inventory levels at the last period τ of the planning horizon are at least equal to the initial ones. The limited time availability of carriers a[v,t] at each period t is represented by (11). We introduce binary variables δ[v,t] to allow for the maintenance of carriers: δ[v,t] takes value 1 if the carrier v is set up at time t, and value 0 otherwise. Together with (13), constraints (12), in which M is a large positive number, enforce that carriers are not used during at least one period, at which they can be maintained. Constraints (14) make the distribution scheme operational, accounting for the distribution time-window constraints due to the impossibility to supply j at period t due to bad weather conditions, the closing of facilities, and the inaccessibility between identified pairs (i,j ) of distributors and suppliers. Denoting by Z + the set of positive integers, constraints (16) enforce the indivisibility requirements of the number of shipments, and (17) are the non-negativity constraints. The stochastic stockout service level constraints enforce a cycle non-stockout service level holding across the entire planning horizon. At this stage in the paper we use the generic formulation (15). The stochastic nature of the model is due to the random customers demand and the need to respond to it with a prescribed service level. The supply chain is not expected to be able to handle all demand levels, since this would force its entities to keep very large safety stocks and to support excessive holding costs. The random variables are in the right-hand sides of the service level constraints which take a different form, depending on whether we consider a cycle service level limiting the probability or the magnitude of a stockout. These types of cycle service levels and the formulation of the associated stochastic constraints, are discussed in the next sections Stochastic stockout constraints Before discussing the formulation of the service level constraints, we introduce the following notation. The product demand d[j,t] at time t and node j is discretely distributed: at each time t and node j, d[j,t] can take l max different levels. We denote by d l [j,t] the realization of the random demand d[j,t] taking level l: p l [j,t] = P(d[j,t] = d l [j,t]), j J,t T,l = 1,...,l max. (18) Denoting by

9 9 ξ l [j,t] = d v 1 [j,1]+d v 2 [j,2]+ +d v t [j,t], 1 v 1,v 2,...,v t l max, t T the realization of the cumulative demand ξ[j,t], it can be seen that the number of realizations of the cumulative demand ξ increases exponentially with the number of periods in the planning horizon. The probability of ξ[j,t] being realized as α is given by P ( ξ[j,t] = α ) = (v 1,v 2,...,v t ) V j (t,α) t p vr [j,r], r=1 where V j (t,α) = { (v 1,v 2,...,v t ) : d v 1[j,1] + d v 2[j,2] + + d v t[j,t] = α } is the set of all demand paths in the first t stages totaling α. The probability p lr [j,r] (18) of the stagewise demand d[j,r] being realized as d lr [j,r] can be considered as independent of the past realizations of the random demand as well as modelled as a conditional probability whose value is contingent of the realization of the stagewise demand at the previous periods (t = 1,...,r 1). This implies that no restrictive independence assumption is imposed on the demand. The proposed methods are applicable to a random demand that can be time-independent as well as autocorrelated, identically distributed or non-stationary, and monotone or not. The stochastic stockout constraints impose the supply to be larger than the demand with a prescribed reliability level. As aforementioned, in order to capture the actual reliability of the supply chain, a cycle service level holding true over the entire planning horizon [22] is required. The attainment of the prescribed cycle service level requires the inventory level z[j, t] to be sufficient to satisfy a large part of customers demand. The formulation between inventory, supply and demand, with respect to which the probability of not having a shortage is defined, is therefore critical. The non-stockout probability can be written in terms of the cumulative demand ξ[j,t] and supply ω[j,t] P([z[j,0]+y[j,t] ξ[j,t], t T), (19) or in terms of the stagewise demand d[j,t] and supply m[j,t] P(z[j,t 1]+m[j,t] d[j,t], t T). (20) We note that both probabilities (19) and (20) are joint probabilities, but despite this commonality, and the very similar appearance of (19) and (20), they differ in a major way, which impacts very much their respective computational tractability. The value taken by (19) is a function of a decision variable ω[j, t], a random variable ξ[j, t], and a fixed parameter, i.e., the initial inventory level z[j, 0]. By contrast, the value taken by (20) is a function of a decision variable m[j,t] and two random variables d[j,t] and z[j,t]. Indeed, the inventory level z[j,t] is stochastic, for its value depends on the inventory level z[j,t 1] at the preceding

10 10 period (t 1), that is itself a random variable whose value is a function of the realization of the stochastic demand at the preceding periods t = 1,...,t 1: z[j,t 1] ( d[j,t ], t = 1,...,t 1 ), t T. Therefore, using (20) would not make possible to give a static formulation for the stochastic programming problem. For a one-year planning horizon with monthly time-periods, a twelve-stage recourse programming problem would result [28], for which a huge number of scenarios should be taken into account, making the problem intractable. We also note that such a formulation assumes that the quantity supplied, and thus the production-distribution scheme, is very flexible, and can be modified at each period, which in the context of this paper is not likely. That is why we use thereafter the formulation (19) that defines a cycle nonstockout measure in terms of cumulative demand and supply. This is as a necessary condition to have a tractable stochastic programming model allowing the construction of a replenishment plan satisfying cycle non-stockout requirements Compact model formulation The probabilistic replenishment planning model enforcing a cycle service level takes the form of a stochastic mixed-integer programming problem, and its compact formulation is given by min c T x subject to Ax b Tx ξ. (21) x = [x x ] R + Z + The expression Ax b represents all deterministic constraints, while the set of stochastic service level constraints is denoted by Tx ξ. The service level constraints are defined with respect to the cumulative demand and supply, with the random cumulative demand being in the right-hand side of the constraint. Their specific form depends on the considered type of service level. The vector x can be partitioned into [x x ]. Elements of x are restricted to be positive integer numbers, representing the number of shipments x[i,j,v,t] from i to j carried out with v at t. Elements of x are real positive numbers, and encompass all the other decision variables. The symbol R + refers to the appropriate multi-dimensional space of positive real vectors, while the symbol Z + refers to the appropriate multi-dimensional space of positive integer vectors. 3. Probability of stockout In this section, we consider the non-stockout, also called ready rate, service level [2, 15] that requires the probability of a shortage to be below a predefined level. It allows stockout to occur with no more than a small probability. Probabilistic constraints will be used to model it.

11 11 Provided that the enforced non-stockout service level is of a cycle nature, the probabilistic programming model contains a number of joint probabilistic constraints [6, 21] guaranteeing that the probability of the joint fulfillment of a system of linear inequalities with random right-hand side variables is above a prescribed probability level p, representing the cycle non-stockout service level. Since customers are not interested in the service level provided to the others, we have a joint probability constraint holding over the entire planning horizon for each j: P(z[j,0]+ω[j,t] ξ[j,t], t T) p j, j J. (22) This formulation requires ξ[j,t] to be larger than or equal to ω[j,t]+z[j,0], with some probability p, for all possible realizations of the right hand side ξ[j, t], and allows the enforcement of a differentiated service level p j. This enables the supply chain to customize the service level provided to each customer, and to account for their respective requests. Substituting (22) for T x ξ in (21), we obtain the stochastic mixed-integer programming problem min c T x subject to Ax b P(z[j,0]+ω[j,t] ξ[j,t], t T) p j, j J. (23) x = [x x ] R + Z + We discussed above why the probabilistic constraints need to be formulated in terms of the cumulative demand and supply. Clearly, the level of the random cumulative demand at time t is not independent of that at time (t 1). Therefore, the joint probabilistic constraints have dependent random variables located in the right-hand side of the constraints, and the probability P(z[j, 0] + ω[j, t] ξ[j, t], t T) is a τ-variate one. In the next sub-sections, we shall propose two different approaches for problem (23) Modeling approach based on the intersection of events bounding scheme The stochastic problem (23) is particularly complex, since it involves multi-dimensional joint probability distributions. Instead, we shall attempt to solve (23) after having replaced the constraints (22) involving a multi-dimensional probability distribution by an expression involving the probability distribution functions of uni-dimensional random variables. We introduce in this section an approximation scheme based on the intersection of events [23], which consists of replacing (22) in (23) by the set of constraints P(ξ[j,t] ω[j,t]+z[j,0]) p jt, t T(T = 1,...,τ), j J τ. (1 p jt ) 1 p j, j J t=1

12 12 The substitution above is allowed, since P ( ξ[j,t] ω[j,t]+z[j,0],t T ) (, j J = 1 P t T ) (ξ[j,t] > ω[j,t]+z[j,0]), j J 1 P ( ξ[j,t] > ω[j,t]+z[j,0] ), j J t T 1 ( 1 P ( ξ[j,t] ω[j,t]+z[j,0] )), j J t T. (24) From (24), constraints (22) in (23) are always satisfied if the inequality 1 ( 1 P ( ξ[j,t] ω[j,t]+z[j,0] )) p j, j J t T holds for each j. This requires that constraints 1 ( 1 P ( ξ[j,t] ω[j,t]+z[j,0] )) 1 (1 p jt ), j J t T t T 1 t T (1 p jt ) p j, j J hold jointly, which is guaranteed if P ( ξ[j,t] ω[j,t]+z[j,0] ) p jt, j J,t T 1 p j t T (1 p jt ), j J. Clearly, constraints 1 p j t T(1 p jt ), j J can only be satisfied if p jt p j, t T, j J. Using the substitution described above, we obtain the problem below min c T x subject to Ax b P(ξ[j,t] ω[j,t]+z[j,0]) p jt, t T, j J, (25) (1 p jt ) 1 p j, j J t T 1 p jt p j, t T, j J x 0 where p jt are decisions variables, p j is the given non-stockout service level for j, with p j close to 1. A branch-and-cut algorithm will be used to solve the problem above p-efficiency model We now propose a model and develop a solution method resting on the p-efficiency concept for a discrete probability distribution [22]. Let p [0,1] and F be the discrete probability distribution function: F(v) = P(v ξ),ξ Z+ s. DEFINITION 3.1. A point v R S is p-efficient point for the probability distribution function F if: F(v) p, and (26) there is no v v,v v such that F(v ) p. (27)

13 13 From (26) and (27), it follows that for every y R n such that F(y) p, y v = (v 1,...,v i,...,v n ), v R n, (28) where v i denotes the p-efficient point of the one-dimensional marginal distribution F i ( ), i = 1,...,n. An illustration of the concept of p-efficiency for a bivariate random variable is given in Figure 1. The only points satisfying (26) are the points located on the bold line or above it (in the grey-shaded area). Among those, the points v (4) and v (5) are not p-efficient, since v (1) v (4), v (2) v (5) and v (1) < v (5), and do not satisfy (27). The only p-efficient points are thus v (1), v (2) and v (3). Figure 1 p-efficient points v (1),v (2),v (3) for z = (z 1,z 2) Contrasting with the intersection of events bounding scheme that provides a conservative approximation of the original problem (23), the approach proposed in this section allows the direct handling of the multivariate probability distribution and the transformation of (23) into an equivalent disjunctive mixed-integer problem. Below, we discuss the transposition of the concept of p-efficiency into the cycle supply chain context and we propose a two-stage solution method Solution method In the first step of the solution method, we identify the p-efficient demand trajectories. As explained above, they are such that the fulfillments of all their requirements guarantee the satisfaction of the cycle nonstockout service level. In the second step, we solve the resulting disjunctive mixed-integer problem Step 1: p-efficient demand trajectory We shall now transpose the p-efficiency concept into the probabilistic supply chain management setting, and introduce the concepts of distributor and supply chain p-efficient demand trajectories, respectively denoted by D p j and D p. DEFINITION 3.2. A distributor p-efficient demand trajectory ψ p j is a τ-dimensional vector D p j Ψ p j : D p j = [ ξ p [j,1],...,ξ p [j,t],...,ξ p [j,τ] ], j J, whose components ξ p [j,t],t T are such that the p-efficiency conditions (26) and (27) hold. The set Ψ p j contains all p-efficient demand trajectories for j.

14 14 It follows that p-efficient demand trajectories are such that, by satisfying all their requirements, the supply chain is able to attain the prescribed cycle non-stockout service level. Since a supply chain serve many customers, we introduce now the concept of supply chain p-efficient demand trajectory. DEFINITION 3.3. A supply chain p-efficient demand trajectory D p is defined as the combination of J distributor p-efficient demand trajectories: D p Ψ p : D p = [ D p 1,...,D p j,...,d p J ], (29) where Ψ p denotes the set of supply chain p-efficient demand trajectories. For any discrete probability distribution, and for any p,0 p 1, the set of p-efficient points is non-empty and finite [10]. This implies that one can find at least one distributor p-efficient demand trajectory for each j, and, therefore (29), at least one supply chain p-efficient demand trajectory. After having generated the distributor p-efficient demand trajectories, we substitute the constraints z[j,0]+ω[j,t] ξ p [j,t],t T, j J (30) for the joint probabilistic constraints (22), therefore transforming the stochastic mixed-integer programming problem (23) into a disjunctive mixed-integer programming one min c T x subject to Ax b Tx D p D p Ψ p, (31) x = [x x ] R + Z + where Tx D p stands for (30) and must hold for at least one p-efficient demand trajectory as enforced by D p Ψ p. The supply chain p-efficient demand trajectory D p is a multi-dimensional unknown vector that must be found prior to the optimization of (31) with respect to x. The definition of p-efficiency makes it rational to use a level search approach to identify p-efficient points. We have here general integer random variables and we use a forward algorithm resembling the enumerative algorithm proposed in [23] to identify the p-efficient demand trajectories. Our algorithm differs in its enumerative scheme: it relies upon (28) and the knowledge of the p th quantile of the cumulative demand at the last period of the planning horizon to eliminate levels from the start and to reduce the enumerative burden Step 2: Optimal decisions Since the number of supply chain p-efficient demand trajectories can be very high, problem (31) is very complex to solve, making the enumeration of all possible solutions impractical. Even if the satisfaction

15 15 of the conditions imposed by any p-efficient demand trajectories allows the attainment of the prescribed service level, it does not imply that they are homogeneous : the satisfaction of the requirements of any of them is not likely to result in supply chain costs of same magnitude. We therefore propose a specific column generation algorithm called the congestion-relief column generation algorithm which selects p-efficient demand trajectories that minimize the risk of congestion, defined as the inability, due to capacity restrictions, to respond to an abnormal demand or to sudden changes in demand. Very often, the requirements are not uniformly spread over the planning horizon, and the risk of having a bottleneck in the resources is much larger at the periods with peak requirements. The congestionrelief column generation algorithm is built on the idea that congestion negatively affects production efficiency [10]. The focus is on the appropriate usage of the distribution resources, for transportation costs are predominant in the type of problems handled in this paper: the chartering of a ship or barge involves a major capital investment, and the daily operating costs are also substantial [8]. The algorithm involves the alternate optimization of a master and an auxiliary problems, which, respectively consist of optimizing the production-inventory-distribution scheme for a given p-efficient demand trajectory and the selection of an alternative one. It is used in conjunction with the CPLEX branch-and-cut algorithm to solve (31). Step-by-step decomposition: Step 0: Select D p (m) Ψp. Step 1: Solve the associated master problem: min c T x subject to Ax b Tx D p (m). (32) x = [x x ] R + Z + Step 2: Identify the most congested period t T. This is the period at which the consumption of the distribution resources is the highest, i.e., the period in the planning horizon which has the lowest value for the slack u[v,t] of the distribution constraint u[v,t] = a[v,t] (l[i,v]+2 n[i,k,v]+u[k,v]) x[i,k,v,t], v V,t T ; i I k K u[v,t] is interpreted as the positive amount of unused distribution resources (unused time of carrier v at t). t = argmin{u[t] = u[v,t], t T }. v V Step 3: Identify the set J of distributors to which shipments are carried out at t. j J if x[i,j,v,t ] > 0 i I v V

16 16 Step 4: Select j J, and check whether the selection of a new demand trajectory D p j,(m+1) for j can reduce the distribution costs. If this is the case, we proceed to a demand trajectory substitution for j : D p j,(m) Dp j,(m+1) Dp (m) Dp (m+1). We go to step 1 and solve the new master problem resulting from the above substitution. The solution of the updated problem (32) indicates whether the reduction in the distribution costs for j can be obtained without affecting for the others j,j J\{j }. If yes, we continue the iterative process and go to step 2. Otherwise, we consider another j J. We stop after having considered all j J and not having found any possible improvement. The iterative process of the column generation algorithm is stopped when no further efficiency gain in the distribution scheme can be identified. 4. Magnitude of stockout We shall consider two cycle stockout service levels limiting the magnitude of the stockout. The fill rate [25, 15] service level allows stockout to occur in no more than a prescribed expected proportion (1 p ). The conditional expected stockout service level bounds the conditional expected magnitude of the stockout Model for fill rate service level The repeated appearance of the fill rate service level in the supply chain literature (see, inter alia, [2, 15]) and the business press attests that it is a major performance indicator. The fill rate service level p with p [0,1] and usually close to 1, requires the expected fraction of product demand that cannot be met from on hand inventory to be lower than or equal to (1 p ); the complement 1 p of p is called the unfill rate. Before providing the formulation of the model allowing the construction of a plan satisfying the requirements of the cycle fill rate service level, we recall the principle of the proposed two-step solution method, since it clearly affects the formulation of the fill rate (normalized expected shortfall) service level constraints. The first step is devoted to the definition the fill rate demand trajectories that are such that the satisfaction of all their requirements, at each period, ensures the attainment of the targeted fill rate service level. The optimal inventory, production and distribution decisions that satisfy the requirements (i.e., no stockout) of the fill rate demand trajectory at the lowest possible cost for the supply chain are determined in the second step. Using this approach, we can formulate the expected shortfall as E [ ξ[j,t] (ω[j,t]+z[j,0]) ] +, where z[j,0] is the known initial inventory level at j. Normalizing the expected shortfall with respect to the cumulative demand ξ[j,t], we obtain the functions [ ] + ξ[j,t] (ω[j,t]+z[j,0]) g jt = E. ξ[j, t]

17 17 The attainment of a cycle fill rate service level p imposes the expected fraction of the stockout to be less than the unfill rate (1 p ) at any period of the planning horizon. The fill rate constraints thus take the form [ ] + ξ[j,t] (ω[j,t]+z[j,0]) E max (1 p t T ξ[j, t] j), j J, and ensure that the expected amount by which each inequality ξ[j,t] (ω[j,t]+z[j,0]) ξ[j,t] 0,t T is violated, is limited from above by (1 p j) at any t. The substitution of the constraints above for Tx ξ in (21) gives min c T x subject to Ax b [ ] + ξ[j,t] (ω[j,t]+z[j,0]) E max (1 p t T ξ[j, t] j), j J x = [x x ] R + Z + (33) where the cumulative supply ω[j,t] is a component of x. We prove below that the functions g jt are piecewise linear and convex. This shows that the problem above is a piecewise linear mixed-integer problem. PROPOSITION 4.1. If the univariate random vector ξ is discretely distributed, taking a finite number of possible values, then the function [ ] + ξ Ty E (34) ξ is piecewise linear and convex in R, with T denoting here the technology matrix. i) Let p i,i = 1,...,r be the probability for ξ taking value ξ i, I {1,2,...,r}, and { Y (I) = y ξ i T i y 0, i I, ξ } T i y i < 0, i / I. ξ i ξ i It can be seen that [ ] + ξ Ty E = ξ i T i y p i ξ ξ i if y cl Y (I), with cl Y (I) denoting the closure of Y (I). The function (34) is therefore linear on each set cl Y (I). Moreover, provided that cl Y (I) is a convex polyhedron, and I {1,...,r} cl Y (I) = R, it turns out that (34) is piecewise linear in R. ii) Convexity. Since the one-dimensional function [z] + is convex, it follows that [ ] + [ ] + [ ] + ξi T i (λy +(1 λ)x) ξi T i y ξi T i x λ +(1 λ) (35) i I ξ i ξ i ξ i for any x,y R, 0 λ 1, 1 i r. Multiplying (35) by p i and summing the inequalities, we get [ ] + [ ] + [ ] + ξ T(λy +(1 λ)x) ξ Tx ξ Ty E λe +(1 λ)e. ξ ξ ξ

18 18 The functions g jt are linear on the successive intervals [ξ (l 1),ξ (l ) ], l = 1,...,(l max ) t (Figure 2), and ξ (l ) represents the l th cutpoint in [ξ (1),ξ (lmax)t ) ]. Problem (33) can thus be rewritten as: min c T x subject to Ax b [ [ p l max t T l L x = [x x ] R + Z + ] + ] ξ l [j,t] (ω[j,t]+z[j,0]) (1 p ξ l [j,t] j), j J, (36) where p (l ) [j,t] denotes the probability for the cumulative demand ξ[j,t] to be realized as ξ l [j,t] Solution methodology for fill rate service level Step 1: Critical fill rate demand trajectory The objective is to determine the minimum amount to be supplied at each node j and at each period t in such a way that it ensures that the fill rate service level p is attained. More precisely, for each j, we identify the distributor fill rate demand trajectories D p j = { y p [j,1],...,y p [j,τ] }, j J which are such that, if all their requirements are met, then constraint (33) is satisfied. In other words, supplying the distributors with product quantities at least equal to the levels y p characterizing the fill rate demand trajectories, allows the satisfaction of the prescribed cycle fill rate service level p. Considering the ordering of the levels l = 1,2,...,(l max ) t,t T that the cumulative demand ξ l [j,t] can take, the (j,t) th component y p [j,t] of a fill rate demand trajectory fulfills the following conditions: l =(l max) ( ) t p l ξ l [j,t] y p [j,t] (1 p ξ s l [j,t] j), j J, ξ s [j,t] y p [j,t] where l =(l max) t s p l ξ l [j,t] refers to the weighted summation in decreasing order (l = l max t,...,s + 1,s). Many different fill rate demand trajectories exist for a given fill rate service level p. The objective is now to uncover the best fill rate demand trajectory, thereafter referred to as the critical fill rate demand trajectory. Since the objective function, i.e., the supply chain costs, is monotone nondecreasing in the supplied quantity, the objective is to determine the minimal supply quantities, whose delivering allows the attainment of the prescribed cycle fill rate service level p. Therefore, we define the critical fill rate demand trajectory D p j = [ y p [j,1],...,y p [j,τ] ], j J as the fill rate demand trajectory whose components y p [j,t],t = 1,...,τ have the lowest demand levels such that, if satisfied, the expected fraction of lost demand is at most equal to (1 p ). The optimal value of y p [j,t],t = 1,...,τ is thus equal to y p ; no better solution can be found in the solution space defined by the cone y p [j,t]+r +,j J,t T, while any quantity strictly lower than y p [j,t] does not allow reaching the prescribed fill rate service level. Consequently, there is a single critical fill rate demand trajectory for each j, and a single fill rate supply chain

19 19 demand trajectory D p = {D p 1,...,D p } which is a combination of the J critical fill rate distributor demand trajectories. The t th component y p [j,t] (Figure 2) of the critical demand trajectory D p j where q is the decision variable. J y p [j,t] =min q l =l t subject to (p l ξl s q 0 [j,t] q ξ l [j,t] ) (1 p j) can be obtained by solving, (37) Figure 2 Critical Fill Rate Demand Trajectory The black-colored area in Figure 2 represents the maximal amount of unmet demand for the prescribed fill rate service level p Step 2: Optimal decisions After having determined the critical fill rate demand trajectory D p j, problem (36) is transformed into the mixed-integer problem min c T x subject to Ax b z[j,0]+ω[j,t] y p [j,t], t T,j J x = [x x ] R + Z + where y p [j,t] is the a priori determined optimal solution of (37) Model for conditional expected stockout service level Another measure of shortage is the conditional expected stockout that defines a limit on the expected amount of unserved demand, and is modeled using conditional expectation constraints. In the stagewise case, provided that each period is considered independently of the others, the conditional expected amount is constrained to be below a prescribed quantity s j,t E [ ξ[j,t] (ω[j,t]+z[j,0]) ξ[j,t] > ω[j,t]+z[j,0] ] s j,t, j J,t T. For a discrete probability distribution, the constraint above can be equivalently written as [ p l ξ l [j,t] (ω[j,t]+z[j,0]) ] + l L P(ξ[j,t] > ω[j,t]+z[j,0]) s j,t, j J,t T. (38)

20 20 In a cycle perspective, i.e., considering an entire horizon of interdependent periods, joint conditional expectation constraints must be used and take the form E [ max (ξ[j,t] (ω[j,t]+z[j,0])) ξ[j,t] > ω[j,t]+z[j,0] at some t T] s j, j J t T [ p l ξ l [j,t] (ω[j,t]+z[j,0]) ] + l =max L s j, j J t T P(ξ[j,t] > ω[j,t]+z[j,0]) which guarantees that the maximum (over all periods in the planning horizon) conditional expected shortage amount is below a constant s j indicative of the cycle expected conditional stockout service level. Replacing Tx ξ in (21) by (39), we obtain: (39) min c T x subject to Ax b [ p l ξ l [j,t] (ω[j,t]+z[j,0]) ] + max t T l L P(ξ[j,t] > ω[j,t]+z[j,0]) x = [x x ] R + Z + s j, j J. The proposed solution method rests on the same approach as that used for the fill rate service level. We determine the critical conditional expected stockout demand trajectory D p j = [ y p [j,1],...,y p [j,τ] ] for each distributor j, such that, if all its requirements are satisfied, then the conditional expected stockout service level is attained at the lowest possible costs. The components y p [j,t] of D p j represent the minimal quantities y satisfying the inequalities l =(l max) t s ( [ ] + ) p l ξ l [j,t] y P(ξ[j,t] > y) are such that they s j. (40) This allows the transformation of (39) into its deterministic equivalent (40), and results in the reformulation of the original problem into a mixed-integer stochastic program. 5. Computational results and validation 5.1. Test laboratory As a test bed for the proposed modeling and solution approaches, we use the data provided by one of the three largest North American chemical companies producing soda ash and calcium chloride. A one-year planning horizon decomposed into monthly time-periods (τ = 12) is considered. We use historical data to derive two probability distributions for the random demand. In the first one, the demand d[j, t] can take 5 (l max = 5) different levels at each time-period, and the probability distribution resembles the normal probability density function. In the second one, d[j,t] can take 10 levels, and the probability distribution is similar to the standard lognormal probability density function. For these two probability distributions, we consider three values 0.9, 0.95 and 0.97 for each service level.

21 21 The problem instances are modeled with the AMPL modeling language and solved with the CPLEX solver. We complement the CPLEX solver by a specific family of binary-integer cover inequalities [18] that are very appropriate to dealing with the distribution constraints (11) containing binary and general integer variables. The following statistics give an idea of the size of the problem; we have about 850 continuous variables, 500 general integer variables, 500 binary integer variables and 2200 constraints. The exact numbers vary with the type and prescribed value of the service level and the modeling approach. Due to confidentiality requirements, the numbers associated with costs and demand levels have been rescaled Probability of stockout In this section, we discuss the results obtained with the two modeling approaches for the cycle non-stockout service level, their computational difficulty and their managerial implications. We conduct a simulation study which allows the estimation of the true service level obtained with a cycle service level, a stagewise service level and an expected value approach respectively, and we conclude with a rolling horizon approach Computational tractability of cycle ready rate modeling approaches Entries in the left side of Table 3 report the value of the total costs for each combination of p and l max when we apply the intersection of events bounding approach. We were able to solve to optimality the 6 problem instances, which shows that this modeling approach is computationally tractable irrespectively of the specifics of the discrete probability distribution and the value of the prescribed cycle service level p. The total costs are a monotonically increasing function of the prescribed value for p. We now evaluate the p-efficiency approach. Applying the level search algorithm to identify the p-efficient demand trajectories, we reduce significantly the number of p-efficient demand trajectories, moving from an initial number of demand trajectory candidates (l max ) 12 to a number S p j. The number of p-efficient demand trajectories is a decreasing function of the value of p, and generally increases with the number of levels l max the demand can take (Table 1). It is important to note that the value of S p j does not increase exponentially with l max. The very different number of p-efficient demand trajectories per distributor is due to the non-stationarity (i.e., the specifics) of the distributor demands. p CLE DAR LIT MON MOR NEW OSH OSW OWE QUE SEP STE THU S p j l max = S p j l max = Table 1 Identification of p-efficient demand trajectories The disjunctive mixed-integer program (31) contains as many columns as the (large) number ( ) of supply chain p-efficient demand trajectories. Thus, it will be difficult to solve it to optimality in a reasonable j J S p j

22 22 amount of time. To evaluate the quality of the solution obtained with our approach, we compute the global integrality gap I ( ) I (m) = UB(m) LB LB, (41) where UB (m) is the value of the best feasible integer solution for (32) associated with the m th p-efficient demand trajectory and the lower bound LB is the optimal value of the linear problem min c T x subject to Ax b S p j T j x λ j,m DT p, j J j,(m) m=1, (42) S p j λ j,m = 1, j J m=1 λ j,m 0, j J,m = 1,..., S p j x R + which is the convexification of the continuous relaxation of (31). We now solve the disjunctive mixed-integer problem (31) with the congestion-relief column generation algorithm. The results obtained for l max = 5 and p = 95% are detailed in Table 2, where Si refers to ship i. Value of I ( ) t b Distributor Distribution objective function considered changes Iteration 1 $ % 2 Montreal 3 trips with S1 instead of 4 with S2 Iteration 2 $ % 3 Little 2 trips with S2 instead of 1 with S1 and 1 with S2 Iteration 3 $ % 3 Oshawa 3 trips with S1 and 2 with S2 instead of 4 with S1 and 1 with S2 Iteration 4 $ % 3 Thunder None Table 2 Operation of the column generation method. The congestion-relief column generation algorithm allows the fast (3 iterations) progressive reduction in the total costs (Column 2 of Table 2) of the supply chain, and in the value of the integrality gap I (Column 3). Column 4 reports the critical time-period t b. The modification in the transportation schedule for the Montreal distributor relieves the congestion at period 2 in such a way that, at the next iteration, the critical time period is the period 3. The following changes in the delivery schedule allow the progressive reduction in congestion at period 3, but leave it as the most congested one. Column 5 reports the terminal j whose distribution costs can be reduced through the selection of another p-efficient demand trajectory. Column 6 reports the changes in the distribution schedule of that terminal j. The distribution costs for Montreal, Little

23 23 and Oshawa can be successively reduced, through the selection of a more appropriate demand trajectory, without affecting the distribution costs associated with the other terminals. The recursive improvement in the objective value stops when it is observed that a reduction in the distribution costs for Thunder can only be attained by increasing the distribution costs of the other terminals. Although the successive reduction in I may seem minimal, the reduction in the total cost is substantial, since the total cost is a 7-digit number. The results displayed in Table 2 show the fast convergence of the algorithm that stops after very few iterations. Clearly, the proposed solution methodology performs well for constructing a probabilistic inventoryproduction-distribution model subject to cycle non-stockout service level. It eventually finds feasible integer solutions with a very low integrality gap (2.87%), while the optimality gap calculated by the CPLEX solver is lower than 0.1% at the last iteration. Similar results (right side of Table 3) extend to all problem instances, regardless of the value taken by p and l max. Table 3 Intersection of Events l max p-efficiency l max Costs Costs Costs I Costs I 90% $ $ % $ % $ % p 95% $ $ p 95% $ % $ % 97% $ $ % $ % $ % Results with intersection of events and p-efficiency approaches Conservativeness of approximation and risk aversion The comparison (Table 3) of the optimal solutions, respectively obtained with the p-efficiency and the intersection of events approaches, clearly shows that using the p-efficiency concept allows the obtaining of a significantly less costly solution. The reason is that the intersection of events approach is more constraining and gives an upper bound, not always very tight, on the optimal solution of (23). Indeed, as it can be inferred from constraints (24), the higher the dimensionality (value of τ) of the random variable, the looser is the bound provided by the intersection of events bounding scheme. To illustrate this, we consider the case in which l max = 5 and p = 0.95, and we deal with the 12- dimensional vectors of random variables. The cycle non-stockout service level p B j obtained with the intersection of events approach, is, for each distributor j, larger (Table 4) than the prescribed cycle service level p j = 0.95; it is at least equal to 0.96 and goes as high as 0.99 for about one third of the distributors. j CLE DAR LIT MON MOR NEW OSH OSW OWE QUE SEP STE THU p B j 99% 98% 96% 98% 98% 99% 99% 97% 99% 97% 96% 96% 99% Table 4 Cycle non-stockout service level obtained with the intersection of events bounding scheme Table 3 also shows that the total cost obtained for p = 0.90 (respectively p = 0.95) with the intersection of events approach is closer to the total cost obtained with the p-efficiency approach when p = 0.95 (p = 0.97)

24 24 than to that obtained with the p-efficiency approach when p = 0.90 (p = 0.95). Thus, using the p-efficiency approach, the 95% cycle non-stockout service level can be reached at a cost that would roughly be the same as that needed to reach a 90% cycle service level were we using the intersection of events approach. This shows that the intersection of events approach is very conservative and can be suited for very risk-averse managers. The conservativeness and the lower computational complexity, however, have to be traded for with the much higher incurred costs. The degree of conservativeness of the intersection of events approach does not only increase with the number of components of the random variable, but also with the magnitude of the cycle service level p. Indeed, the importance of the savings due to the p-efficiency approach increases with the magnitude of p. This is shown in Figure 3 that reports the ratio of the optimal solution (i.e., total costs) of the intersection of events approach to that of the p-efficiency approach. Clearly, this ratio is an increasing function of p. Figure 3 Ratio of total costs 5.3. Simulation study We conduct here a simulation study that provides an estimation of the cycle probability of not having a stockout obtained with four modeling approaches: IE: enforcement of a cycle service level p using the intersection of events approach (Section 3.1); P-EFF: enforcement of a cycle service level p using the p-efficiency approach (Section 3.2); IPC: enforcement of a stagewise service level p t at each period of the horizon: this consists of assuming that the planning horizon is composed of periods that are independent of each other, and is modeled using individual probabilistic constraints (43): min subject to c T x Ax b P(z[j,t 1]+m[j,t]+ d[j,t]) p j,t, t T,j J (43) x = [x x ] R + Z + Replacing d[j, t] by its uni-dimensional p-quantile, the problem above turns into a deterministic mixedinteger programming problem; EXP: replacement of the random demand by its expected value.

25 25 We now discuss the results of a simulation study comparing these four approaches. We assume that the random demand can take 5 different levels at each period, and set the stagewise p t,t = 1,...,τ and the cycle non-stockout service levels p are all equal to We determine the optimal replenishment plan for each approach, simulate 100 demand trajectories according to the above assumptions, and examine in how many cases per approach the supply chain is in stockout. Figure 4 Comparative study The chart on the left side of Figure 4 shows that the supply chain experience at least one stockout over the entire planning horizon in many scenarios (at least 42%, at most for 59%) with the replenishment plan associated with the expected value approach (EXP). This approach is clearly not viable, since it entails a high risk of violating the constraint of not having a stockout. The approach enforcing stagewise service levels (IPC) does not reflect the desired safety requirements either. It guarantees that, at each period, the probability of negative inventory level is below 5%. Yet, the probability of having a shortout at least once in the planning horizon varies between 10% and 27%, and remains much higher than the probability (5%) allowed by the prescribed cycle non-stockout service level. Both the p-efficiency and the intersection of events approaches allow the attainment of the prescribed cycle non-stockout service level. As underlined in the previous section, the intersection of events approach is more demanding than the p-efficiency one. The chart on the right side of Figure 4 displays the cumulative (up to period t) number of stockouts experienced by the Oshawa-based distributor for the four evaluated approaches. We can see that the cumulative number of stockouts increases fast (almost attaining its maximum) at the periods 4, 5, 6 when the demand reaches its peak and the risk of congestion is the highest. This simulation study illustrates the non-suitability of using individual probabilistic constraints and stagewise service level if one wants to have a small probability of stockout over the entire planning horizon, and the importance of focusing on the risk of congestion Rolling horizon approach A rolling horizon approach allows us to take advantage of favorable realizations of uncertain variables, and to amend the replenishment plan accordingly. We perform a simulation study, in which, using the p- efficiency approach, we design a new plan every 4 months. The first plan covers months 1 to 12 and only

26 26 the front end decisions (i.e., related to the first 4 months) are implemented. Then we construct a new oneyear plan covering months 5 to 12, and we implement its decisions for months 5 to 8, to finally construct a plan covering months 9 to 12. We repeat the experiment 100 times, to assess the service level attained by the method. We describe below the results obtained for the probability distribution in which the random variables can take 5 levels and for a non-stockout level of 95%. Since the constructed plans hedge against most (95%) possibilities of shortfall, the planning updates allow making the economical adjustements in order to take advantage of possibly lower realizations of the random demands. This translates into a slightly lower average number of shipments for certain locations (lower distribution costs), into lower average production costs, and into lower average total costs. However, since, on one hand, the level of the simulated demand is in most cases lower than that against which the plan hedges, and, on the other hand, this reduction is not necessarily sufficient to reduce the number (and thus the supplied amounts) of the deliveries (full load shipments) or to switch to a carrier with lower loading capacity, the average ending inventory levels do not always decrease. Table 5 below reports the ratio (R s ) of the average number of shipments in the simulation study to the number of shipments calculated initially, and the ratio (R e ) of the average ending inventory level in the simulation study to the ending inventory level initially planned by the method. CLE DAR LIT MON MOR NEW OSH OSW OWE QUE SEP STE THU MAN R s R e Table 5 Distributor shipments and inventory levels in the simulation study, relative to initially planned. The estimates of the true service levels (i.e., percentage of stockout in the 100 experiments) with our approach are given in Figure 5. The estimates of the expected cost and of the standard deviation of the cost are respectively equal to and Figure 5 Distributor service level estimates 5.5. Fill rate service level Entries in Table 6 report the total costs for each combination of p and l max. The computational tractability of the proposed approach is attested by the fact that the six problem instances were solved to optimality. The total costs associated with the fill rate service level (Table 6) are on average 5.72% smaller than those