On the equivalence of economic lot scheduling and switched production systems

On the equivalence of economic lot scheduling and switched production systems K.G.M. Jacobs, Dieter Armbruster,, Erjen Lefeber and J.E. Rooda January, 2 Abstract Scheduling production of several product types in one machine such that it satisfies known demands is known as the economic lot scheduling problem. Finding the schedule of a machine that can produce different types of products that optimizes a particular production measure (work in progress, throughput, cycle time) is known as the switched production problem. It is shown that the two problems are completely equivalent by reversing the flows and replacing outside demand by an infinite buffer of raw material. As an example, two results from switched production systems are applied to the economic lot scheduling problem: i) A feedback controller that allows recovery of a cyclic production schedule from two products on one machine from an arbitrary perturbation is derived. ii) It is shown that inventory refill policies for networks based on producing with maximal rate until an order up to point is reached can lead to starvation of the network and suboptimal behavior. An alternative policy that is optimal is also presented. Introduction The traditional approaches to the problem of scheduling production for a set of items in a factory or a whole supply chain is known as Economic Lot Scheduling Problems (ELSP) [6, 7, 5]. An ELSP addresses production scheduling of several part types, each with a known demand occurring at a constant rate, typically on a single machine. Production rates are known and either constant or controllable. Backorders are either not allowed or Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 53, NL-56 MB, Eindhoven, The Netherlands, email:k.g.m.jacobs@student.tue.nl, A.A.J.Lefeber@tue.nl Department of Mathematics and Statistics, Arizona State University Tempe, AZ 85287-84, USA, email: armbruster@asu.edu

incur backorder costs, holding costs are charged on average inventory levels and setup costs for switching between different products arise. A specific subset of ELSP determine optimal Common Cycle Schedules (also known as Rotation Schedules), restricting the problem to policies that cycle production through the items every T units of time. The objective is to minimize the sum of the average inventory holding costs, backorder costs and setup costs per unit time. A solution must specify the sequence in which parts are to be produced; the production quantities, or, equivalently, the productionrun durations; and the time at which each setup is to begin ([] for a good overview). A complementary approach to ELSP comes from job shop, flow shop or re-entrant production scheduling where parts arrive into the manufacturing system with certain (random) arrival rates and compete for available manufacturing capacity along their flow through the shop. We call this approach switched production systems. The goal of scheduling switched production systems is to improve performance measures such as mean cycle time, the variance of the cycle time or the amount of work in progress (WIP) in the system. ELSP is a form of inventory management and focusses on costs. Switched production systems focus on throughput as a surrogate for production costs per unit. The smallest examples of a switched production system and an inventory management system are depicted in Figure a and b, respectively. In inventory management systems, see Figure a, buffers are drained at a constant rate. A server can fill up the buffers one at a time. In switched production systems Figure b it is assumed that products enter the system at buffers at a constant rate. A server can take lots out of a buffers one at a time. Switching between serving two types takes setup time. Products leave the system after the server. Figure illustrates the main point of this paper: Economic lot scheduling problems and switched production problems are completely equivalent by reversing their flows and replacing outside demand by an infinite buffer of raw material. While it was obvious that both approaches to scheduling deal with similar problems (see eg. comments in [3]), the exact equivalence between the two approaches has so far escaped notice. As a consequence, results that are known in one field have not been transfered to the other. The current paper illustrates the importance of this equivalence by transfering two recent results on scheduling switched production systems by Lefeber et al [9,, 2] into the realm of inventory management: We will show that a feedback controller developed for switched production can be used to generate a feedback control system that recovers a 2

(a) Simplest cyclic inventory management system (b) Simplest switched production system Figure cyclic production schedule for two inventories from an arbitrary perturbation. We will show that using order up to policies where production is executed with a maximal possible rate may lead to starvation of inventories in certain supply chains. In section 2 we will develop a controller using Lyapunov s direct method to recover a cyclic production schedule from an arbitrary perturbation. The resulting controller is related to the dynamic PUT policy discussed by Eisenstein [3]. Extending Eisenstein s result, it is a state feedback controller, that can decide to terminate production at any inventory level, independent of a fixed order-up-to point. In particular the feedback controller may require other than maximal production rates, e.g. it may have a mode where production is reduced to the demand rate [, 4] (called a slow mode.in switched production systems). In addition, we will show that the feedback is robust (section 2.2) and performs well for the Stochastic Lot Scheduling Problem (section 2.3). In section 3 we will extend the ELSP to networks of inventories and the question of inventory refill policies and their scheduling. A famous system 3

dealing with re-entrant production is the Kumar-Seidman system [8]. The startling result of this system is that, although the production capacities of the machines are bigger than the combined influx rates, any policy that switches production when a queue has been emptied (a clearing policy) will always lead to an instability, i.e. a queue that grows unbounded. The reformulated K-S result in the ELSP context is that there are demand levels where, although the production capacities of the factories in the network exceed the demand, any PUT policy that replenishes inventories with maximal possible rates will starve some inventories and hence the supply chain will not be able to satisfy the demand. We will show that a cyclic schedule that contains slow modes will allow maximal demand rates. 2 Controller design via Lyapunov s direct method for a single server system We illustrate the impact of the equivalence between ELSP and switched production by deriving a feedback controller for periodic schedules of a two product inventory management system. 2. State space and dynamics We consider the ELSP for the system shown in Figure a. We assume that an optimal target schedule is given. It is worth commenting on the optimality of the target schedule: Given an period T and a production capacity that is high enough to satisfy the demand, a target schedule with minimal mean inventory may have idle times and one or more slow modes. Idle times can be forced due to setup times or may be part of the optimality condition of the target schedule. More than one slow mode and unforced idle times are an indication that there is slack production capacity that will allow us to recover from perturbations of the system away from the desired orbit. However given a production capacity, there is always a minimal period T in which the production can satisfy the demand. Any perturbation from that orbit can at best recover to the schedule with a phase shift. In the following, we assume that there is slack capacity in the system. We consider two lot types and two inventories x i,i =,2. The time evolution of the two inventories is described through a fluid model that continuously drains the inventories and refills them when the machine M produces the appropriate lot type. The maximal production rates are given as µ i,i =,2 and the drain rates are given as λ i,i =,2. The state of the system is characterized through the size of the inventories x i which are continuous state variables and the current mode m {, 2}. Negative buffers are allowed which will be treated as backorder that will be filled within the cyclic schedule. In the mode m = the machine produces type lots with 4

production rates u, u µ, in mode m = 2 it produces type 2 lots with production rates u 2, u 2 µ 2. Production rates can be zero, i.e. we allocate idle times to a production mode and without loss of generality we start a mode with the idling period. Finally, x is a continuous state variable that describes the elapsed time in the current mode. Hence the state of the system is fully described by X = (m,x,x,x 2 ) = (m,x,x) {,2} R + R 2. The input of the system describes the time evolution of the production rates u i (t),i =,2,(u,u 2 ) R 2 +. and the switches between the two production types u (t) where u takes the values or 2. The switching dynamics is given by the rule: If u (t) m(t) then change m(t) and reset the phase x (t) to zero. The dynamics of the buffers are given as a fluid model: ẋ (t) = () ẋ (t) = u (t) λ, (2) ẋ 2 (t) = u 2 (t) λ 2. (3) Furthermore, at each time instant the input (u,u 2 ) is subject to the following constraints: u µ, u 2 = if m = u = u 2 µ if m = 2 (4) i.e. the server can only produce one type of lot. Any time a production rate changes from zero to a nonzero value, the state variable m is updated to the current production type. 2.2 Controller Consider a given periodic orbit (m (t),x (t),x (t),x 2 (t)) and assume the orbit can be written as (x (m,x ),x 2 (m,x )). The set of values of m and x for the desired orbit is denoted by D describing the total time (idling and production) in each mode. Definition. For an arbitrary point X = (m,x,x,x 2 ) D we define If x D then V (X) = x x (m,x ) µ + x 2 x 2 (m,x ) µ 2 V (,,x,x 2 ) = x x (2,) + x 2 x 2 (2,) µ µ 2 V (2,,x,x 2 ) = x x (,) + x 2 x 2 (,) µ µ 2 5

V measures the difference between the scaled inventory at the current point and the inventory at the point in the cycle where the system should be. If x D then the system has been in a specific mode for too long and it should immediately switch modes. Hence the difference is with the inventory at the starting point of the new mode. Lemma. The scalar function V as defined in Definition is a Lyapunov function candidate. Proof. Clearly V (X) and V (X) = if and only if X is at the desired periodic orbit. In addition, V is obviously positive definite. Figure 2: Trajectory of the target periodic orbit associated with the system given in Equation 8 Given a specific target periodic orbit we can now choose an input function in such a way as to minimize V (X) over the set of allowed inputs. Assume that initially, at t =, the server is in a state X = (m,x,x,x 2 ) D corresponding to a uniquely defined point (x (m,x ),x 2 (m,x )) on which the system should be on the desired periodic orbit. Associated with the current state and this desired point is a value V (X) of the Lyapunov function. The Lyapunov function does not change, if the system and the target trajectory are in the same mode (idling or producing) since the difference 6

between x i (t) and x i (t) stays constant for all i. The Lyapunov function can be reduced however by reducing or enlarging the idle time. Synchronization will be maintained by keeping the schedule for the mode switching intact, adjusting the idle times. Reducing the idle time is appropriate, if a buffer is empty and the other buffer is higher than the desired buffer level. In that case we can keep the empty buffer empty by supplying an input that is equal to the demand (a slow mode). Prolonging the idle time is appropriate if at the time to switch to producing product i, buffer x i it not empty. In that case, production is delayed until the inventory x i = x i. Hence the resulting feedback control rule for a state X = (m,x,x,x 2 ) D can be stated very simply: Continue in the current mode of the system until the next scheduled switching event. If you are in the idle time interval for mode i and buffer x i runs empty before going into production mode maintain the empty buffer by supplying an input that is equal to the demand until the next scheduled production time. If you are at the time to switch to production of product i and x i, then delay switching until x i = x i. If you are in the production time interval of mode i and x i > x i, then idle until x i = x i otherwise produce with maximal rate until the next switching time. It only remains to discuss what to do if the state X / D. If at any point in time the buffer contents is negative, i.e. backlog occurs, this should directly be compensated by serving at full rate. If the inventories are too high, the system should immediately switch to the desired production schedule and drain the inventories until X = (m,x,x,x 2 ) D. This input schedule reduces the Lyapunov function and leads to convergence in state space, i.e. convergence to the target schedule with zero phaseshift and zero backlog (see [9] for a formal proof). Note that Eisenstein s Dynamic PUT policy [3] also recovers the signal without phase shift, if the token velocity v i = µ i for i {,2}. However, dynamic PUT policies create backlogs instead of using a slow mode. We illustrate our development of a state space controller with the following example. Example : We assume a demand rate for product of λ = lots/sec, λ 2 = 2 and production capacities of µ = 6 and µ 2 = 2. The desired periodic orbit is shown in Figure 2 in the (x,x 2 ) phase plane. Figure 3a) shows a perturbed trajectory in the inventory phase space and Figure 3b) shows the associated time signals of the inventories and the recovery of the periodic orbit. The starting point corresponds to inventories that are too high in x and too low in x 2. The control policy is to stay in mode m = 2 draining both inventories until x 2 is zero, to switch to a slow mode until the regular time to start production for product 2 and to continue with a regular switches to m =. At that time x 2 is on target but x is too high. Idling longer than ususal until the target trajectory is reached completes the 7

4 25 x2 [lots] 35 3 25 2 5 5 5 5 2 25 x [lots] x [lots] x2 [lots] 2 5 5 5 5 2 25 4 3 2 t [sec] Recovery Lyapunov Desired trajectory 5 5 2 25 t [sec] Figure 3: Target trajectory and recovery in phase space (left) and as time signals for the inventories x (t) and x 2 (t) (right). Note the slow mode at t = 3..6 where x 2 (t) stays zero thus avoiding backlog. recovery. The equations for the periodic orbit, the Lyapunov function candidate and the feedback control on the input signal are given in the Appendix. 2.3 Stochasticity Having a state feedback controller is the first step to be able to systematically deal with stochastic perturbations of the demands for the different products. Figure 4 shows a sample run. We assume that both buffers are drained with demand sampled daily in an iid fashion from a Γ-distribution with mean of lot lot day and a variance of.25 for buffer and 2day and a variance of.5 for buffer 2, respectively. We also add an intial perturbation starting buffer much higher than on schedule and starting buffer 2 with a backlog. Figures 4a and b) show a sample path for the two drain rates on a time interval of 2 days. Figures 4c and d) show a desired schedule with significant idle times and the actual buffer levels for a controlled system run where the drain rates vary according to the sample in Figure 4a and b). We see that i) the 8

initial perturbation can easily be absorbed, and ii) the controller is able to maintain the schedule. In particular, after the initial perturbation has been absorbed, peak inventory occurs always at the exact time of the schedule. Figure 5 a and b) show the behavior of the controller and the inventories for a different sample path of the drain rates and for a schedule that has no idle times but two slow modes, a short one for buffer and a longer one for buffer 2. The advantage of such an orbit is that the total amount of inventory is much less (average inventory is 9.6) for this orbit than for the previous target orbit (average inventory 3). Again, the controller can deal with the very significant initial perturbations and with the ongoing stochasticity in the drain rates. However, the fit to the original schedule is not quite as good as before and there are occasions where the system creates small backogs during the slow modes. A more detailed analysis of the behavior of the controller for stochastic drain rates requires the use of filters. Clearly, for high frequency variations, the controller should not follow the noise but perform some sort of smoothing. 3 A re-entrant supply chain The second transfer of results based on the equivalence between ELSP and switched production systems relates re-entrant production and inventory management in supply chains. Produce up to policies (PUT) in its various guises (static - fixed or ignore [5] or dynamic [3]) can be shown to be optimal for single inventories under certain circumstances. They are characterized by events that trigger production runs using maximal production capacity to refill inventories from zero up to a specific predetermined level. It is therefore tempting to try to extend them to schedule inventory replenishment for production networks with more than one inventory. We will show by a specific example of a reentrant supply chain, that production switches between maximal production rates of different lots, triggered by the drainage of an inventory to specific predetermined level (typically zero) may lead to a drastically reduced overall capacity of the network and hence to suboptimal production cycles. We will show that optimal production cycles can be found containing slow modes and idle times. Furthermore, we will show that there exist robust recovery policies that stablize those optimal cycles under perturbations. Figure 6 shows the inventory management system equivalent to the Kumar-Seidman (K-S) system [8] in switched productions. The system serves lots of one type, which first visit server, then server 2, then server 2 again and finally server again. The successive buffers are denoted by x i, i {,2,3,4}. A constant demand rate λ > is assumed at buffer x 4, while the maximal processing rates at the machines to refill the buffers are 9

2.5 4 3.5 2 3 λ(t) [lot/sec].5 λ2(t) [lot/sec] 2.5 2.5.5.5 35 3 2 4 6 8 2 4 6 8 2 time [sec] (a) J = 3 DES Simulation Schedule 2 4 6 8 2 4 6 8 2 4 time [sec] (b) 3 25 2 2 x [lots] 5 x2 [lots] 5 2 3 5 2 4 6 8 2 4 6 8 2 time [sec] (c) 2 4 6 8 2 4 6 8 2 time [sec] (d) Figure 4: a) and b) shows a sample path for the drain rates of λ and λ 2, c) and d) show a target schedule and time evolution of the inventories under the control action. µ >, µ 2 >, µ 3 >, and µ 4 >. 3. Maximal rate PUT policies reduce capacities for a supply chain We are interested in periodic schedules for a given period T and define a Maximal Rate PUT policy (MR-PUT) Definition 2. Maximal Rate Produce-Up-To (PUT) policy. Whenever a server is serving parts in a particular buffer, it continues serving at maximal rate until it has produced λt lots. Next, it switches to one of the buffers that is empty (or has a negative value in case of buffer x 4 ). In one period, all buffers are served exactly once.

35 J = 9.6 DES Simulation Schedule 2 3 5 25 5 2 x [lots] 5 x2 [lots] 5 5 5 2 5 2 4 6 8 2 4 6 8 2 time [sec] 25 2 4 6 8 2 4 6 8 2 time [sec] (a) (b) Figure 5: a) and b) shows a target schedule with two slow modes and the time evolution of the inventories under the control action. The drain rates λ i were chosen stochastically with the same statistical properties as in Figure 4 but with a different sample The standard definition of a PUT policy is only concerned with the buffer levels that should be attained and not in the way to get there. However, almost always, they are implictly using an MR-PUT policy. We will show Figure 6: Flow in a re-entrant supply chain that MR-PUT policies cannot satisfy certain demands even though there should be enough production capacity. Define ρ i = λ µ i. Assume that every machine by itself has enough production capacity to satisfy the demand, i.e. ρ + ρ 4, ρ 2 + ρ 3. (5) Consider in addition µ < µ 2 and µ 3 < µ 4, i.e. ρ > ρ 2 and ρ 3 > ρ 4 and assume that initially, the system is in state x(t ) := (x (t ),x 2 (t ),x 3 (t ),x 4 (t )) = (,,,ξ). (6)

During [t,t ] server M serves type at full rate, µ, and server M 2 serves type 2 at reduced rate, µ, since µ < µ 2. At t = λt µ both servers M and M 2 ( have processed λt ) lots. This brings the system into the state x(t ) =,λt,,ξ λt µ λ. Since both M and M 2 have processed exactly λt, the servers can switch to serving type 4 and type 3, respectively. During [t,t 2 ], M 2 serves type 3 at full rate, µ 3, and M serves type 4 at reduced rate, µ 3, since µ 3 < µ 4. At t 2 = t + λt µ 3 = (ρ +ρ 3 )T both servers have produced λt lots and buffers x, x 2 and x 3 are empty, i.e. they are in same state as initially. Buffer x 4 contains x 4 (t )+θλ, with θ = T t 2 = T( (ρ +ρ 3 )). If θ, i.e. ρ + ρ 3, the supply chain with this policy satisfies the demand λ. The system idles for θ units of time until t 3 = T. During idling exactly θλ products are drained out of buffer x 4, thus bringing the system back into the intial state (Equation 6). In case θ >, i.e. ρ + ρ 3 >, the system cannot satify the demand λ. At t 2 > T, all buffers are in the same state as the initial one, except for buffer x 4 which is smaller than at t =, x 4 (t 2 ) x 4 (t ) <. In order to produce λt lots, we need a period longer than T. Even though we are draining only the amount allowed during a cycle of length T, the system is drained of its WIP at the end of the cycle. Hence, repeated iteration with a cycle length of t 2 will lead to starvation of the final inventory and hence to lower output. Notice that the instability can also be viewed as an increase in the period of production which will iterate to infinity. The other cases for the relationships between µ and µ 2 and µ 3 and µ 4 can be discussed in a similar way. Overall we get the following theorem: Theorem. Consider the K-S supply chain with an MR-PUT inventory policy and a fixed drain rate λ. Then demand that can be satisfied in a stable way has to obey the capacity limits of any one of the individual machines serving inventory one and four and two and three respectively, and the capacity limits of the virtual machines [2] relating step two and four and step one and three respectively, i.e. λ min ( µ µ 4 µ + µ 4, µ 2 µ 3 µ 2 + µ 3, 3.2 A policy that minimizes WIP µ 2 µ 4 µ 2 + µ 4, µ µ 3 µ + µ 3 The Kumar Seidman problem for switched production has been studied in [] and a schedule that minimizes WIP J = lim sup t t t x (τ) + x 2 (τ) + x 3 (τ) + x 4 (τ)dτ has been found. The transfer to the ELSP is straightforward and leads to scheduling policies with minimal inventory which satisfy demand up to the 2 ).

capacity limit of the real machines. The resulting schedule involves idling and slow modes as shown in the following example. Example 2 Consider ρ =.6, ρ 2 =.4, ρ 3 =.5, ρ 4 = 3, λ = 4 [lots/s], and T = [s] and an initial buffer content (x,x 2,x 3,x 4 ) = (τ µ 2 (µ 2 µ ) τ2 µ,,,τ µ λ). Then the production modes and their swithing times for a WIP-optimal cycle are given through the D = (m,x x 5 if m = (,2),x,x 2,x 3,x 4 ) x if m = (,3). x 4 if m = (4,3) where m = (i,j) indicates that machine M serves inventory i and machine M 2 serves inventory j. The related time evolution of the inventories is given by m = (,2) and x : { x = 66 2 3 + 62 3 x x 2 = x 3 = x 4 = 24 4x m = (,2) and x 5 : { x = 33 3 3 3 (x ) x 2 = (x ) x 3 = x 4 = 24 4x m = (,3) and x : { x = 6 2 3 x x 2 = 4 8x x 3 = 8x x 4 =. m = (4,3) and x : { x = 66 2 3 x 2 = 32 8x x 3 = 8 + 4x x 4 = m = (4,3) and x 4 : { x = 66 2 3 x 2 = 32 8x x 3 = 2 4(x ) x 4 = 8(x ) (7) Figure 7 gives a graphical represention of the inventories. Notice that for the first timesteps machine M 2 is idling and that for 6 t 8 machine M serves in slow mode producing just enough to meet the external demand λ. 4 Conclusion We have pointed out the equivalence of ELSP problems and switched production systems. We have illustrated the usefulnes of this observation by deriving a feedback controller for a cyclic production sytem that allows the 3

x x 4 2 Inventory [lots] 5 5 2 4 6 8 t[s] (a) Inventory levels for the inventories refilled by machine M in example 2 as a function of time: x (t) (dashed-dotted) and x 4(t) (full line). 4 35 x 2 x 3 3 Inventory [lots] 25 2 5 5 2 4 6 8 t[s] (b) Inventory levels for the inventories refilled by machine M 2 in example 2 as a function of time: x 2(t) (dotted) and x 3(t) (dashed). Figure 7 4

schedule to recover from arbitrary perturbations. In addition we have discussed the implications of the Kumar-Seidman problem of switched production systems for the refill policies of production and inventory networks. We show that refill policies based on order up to points and maximal production rates to achieve those inventory levels may lead to suboptimal throughput for the network. In addition, we showed that we can recover optimal behavior using a slow mode production. Generalizing from this specifc example we propose that any approach to scheduling inventory refills in topologically complicated supply networks that restricts the inventory refill to MR-PUT policies or similar policies based on the maximal possible refill rates is highly likely to lead to suboptimal responses of the supply chain. Acknowledgement We would like to thank Esma Gel for introducing us to the Eisenstein paper [3]. D.A s research was partially supported by NSF grant DMS-64986 and a grant from the Stiftung Volkswagenwerk through the program on complex networks. 5 Appendix 5. Periodic orbit for Example The desired target periodic orbit is given as for m = : { x = x x 2 = 4 2x if x, x = 5(x ) x 2 = 4 2x if x 4, for m = 2 : { x = 2 x x 2 = 2 2x if x 6, x = 2 x x 2 = (x 6) if 6 x. (8) The set of values of m and x for the desired orbit is given by { D = (m,x,x,x 2 ) x } 4 if m =, x if m = 2. 5

5.2 A Lyapunov function candidate for Example The Lyapunov function candidate for Example is given by: 6 x x + 2 4 2x x 2 if m = x, 6 5(x ) x + 2 4 2x x 2 if m = x 4, V = 6 2 x + 2 2 x 2 if m = x > 4, 6 2 x x + 2 2 2x x 2 if m = 2 x 6, 6 2 x x + 2 (x 6) x 2 if m = 2 6 x, 6 x + 2 4 x 2 if m = 2 x >. 5.3 The input signal for a feedback controller for Example (,,) if m =, x >, x <, (,λ,) if m =, x =, x <, (,µ,) if m =, x <, x < 4, (,,) if m =, x < x, x < 4, (,µ,) if m =, x x, x < 4, (2,,) if m =, x (u,u,u 2 ) = 4, () (2,,) if m = 2, x 2 >, x < 6, (2,,λ 2 ) if m = 2, x 2 =, x < 6, (2,,µ 2 ) if m = 2, x 2 <, x < (2,,) if m = 2, x 2 < x 2, 6 x <, (2,,µ 2 ) if m = 2, x 2 x 2, 6 x <, (,,) if m = 2, x, References [] Karla E. Bourland and Candace A. Yano. A comparison of solution approaches for the fixed-sequence economic lot scheduling problem. IIE Transactions (Institute of Industrial Engineers) 29 (2), 3-8, 997. [2] J. G. Dai and J. H. Vande Vate. The stability of two-station multi-type fluid networks. Operations Research, Vol 48, 72-744, 2. [3] D. Eisenstein. Recovering cyclic schedules using dynamic produce-up-to policies. Operations Research 53 (4), 675-688, 25. [4] Mohsen Elhafsi and Sherman X. Bai. The Common Cycle Economic Lot Scheduling Problem with Backorders: Benefits of Controllable Production Rates. Journal of Global Optimization : 283-33, 997. [5] G. Gallego. Scheduling the Production of Several Items with Random Demands in a Single Facility. Management Sci., 36, 579 592, 99. (9) 6

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