Economic lot scheduling with uncontrolled co-production

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European Journal of Operational Research 188 (2008) 793–810 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Economic lot scheduling with uncontrolled co-production ¨ ner, Taner Bilgic¸ Selma O

q

*

Bog˘azic¸i University, Department of Industrial Engineering, Bebek, Istanbul 34342, Turkey Received 30 October 2006; accepted 9 May 2007 Available online 18 May 2007

Abstract The aim of this paper is to analyze the effects of uncontrolled co-production on the production planning and lot scheduling of multiple products. Co-production occurs when a proportion of a certain production comes out as another product. This is typical in the process industry where quality and process specifications can lead to diversified products. We assume that there is no demand substitution and each product has its own market. Furthermore, we assume that co-production cannot be controlled due to technical and/or cost considerations. We introduce two models that extend the common cycle economic lot scheduling (ELSP) setting to include uncontrolled co-production. In the first model we do not allow for shortages and derive the optimal cycle time expression. In the second model, we allow for planned backorders and characterize the optimal solution in closed form. We provide a numerical study to gain insight about co-production. It seems that the cycle time increases with co-production rate and utilization of the system. The effect of co-production on long-term average cost does not exhibit a certain characteristic.  2007 Elsevier B.V. All rights reserved. Keywords: Production; Scheduling; Uncontrolled co-production

1. Introduction There are manufacturing environments where some processes are not well understood and/or not exactly under control. In such systems, while producing a certain product other products might also be generated due to some physical or chemical phenomena in the manufacturing process. This is called co-production. Co-production is encountered in both low and high technology production environments. Typical examples are glass and semiconductor industries. In some cases, co-production is desirable and is planned for (e.g., power co-generation at an industrial site). In this paper, we study the effects of (uncontrolled) co-production on the production schedules of multiple products whose demands are not substitutable. When the effect of co-production is not accounted for, there could be excess inventories or unexpected shortages. We assume that co-production is either not controllable or too expensive to control.

q *

This work has been supported by Bog˘azic¸i University Research Fund under Grant 01A302. Corresponding author. Fax: +90 212 265 1800. E-mail address: [email protected] (T. Bilgic¸).

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.016

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A typical example of this situation arises in glass manufacturing. Continuous flow of glass is cut to different sizes. Each size is an independent product with its own market. The ‘‘quality’’ (which is measured as the number of imperfections per unit area) is better at the beginning of the process and therefore large size products are produced earlier. As the process continues, the quality deteriorates further but one can compensate for it by cutting smaller size glass with the help of an automated vision system to detect where imperfections are. Therefore, while producing a certain product (large size glass), some of it comes out as another product. A depiction of this is shown in Fig. 1. Note that as shown in Fig. 1, when the setup is performed, the cutting can be of two types: size of the current product in production and one size smaller. To cut smaller sizes one needs to wait until the next setup. In the figure, product A is the largest size product and during its production only product B can be co-produced. Similarly, while product B is being produced, only product C can be co-produced. There is a point where the line is shut down and the process is re-calibrated to start all over again. But this setup is exceedingly expensive and is avoided as much as possible. Note that each size of glass has an independent market and the demand need not be substitutable. Although it is possible that the process deteriorates with time the yield of smaller size products might be higher than that of the larger size products. Furthermore, smaller size products are produced towards the end of the process where imperfections are higher in density, it is not reasonable to assume that the demand for smaller size products is lower than that of the larger size products. In the literature there are studies on imperfect production processes where the process shifts from the incontrol state to the out-of control state and begins to produce defective products or scrap. There are also models that study demand substitution and imperfect production processes. But co-production has remained unrecognized by the academic community at large. Only a few papers consider co-production and random yield which are relevant to this paper. But in all of these demand is substitutable. This particular problem arises in the semi-conductor industry. The underlying assumption is that there is a serially nested product structure which implies that the demand for a given product can be satisfied with products that fall into the category of the product or higher level products in the hierarchy. Deuermeyer and Pierskalla (1978) propose an optimal control model to minimize the costs of production, inventory holding and backorders in a two-product system for which two production processes are available. Process one produces both products A and B; while process two can only produce product B. They show that depending on the current inventory positions it can be optimal to produce A or produce B or produce both. Ou and Wein (1995) examine a case in which a family of products are serially ordered in terms of quality. For each product, there is a process which yields both that product and those of lower quality as by-products. The yields are assumed to be random. They derive scheduling policies from the exact solution to a Brownian motion control model of the production and inventory system. Bitran and Dasu (1992) consider a situation in which a single process has output with different grades of quality, in which higher grades of output can be used to satisfy the demands for lower grades of the output. The fraction of output is random. The objective is to maximize the expected profit. The problem is developed as an infinite-horizon stochastic program, but because of the computational considerations, they use simpler approximate procedures. Since the demand in their case is substitutable, they also consider finding the best downgrading policy. Bitran and Gilbert (1994) study a version of the same problem where the objective is Flow of Glass

Imperfections in production

A

B B

A

A

B B

Production time for A

A

Setup

B B B

C C C C C B B Setup C C C C C

Production time for B

Fig. 1. An example of co-production in the glass industry.

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to minimize the expected cost comprised of production, inventory holding, and shortage costs. Alternative lot sizing policies are considered, from simple to those that consider the impact of downgrading and production smoothing under different downgrading policies. The models in this paper are multi-item models where all of the items are to be produced on a shared facility. There are significant setup times and setup costs that cannot be neglected in modelling. The production plan is cyclical. Our study differs from the literature in several aspects: (i) co-production is not scrap, (ii) there is a hierarchy among the products but the hierarchy is not only in the quality aspect, (iii) a customer demanding a certain product will not be happy with another product from higher levels because every product has its own market and demand is not substitutable. The policy must balance the trade-off between constant and variable costs of production, inventory holding and shortage. We base our model on the Economic Lot Scheduling Problem (ELSP) where demand is a known deterministic rate and the objective is to schedule production of multiple products on a single facility that incurs significant change-over costs and times (Bomberger, 1966; Delporte and Thomas, 1977; Doll and Whybark, 1973; Elmagrabhy, 1978). In ELSP, the facility produces to meet the demand in the cheapest possible way with respect to the trade-off between setup and inventory costs. For two or more products, a schedule called the independent solution (IS) can be constructed by scheduling each product according to its economic order quantity (EOQ) solution independent of other products. However, this is not necessarily feasible. Even in the two-product IS schedule, the problem of checking feasibility is quite complicated (Vemuganti, 1977). Furthermore, checking feasibility of the N-product IS schedule is NP-complete. Hence, the general ELSP is NP-hard (Hsu, 1983). Most of the attention has been on cyclic schedules where a production sequence repeats itself indefinitely. A simple scheduling policy which guarantees feasibility is the Common Cycle (CC)policy. Unlike the IS schedule, the solution is always feasible. The CC schedule simply schedules one lot of each product in a time interval called the ‘‘Common Cycle’’ and repeats itself. The best common cycle time is the minimum cost common cycle time, but if it is not long enough to accommodate the setup times, feasibility requires it to be increased to a sufficient level. Although CC schedule is always feasible, it is not necessarily optimal. However, when the ratio of setup cost to holding cost is equal for each item the CC schedule is optimal. When these ratios are close to each other, the CC schedule is also close to the optimal (Jones and Inman, 1989). Gallego and Roundy (1992) introduce backorders and related costs to the ELSP where inventory and the backorder costs are charged as a linear time weighted average. The resulting model yields less average cost depending on the level of backorder cost relative to holding cost. Furthermore, ELSP with backorders is used in a problem of scheduling production in real time when demands are random or the facility is subject to sudden failures but they do not consider co-production. Silver and Schweitzer (1983) take up the problem of scheduling multi-products on a facility with controllable production rates and backorders. The motivation for studying such a facility is twofold: first, improving the utilization of the bottleneck facility may be equivalent to the efficiency improvement of the flow line; second, the single facility system is a starting point for the study of the multi-stage flow lines. They also concentrate on a common cycle policy to achieve analytical results. In our setting, we do not have control on co-production. Zipkin (1991) postulates that the optimal schedule of a large percentage of randomly generated problems are cyclic common schedules. Then he treats production cycle time, the maximum inventory level, and the production rate for each product as decision variables. He shows that an optimal schedule characterized by these decision variables for each product improves the solutions provided by ELSP. We also use cyclic schedules in our models but we assume co-production cannot be controlled. Alle et al. (2004) consider ELSP with performance decay and formulate this problem as a non-linear mixed integer programming problem and then show a way of linearizing it. There is no co-production assumption. Cooke et al. (2004) also give a mixed integer linear programming formulation for ELSP. Coupled with an efficient sequencing heuristic, they show optimal schedules can be obtained most of the time. Lin et al. (2006) considers ELSP where items produced are subject to continuous deterioration. The deterioration is characterized by an exponential distribution. Under the common cycle assumption where no shortages are allowed a near optimal production cycle time is suggested.

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There has been considerable amount of effort spent on managing production systems with (random) yield. In the co-production setting, there is a yield problem but the extra production has its own market (i.e., demand is not substitutable). Furthermore most of the random yield literature focuses on one-stage, one-product models (Yano and Lee, 1995). Another related line of research is production models with imperfect production processes where product quality is a function of the state of the production process. When the process is in in-control state, the items produced are conforming, when the process deteriorates from the ‘‘in-control’’ state to the ‘‘out-of-control’’ state; the items produced are non-conforming or of substandard quality. The relationship between the lot size and the quality is significant. A larger lot requires a longer production cycle; so that it is more likely to contain more defective items. Majority of the models in this literature are also single item models and they explicitly model the process of deterioration. Porteus (1986) and Lee and Rosenblatt (1985) incorporate the effects of imperfect production processes into the basic EOQ model. Former paper assumes that the time-to-failure distribution is geometric. Whereas in the latter paper, time-to-failure is assumed to be exponentially distributed. The optimal production cycle is derived, is shown to be shorter than that of the EOQ model. The analysis is extended where the deterioration rate is a function of the set-up cost, for which the setup cost level and the production cycle time are jointly optimized. Linear, exponential, and multi-state deteriorating processes are studied. The defective items are eventually reworked, replaced, or passed onto the customer. The critical point is that the process is not restored until the next setup. For multi-state deterioration, the paper considers the case where there are many states in the system, corresponding to different levels of defective rates during the production cycle. The duration in each state before the process shifts to another one is random. Daya and Hariga (2000) use the classical ELSP common cycle approach to model the multi-item scheduling with a deteriorating production process. The deterioration process is modelled by an exponential distribution. They consider models with restoration and where the restoration cost depends on detection of the delay. The inventory related costs are evaluated by an approximation. Our model also utilizes the common cycle approach but we assume that affecting co-production is not possible or is too expensive. Motivated by the semi-conductor industry, Lee (1992) incorporates the time required for rework explicitly in determining the lot size for wafer fabrication. Wafer probe process starts in the in-control state and then deteriorates to the out-of-control state with geometric distribution. A closed-form expression for the optimal batch size is derived. Of interest is the observation that the fixed rework times tend to favor small lots. We use a continuous review inventory model where both the demand and the co-product rate are deterministic. We restrict attention to cyclic schedules and particularly to common cycle schedules. We explicitly include setup times and costs in the model. We assume that there is a product hierarchy in terms of an attribute (like size or quality) and each product category has its own price and demand. Thus, there is no demand substitution. The production sequence is also determined along this hierarchy. We call our models ‘‘asymmetric co-product’’ ELSP models to emphasize the fact that a product with a higher place in the hierarchy can only cause the co-production of a product which is lower in the hierarchy and not vice versa. The rest of the paper is organized as follows. In Section 2.1, we formulate the model where shortages are not allowed. We derive the optimal common cycle time and show that it corresponds the ELSP common cycle when there is no co-production. In Section 2.2, we extend the model to allow for planned backorders. We obtain optimal common cycle and backorder levels for two-items and discuss the limiting cases when shortage costs are too high and when co-production is not allowed. In Section 3, we present numerical illustrations and in Section 4 we conclude. All proofs are given separately in Appendix. 2. Modelling We work with a continuous review inventory model where the demand is deterministic and it is constant. There is a constant holding cost rate for carrying inventory of each product and a fixed, sequence-independent setup cost. There is a fixed and known co-product rate for each product denoted by qi ð0 6 qi < 1; i ¼ 1; . . . ; N Þ, i.e., when producing item i, a fixed proportion, captured by qi , of that production comes out as item i + 1. Hence, the effective production for item i is only proportional to ð1  qi Þ.

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The deterministic co-product rate may be considered as an average value from historical data. We restrict attention to cyclic schedules, i.e., within a certain cycle time each product is produced in the same sequence. Among cyclic schedules we restrict attention to a special class called the common cycle schedules where each product is produced exactly once in every cycle. We also assume that the sequence of production is fixed (from large size products towards smaller sizes as in Fig. 1). This is dictated by the deterioration of quality as the process goes on. Whereas in general, choosing the production sequence can also result in benefits in terms of holding costs. We do not consider the sequencing issue in this paper. The problem is to determine a production schedule, a complete specification of which items are to be produced, when, and in what quantities. The objective is to minimize the long-run average cost which is composed of setup, inventory holding, and production costs. Table 1 shows the parameters and decision variables of the model. All cost parameters are constant and independent of the cycle time. The determination of the optimal policy relies on the following assumptions: Assumption 1. P i ð1  qi Þ > Di ; Assumption 2.

D1 P 1 ð1q1 Þ

þ

i ¼ 1; . . . ; N .

PN

Di i¼2 P i ð1qi ÞþP i1 qi1

Assumption 3. Diþ1  P i qi > 0;

6 1.

i ¼ 1; . . . ; N .

Assumption 4. All problem parameters, ðDi ; P i ; Ai ; C i ; hi ; S i ; qi Þ are non-negative. Assumption 5. 0 6 qi < 1;

i ¼ 1; . . . ; N .

Assumption 6. q0 ¼ qN ¼ 0. Assumption 1 implies that the production of product i will be intermittent (in lots). Assumption 2 indicates that the facility has enough capacity to meet all the demand for all products. Assumption 3 implies that the demand of the next product in sequence is always greater than the supply coming as co-production. Otherwise, the demand for the next product will be trivially met by the co-production from the previous product. Assumption 4 indicates that no cost can be negative. Assumption 5 prohibits the extreme case of having a coproduct rate of 1 in which case there is no way to meet the demand of that product. Without loss of generality, Assumption 6 normalizes the co-product rates from the first and the last product. It is true that all processes produce scrap at some point. In fact both during the setups and during co-production some portion of the production may be scrapped. But in some production processes (e.g. glass production) this scrap can be recycled. So we do not consider scraps within the scope of the model to isolate the effects of co-production only. Therefore, in the co-production setting q = 0 makes reasonable sense. It means a process state with no co-production (which might still produce scrap).

Table 1 Parameters and decision variables i

Index of products, i ¼ 1; . . . ; N

Di Pi Ai ci hi

Demand rate in units per year for product i Production rate in units per year for product i Fixed cost of setup per one cycle for product i Unit variable cost of production for product i Inventory carrying cost per unit per year, usually expressed as hi ¼ rci , where r is the annual inventory carrying cost rate for product i Sequence-independent setup time per cycle for product i Cycle length (in years) (decision variable) The production time for product i (decision variable) Minimum feasible cycle time Co-product rate of product i + 1

Si T Ti T min qi

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2.1. Model without shortage In this section, we outline the basic model where no shortages are allowed. We assume there are N (N > 1) independent products. The schematic illustration of the model is given in Fig. 2. There are some structural constraints to be satisfied in order for the solution to be feasible. The total cycle time T cannot be smaller than the sum of individual cycle times and setup times: T P

N X

ðT i þ S i Þ:

i¼1

The total demand of product i must be met with the total supply of product i during the cycle: TDi ¼ T i P i ð1  qi Þ þ T i1 P i1 qi1 ;

i ¼ 1; . . . ; N :

ð1Þ

Hence by using (1) total production length of product i is given as T Ti ¼ P i ð1  qi Þ

Di 

qi1 Di1 ð1q i1 Þ

þ



i2 qi1 Di2 ð1qqi2 Þð1qi1 Þ



i1

þ    þ ð1Þ

! Qi1 q j¼1 j D1 Qi1 : j¼1

ð1qj Þ

which can be rewritten as ! i1 i1 Y X qj T k1 Di þ Ti ¼ ð1Þ Dk ; P i ð1  qi Þ ð1  qj Þ j¼k k¼1

i ¼ 1; . . . ; N :

ð2Þ

Maximum inventory position of product i denoted by Ii is given as I i ¼ T i ðP i ð1  qi Þ  Di Þ;

i ¼ 1; . . . ; N :

We can derive the maximum inventory position by substituting Ti from (2) as   Di I i ¼ T ni 1  ; i ¼ 1; . . . ; N ; P i ð1  qi Þ

ð3Þ

where ni is the following expression: ni ¼

Di þ

i1 X

ð1Þ

k1

Dk

i1 Y j¼k

k¼1

! qj ; ð1  qj Þ

i ¼ 1; . . . ; N :

ð4Þ

Total cost is the sum of setup, inventory holding, and production costs. Average cost is total cost divided by the cycle time. Using all cost terms we obtain the average cost as I I2

I1 -D2 P2(1-ρ 2)-D2 -D1

P1(1- ρ1)-D1

-D2+P1ρ 1 -D2

s1

T1

s2

T2

T

s1

T1

s2

T2

Fig. 2. Inventory-time graph for two products with deterministic co-product rates.

t

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0

799

1

N P

1 hI T 2 1 1

C B ðAi þ ci Di T Þ þ C i¼1 1B ! C; B KðT Þ ¼ B 2 2 0 C N T@ 1P I i T i þ Di S i þ 2Di S i T i1 þ T i1 Di A þ2 hi 0 0 0 0 þ2½D S þ D T T þ ½I  D S  D T T i¼2 i i i i i i i1 i i i1 i

ð5Þ

where T 0i is given by T 0i ¼ ðT  T i  S i  T i1 Þ;

i ¼ 2; . . . ; N :

ð6Þ

and D0i satisfies D0i ¼ ðDi þ P i1 qi1 Þ;

i ¼ 2; . . . ; N :

ð7Þ

Our aim is to find the optimal cycle length, T*, that minimizes the average cost: KðT  Þ ¼ minfKðT Þg;

ð8Þ

T >0

where T* is the optimal cycle length. In the following theorem, we derive an explicit characterization for the optimal cycle length. Furthermore, we show that the stationary point is the global minimum. Theorem 1. The optimal cycle length T*of the problem in (8) is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN u 2 i¼1 Ai u  ;   P T ¼ t N D1 0 h1 D1 1  P 1 ð1q h þ i ðuðiÞð1  sði  1ÞÞ þ Di sði  1Þð1  sðiÞÞÞ i¼2 Þ 1 where for i ¼ 1; . . . ; N , ni is as given in (4), uðiÞ, and sðiÞ are given as   Di uðiÞ ¼ ni 1  ; P i ð1  qi Þ 1 : sðiÞ ¼ ni P i ð1  qi Þ

ð9Þ

ð10Þ ð11Þ

Proof. See Section A.1 in Appendix. h The best cycle time, TCC is the minimum of the lower bound and the cost-optimal cycle times: T CC ¼ maxfT  ; T min g; where T min  1

PN

i¼1



1 D P i ð1qi Þ i

PN þ

i¼1 S i Pi1

k¼1 ð1Þ

k1

Dk

Qi1

:

qj j¼k ð1q Þ j

The common cycle schedule simply schedules one lot of each product in a common cycle and repeats itself every TCC time units. Thus, the best common cycle is always feasible. The best common cycle policy of the problem when qi ¼ 0 for all i is the same as the classical ELSP common cycle policy. Corollary 2. When qi ¼ 0 for all i the optimal cycle time is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P  u 2 Ni¼1 Ai u    : T ELSP ¼ tP N Di h D 1  i i i¼1 Pi

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2.2. Model with planned backorders Since the demand is deterministic it seems reasonable not to allow for any shortages. However, allowing for planned backorders might be beneficial depending on the trade-off between backorder costs and inventory holding costs. Therefore, we consider a generalization of the model in Section 2.1. Better average cost value can be achieved in case where the backorder cost is not so high relative to the cost of holding inventory. In this section, a model based on the determination of the optimal production cycle lengths of two products sharing the same facility with co-product rates and backorders is presented (N = 2). Extension to N-products where N > 2 turns out to be very tedious and is not treated in this paper. Once again we need to satisfy all demand which is constant and known. But the demand need not be met immediately, it can be backordered. For the backordered products, we assume that there is loss of goodwill cost which may be very difficult to establish in practice. However, it is realistic to assume that this loss is proportional to the time required to fill the backorder. The problem is handled in two models for two different cases. The main difference between the two respective models is the time the inventory position goes to the backorder position. The slopes of the demand rate for product 2 are different during the setup time and the production length of the first product. This difference results in different cost functions and different optimal solutions. The schematic illustrations of the cases are depicted in Fig. 3. The inventory of the second product goes to backorder position during the setup time of the second product (Fig. 3a) and the inventory of the second product goes beyond zero before the setup time of the second product (Fig. 3b). The demand slope of the second product during the production of the first product and the setup time of the second product are different from each other due to the co-production from the first product. Thus, the time the inventory of the second product goes below zero is critical. This critical time defines two different cases as we study separately. The two models are solved separately and the resulting optimum solution is the one with the minimum cost as illustrated in Eq. (12). Let us denote the optimal long-run average cost of Case 1 by K 1 ðT  ; b1 ; b2 Þ, whereas  the optimal average cost of Case 2 by K 2 ðT  ; b 1 ; b2 Þ. Two different policies are evaluated and the resulting solution is KðT ; b1 ; b2 Þ:  KðT ; b1 ; b2 Þ ¼ minfK 1 ðT  ; b1 ; b2 Þ; K 2 ðT  ; b 1 ; b2 Þg:

ð12Þ

Thus, two independent non-linear optimization problems are solved and the one with the lowest cost is selected as the resulting optimal solution. For both models, parameters and decision variables are as in Table 1 for N = 2. We also define two new cost parameters for backorders. Let pi be the shortage cost per unit short, independent of the duration of I

I

I2

I2

I1

I1 -D1

-D1

P1(1-ρ1)-D1

0 b1 b2

D2-P1ρ1

s1

P1(1-ρ1)- D1

P2(1-ρ2)-D2

-D2

T1

t

s2

T2 T

(a) Case 1

0 b1 b2

P2(1-ρ2)-D2

-D2

t D2-P1ρ1

s1

T1

s2

T2 T

(b) Case 2

Fig. 3. Inventory-time graph for two products with deterministic co-product rate and planned backorders

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^i be the shortage cost per unit per short per year for product i. Level of backthe shortage for product i, and p orders, bi is also a decision variable along with T, and Ti for i ¼ 1; 2. Although we allow the cost component pi (shortage cost per unit short) in developing the average cost functions, we end up having to assume pi ¼ 0 to preserve convexity of the costs. Therefore all results in this section are valid when pi ¼ 0. Assumptions 1–6 are still valid in this section with N = 2. 2.2.1. Case 1: Backorders during the setup time In this case, net inventory drops below zero during the setup time for product two. Once again there are structural constraints. The total cycle time T has a lower bound: T P T 1 þ T 2 þ S1 þ S2: The total demand of product i must be met with the total supply of product i. TD1 ¼ T 1 P 1 ð1  q1 Þ; TD2 ¼ T 2 P 2 þ T 1 P 1 q1 : Notice that these equalities are the same as the ones given in Section 2.1. The demand is backordered but nevertheless it has to be fulfilled. Total production lengths are derived as   TD1 T q1 ; T2 ¼ T1 ¼ D2  D1 : P2 P 1 ð1  q1 Þ ð1  q1 Þ Maximum inventory positions are derived as      D1 q1 D2  b2 : I 1 ¼ TD1 1   b1 ; I 2 ¼ T D2  D1 1 P 1 ð1  q1 Þ ð1  q1 Þ P2 To derive the cost function, we define intermediary quantities.   D1 : w1 ¼ D1 1  P 1 ð1  q1 Þ

ð13Þ

D1 . So w1 is the demand for product 1 during the time product 1 Utilization of the system for product 1 is P 1 ð1q 1Þ is not in production:   1 1 1 1 ¼ ; þ ð14Þ w2 ¼ ¼ D1 D1 P 1 ð1  q1 Þ  D1 D1 ð1  P 1 ð1q1 ÞÞ w1    q1 D2 w3 ¼ D2  D1 : ð15Þ 1 ð1  q1 Þ P2   We know that 1  DP 22 is the proportion of cycle time during which product 2 is not in production.

q1 Þ is the rate of demand for product 2 during this time. Then, w3 is the net demand for product ðD2  D1 ð1q 1Þ 2 during the time it is not in production: !   1 1 1 w4 ¼ þ ¼ : ð16Þ D2 P 2  D2 D2 ð1  DP 2 Þ 2

All wi P 0 by definition and they are independent of decision variables T ; b1 ; b2 . Then we can redefine the maximum inventory positions for products 1 and 2: I 1 ¼ T w1  b1 ;

I 2 ¼ T w 3  b2 :

Maximum backorder level for product 1 is denoted by b1 and total number of backordered units of product 1 per unit time is given by:   1 b21 b21 þ : 2 D1 P 1 ð1  q1 Þ  D1

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Maximum backorder level for product 2 is denoted by b2 and total number of backordered units of product 2 per unit time is given by:   1 b22 b22 þ : 2 D2 P 2  D2 Total cost is derived as a function of cycle time T, and backorder levels b1 and b2 as TC 1 ðT ; b1 ; b2 Þ in Appendix A.2. Our problem is to find the optimal cycle length in order to minimize the long-run average cost function: K 1 ðT ; b1 ; b2 Þ ¼

TC 1 ðT ; b1 ; b2 Þ : T

ð17Þ

The objective is to find the optimal cycle length and backorder levels in order to minimize the average cost function in (17): K 1 ðT  ; b1 ; b2 Þ ¼ min fK 1 ðT ; b1 ; b2 Þg; T ;b1 ;b2 >0

ð18Þ

where T  ; b1 ; b2 is the optimal cycle length and backorder levels, respectively. In the following theorem, we derive an explicit characterization for the optimal cycle length and backorder levels. Furthermore, we show that the point is the global minimum. Theorem 3. The optimal set of ðT  ; b1 ; b2 Þ for the problem in (18) is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u 2 p1 Þ þ 12 b22 w4 ðc p2 þ h 2 Þ u 2 A1 þ A2 þ 12 b1 w2 ðh1 þ c  0 T ¼u    2 1; u D1 D1 u B 2w3 P 1 ð1q1 Þ  ðD2  P 1 q1 Þ P 1 ð1q1 Þ C u C B   u C B u D1 2 D12 ðD2  P 1 q1 Þw3 P 1 ð1q C uh1 w1 þ h2 B 1Þ C B u A @ t  2 2 2 D1 1 þ D2 ðD2  P 1 q1 Þ P 1 ð1q1 Þ þ w3 w4 b1 ¼ b2 ¼

ðw1 h1 Þ  T ; ðh1 þ c p1 Þ h  h2 w3 w4 þ DD12

ð19Þ

ð20Þ q1 ð1q1 Þ

w4 ðc p2 þ h 2 Þ

i T ;

ð21Þ

where w1 ; w2 ; w3 ; w4 are as in (13)–(16) and when pi ¼ 0 for i ¼ 1; 2. Proof. See Section A.3 in Appendix.

h

The best common cycle time, T ¼ maxfT  ; T min g where P2 S i¼1  i : T min ¼ q1 D1 D1 D2 1 þ P 2 ð1q1 Þ  P 1 ð1q þ P2 1Þ CC

ð22Þ

^i ! 1 and as a result bi ¼ 0 for all i is the same as the The best common policy of the problem when q1 ¼ 0, p classical ELSP common policy. ^i ! 1 for i ¼ 1; 2 then bi ¼ 0, and the optimal cycle time is given by Corollary 4. When q1 ¼ 0, and p vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2ð A1 þ A2 Þ u      : T ¼ T ELSP ¼ t   h1 D1 1  DP 11 þ h2 D2 1  DP 22

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2.2.2. Case 2: Backorders before the setup time In this case, the net inventory goes to the backorder position before the setup time of product two (cf. Fig. 3b). The maximum inventory levels are exactly the same as in Case 1. However, the carried inventory and units on backorder differ which result in a slightly different cost function. In addition to w1–w4 from (13)–(16), we define w5 ¼

1 D2  P 1 q1

ð23Þ

which corresponds to the inverse of demand of product 2 to be satisfied by production and not using the coproduction from product 1. Our assumptions guarantee that w5 P 0. Also let us define w6 ; w7 as w6 ¼ w5 D2  1;   1 w7 ¼ w5 þ : P 2  D2

ð24Þ ð25Þ

Total cost, TC 2 ðT ; b1 ; b2 Þ, is derived in Appendix A.4. Our aim is to solve the following problem:  K 2 ðT  ; b 1 ; b2 Þ ¼ min fK 2 ðT ; b1 ; b2 Þg; T ;b1 ;b2 >0

ð26Þ

 where K 2 ðT ; b1 ; b2 Þ ¼ TC 2 ðT ; b1 ; b2 Þ=T , and T  ; b 1 ; b2 are the optimal cycle length and the backorder levels, respectively. The following result characterizes the optimal solution for Case 2.

 Theorem 5. The optimal set of ðT  ; b 1 ; b2 Þ of the problem given in (26) is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u p2 S 22 þ h2 S 21 Þw6 þ b2 ðc p2 S 2  h2 S 1 Þw6 u A1 þ A2 þ 12 D2 ðc u2 u þ 1 b2 w ðh þ c p1 Þ þ 12 b22 w7 ðc p2 þ h 2 Þ t 2 1 2 1 ; T  ¼ h1 w1 þ h2 w23 w7 h1 w1 T  b ¼ ; 1 h1 þ c p1 1 ðh2 w3 P 2 D ÞT   ðc p2 S 2  h2 S 1 Þw6 2 ; ¼ b 2 w7 ðh2 þ c p2 Þ

ð27Þ ð28Þ ð29Þ

where w1 ; w2 ; w3 are as given in (13)–(15), w5 ; w6 ; w7 are given by (23)–(25) when pi ¼ 0 for i ¼ 1; 2. Proof. See Section A.5 in Appendix. h Once again, feasibility requires that T CC ¼ maxfT  ; T min g, where T min is as given in (22). The best common ^i ! 1 results in bi ¼ 0 for all i and again reduces to the classical common policy of the problem when q1 ¼ 0, p cycle ELSP solution. With the same reasoning as in Case 1, the optimal solution is the unique solution of the given problem. 3. Numerical illustration The characterization of the optimal decision variables relies on some tedious parameters which are not intuitive. In this section, we provide an illustration of the methods proposed with the hope of obtaining more insight about planning with co-production. We restrict attention to two products (N = 2) to be able to meaningfully compare models with and without backorder. For the model without backorders the illustration is classified into three categories depending on the utilizations (total demand over total production) of the system: low, medium, and high. The ratio of setup and the holding costs are taken close to each other. This stands to guarantee that the common cycle policy is the near-optimal solution of the general ELSP.

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There are two sets of data illustrated as in Tables 2 and 3. In the first data set, product 1 is cheaper to produce and to hold in inventory, its setup time and setup cost are also lower than product 2 In the second data set the reverse is true. The results obtained are illustrated in Table 4 for data set 1, and in Table 5 for data set 2. As an example, take a look at the case with low utilization and where q ¼ 0:2 in Table 4. With these parameters had the company planned disregarding co-production, they would find the ELSP solution with T ELSP ¼ 0:7219 and with the presumed average cost of 1053.3 which seems better than that obtained by taking co-production into consideration. However, we know that if q > 0 this plan will result in a shortage from product 1 and an excess inventory from product 2 and this shortage and excess will be compounded for each cycle if no corrective action is taken. Hence the true average cost will be much higher than 1053.3. We observe that: • The long-run average cycle time increases as co-product rate increases, for each data set. • The cycle time of respective co-product rates increases from the low utilization environment to the high utilization environment. Table 2 Common data set for all cases Data set 1 Low

Medium

High

Products

1

2

1

2

1

2

Di ci hi Pi Si Ai

10 5 6 100 0.01 60

70 8 10 350 0.03 100

10 5 6 30 0.01 60

70 8 10 280 0.03 100

10 5 6 60 0.01 60

70 8 10 100 0.03 100

Table 3 Data set 2 for deterministic case without shortage Data set 2 Low

Medium

High

Products

1

2

1

2

1

2

Di ci hi Pi Si Ai

10 8 10 100 0.03 100

70 5 6 350 0.01 60

10 8 10 30 0.03 100

70 5 6 280 0.01 60

10 8 10 60 0.03 100

70 5 6 100 0.01 60

Table 4 Optimal cycle times and average costs for data set 1 Data set 1

q

Utilization

Decision variables

0.1

0.2

0.3

0.4

0.5

Low Low

T* KðT  Þ

0.723 1058.65

0.725 1065.26

0.727 1073.55

0.73 1084.25

0.736 1098.6

Medium Medium

T* KðT  Þ

0.757 1038.77

0.763 1042.83

0.772 1047.44

0.787 1052.55

0.815 1057.9

High High

T* KðT  Þ

1.116 900

1.125 901.47

1.137 902.59

1.156 902.69

1.188 900.08

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Table 5 Optimal cycle times and costs for data set 2 Data set 2

q

Utilization

Decision variables

0.1

0.2

0.3

0.4

0.5

Low Low

T* KðT  Þ

0.869 802.97

0.871 807.57

0.875 813.35

0.88 820.81

0.888 830.82

Medium Medium

T* KðT  Þ

0.923 781.17

0.933 783.02

0.949 784.96

0.973 786.82

1.015 788.11

High High

T* KðT  Þ

1.245 689.28

1.257 689.54

1.274 689.35

1.299 688.15

1.338 684.57

• The long-run average costs are decreasing from the low utilization environment to the high utilization environment for the same co-product rate. ^1 ¼ 60 and When we consider the model with backorders, we use data set 1 from Table 2 and assume that p ^2 ¼ 100. Note that backorder costs are significantly higher than holding costs. We furthermore let the fixed p backorder costs p1 ¼ p2 ¼ 0 as is required in our results. The results are illustrated in Table 6 from which we make the following observations: • In low and medium utilization environments the cycle time tends to be smaller than for the high utilization environment. Smaller cycle time also means low backorder positions. Also we know that, Case 2 goes to the backorder position earlier than Case 1. Thus, in low and medium utilization environments it is optimal that the inventory position of the second product goes to the backorder position during the setup time of the second product regardless of the co-product rate. • In all environments, with increasing co-product rate the cycle time increases, so the average cost decreases as was the case where no backorders are allowed. • In the high utilization environment, the backorder levels are higher than those in the low and the medium utilization environments. • In the high utilization environment, Case 2 produces the minimum cost solution. To investigate this further we used three different levels of backorder costs: pbi ¼ 90h, pbi ¼ h, and pbi ¼ 1.

Table 6 Results for data set 1 with planned backorders Cases

Case 1

Case 2

Utilization

Decision variables

0.3

0.4

0.5

0.3

0.4

0.5

Low Low Low Low

T* b1 b2 Cost

0.70 0.28 2.05 1031*

0.71 0.28 2.10 1030*

0.73 0.28 2.19 1029*

0.65 0.25 3.35 1107

0.53 0.14 3.31 1200

0.53 0.14 3.31 1200

Medium Medium Medium Medium

T* b1 b2 Cost

0.59 0.12 1.40 951*

0.59 0.104 1.38 939*

0.59 0.08 1.38 922*

0.81 0.23 3.17 1017

0.82 0.20 3.20 1008

0.86 0.16 3.20 1040

High High High High

T* b1 b2 Cost

1.04 0.47 1.41 887

1.05 0.45 1.44 882

1.08 0.60 1.48 876

1.20 0.58 2.05 884*

1.22 0.58 2.11 879*

1.26 0.56 2.16 871*

Cost values that are shown with * are the minimum between Case 1 and Case 2.

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Table 7 Numerical examples with different backorder costs and with co-product rate 0.4 Decision variables

pbi ¼ 90h Case 1

T* b1 b2 Cost

1.02 0.054 0.166 888*

pbi ¼ h Case 2 1.28 0.075 0.75 911

pbi ¼ 1

Case 1

Case 2

1.30 3.49 10.51 887

1.60 4.59 12.75 836*

Case 1 1.00 0 0 888

In the original data set, the backorder cost was 10 times larger than the holding cost. So by increasing the backorder cost to 90 times larger and lowering it to the level of holding cost we find the results given in Table 7. We can make the following observations regarding Table 7: • In the high utilization environment, Case 2 produces the optimal values because the optimal solution tends to increase the cycle time. However, if the backorder cost is far away from the holding cost, Case 1 produces the solution with the minimum cost. • If the backorder cost is equal to the inventory holding cost, as expected, Case 2 gives the solution with the minimum cost. When the backorder cost is equal to the holding cost, we can lower the optimal cost by increasing the backorder levels. • When the backorder cost is very large, pbi ! 1 backorder levels are nearly zero and the cycle time approaches the value of the model without shortage. The optimal cost decreases as expected. 4. Conclusion We model the effects of co-production in a deterministic setting where all related costs are treated explicitly. We first develop a common cycle solution for the N-product problem. The long-run average cost function is derived and a closed-form optimal solution is characterized. When the co-product rate is equal to zero, the model reduces to the classical ELSP model. When we introduce backorders, the model yields tedious cost functions. By restricting the number of products to two, we characterize the optimal cycle time and show that the problem reduces to the classical ELSP when backorder costs are exceedingly high and no co-production is allowed. We also illustrate the model numerically to gain insight in the dynamics of the models. We observe that the cycle length increases with increasing co-product rate. Whereas the long-run average cost may or may not decrease with increasing co-product rate. In the numerical examples we report in this paper, the sensitivity of the long-run average cost to the coproduct rate has been relatively low. However, with different parameter settings we have encountered longrun average cost deviation of 50% with changing co-product rate. Numerical studies of the model with backorders, have yielded shorter cycle times, and smaller long-run average costs than the model without shortage. Long-run average cost can be minimized by increasing backorder levels when backorder costs are relatively low. As expected, if backorder costs are very high (i.e., ^ ! 1) then the optimal backorder levels are very small (i.e., b ! 0). As a consequence the cycle length gets p longer. This study shows that planning for co-production may be desirable and if it is not accounted for properly excess inventories might build up. Appendix A. Proofs A.1. Proof of Theorem 1 Proof. Using uðiÞ; sðiÞ from (10) and (11), we obtain the average cost function as

¨ ner, T. Bilgic¸ / European Journal of Operational Research 188 (2008) 793–810 S. O

0

N  P Ai



1 h TD1 2 1



D1 P 1 ð1q1 Þ



807

1

þ ci D i þ 1 T C B i¼1 C B 0 1C B B T uðiÞ  uðiÞS i  T uðiÞsði  1Þ C C KðT Þ ¼ B B þD S  D S sðiÞ C C: B N i i i i B CC Bþ 1 P hi B CC B 2 0 0 @ i¼2 @ þTDi sði  1Þ  TDi sði  1ÞsðiÞ A A þP i1 qi1 sði  1ÞS i First-order condition results in 0

1   D1 1 þ h D 1  2 1 1 P 1 ð1q1 Þ C dKðT Þ B B i¼1 C ¼B   C ¼ 0: N @ 1P A uðiÞ  uðiÞsði  1Þ dT þ2 hi 0 0 þD sði  1Þ  D sði  1ÞsðiÞ i¼2 i i N P

Ai T2

Algebraic manipulations yield T* as given in (9). To show that T* is the global minimum of the given problem, the second-order condition has to be checked: X N Ai d2 K ¼ 2 P 0: i¼1 T 3 dT 2 Since the second derivative is non-negative for all i ¼ 1; . . . ; N , K(T) is convex and hence the stationary point is indeed global optimum. For T* to be well defined, the denominator in (9) should be non-zero. It is clear that both uðiÞ and sðiÞ are non-negative. By substituting uðiÞ; sðiÞ and after algebraic manipulations we obtain the denominator of T* as  h1 D1

 X N D1 1 hi ðuðiÞð1  sði  1ÞÞ þ D0i sði  1Þð1  sðiÞÞÞ > 0; þ P 1 ð1  q1 Þ i¼2

where D0i is as given in (7). Note that D0i > 0 by Assumption 3 and sðiÞ < 1 by Assumptions 1 and 2. h A.2. Total cost function with backorders for Case 1 The total cost function, TC 1 ðT ; b1 ; b2 Þ is derived as   1 0   D1 2 1 1 2 p1 Þ þ T c1 ð1q  h h p A1 þ p1 b1 þ 12 b21 w2 ðh1 þ c 1 b1 þ T 1 w1 þ A2 þ p2 b2 þ 2 b2 w4 ðc 2 þ h2 Þ 2 Þ 1 C B B     1C 0 C B q1 D1 D1 1 C B c2 ðD2  D1 ð1q Þ  b2 h2 P 1 ð1q w3  h2 D12 ðD2  P 1 q1 Þ P 1 ð1q þ h2 D12 w3 þ h2 ðP 2 D B 2Þ 1Þ 1Þ 1Þ CC C BþTB @ A     C B D1 D1 C B S ðD  P q Þ D S þh  h 2 1 2 1 1 2 2 1 P 1 ð1q Þ C: B P 1 ð1q1 Þ 1 C B C B 0 1     2 C B D D 1 1 1 1 C B h w ðD  P q Þ  h ð D  P q Þ 1  2 3 P 1 ð1q Þ 2 1 1 2 1 1 D2 2 2 P 1 ð1q1 Þ C B B C 1 C B þ T 2B C A @ @ A  2 2 2 D1 1 1 1 þ 2 h2 D2 ðD2  P 1 q1 Þ P 1 ð1q1 Þ þ 2 h2 w3 w4

A.3. Proof of Theorem 3 Proof. The long-run average cost function is given by K 1 ðT ; b1 ; b2 Þ ¼ TC 1 ðT ; b1 ; b2 Þ=T which is given by

808

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  3 1b 2 h w þ p 1 2 b1 w2 þ p1Tb1 þ 2 1 T 7  7 6    7 6 þ c1 D1  h1 w w b1 þ T 1 h1 w w2 2 1 2 1 7 6 2 ð1q1 Þ 7 6   7 6 1b 2 p 7 6 þ A2 þ p2 b2 þ 2 2 b2 w4 þh2 w4 7 6 T T T 6 0 17 q1 7 6 ðD  D Þ c 2 2 1 ð1q1 Þ 7 6 0  1 C7 6 B  7 6 B D1 C 1 1 h2 P 1 ð1q1 Þ þ h2 D2 w3 þ h2 ðP 2 D2 Þ w3 7 6 B C C 7 6 B b B C   @ A 2 6 C7 K 1 ðT ; b1 ; b2 Þ ¼ 6 þB D1 1 7: C B h2 D2 ðD2  P 1 q1 Þ P 1 ð1q1 Þ 7 6 B C 6 @    A7 7 6 D1 D1 7 6 þh2 S 1 ðD2  P 1 q1 Þ P 1 ð1q  h 2 D2 S 1 P 1 ð1q Þ Þ 7 6 1 1 7 6 1 0    2 7 6 D1 D1 1 7 6 h w h ðD  P q Þ  2 1 1 2 2 P 1 ð1q1 Þ 6 C 7 B 2 3 P 1 ð1q1 Þ 6 C 7 B   6 C 7 B D1 6 þ T B h2 D12 ðD2  P 1 q1 Þw3 P 1 ð1q C 7 1Þ 6 C 7 B 4 A 5 @  2 2 2 D1 1 þ 12 h2 D12 ðD2  P 1 q1 Þ P 1 ð1q þ h w w 2 4 3 2 1Þ 2

A1 6T

ðA:1Þ

The first-order condition with respect to T, oK 1 ðT ; b1 ; b2 Þ=oT yields 1 0 1b2 ðh w þ p bw Þ  TA12  pT1 b21  2 1 1 T22 1 2 C B   C B 1 C B þ 2 h1 w2 w21 C B   C B 1b2 p C B  A2  p2 b2  2 2 b2 w4 þh2 w4 C B T2 T2 T2 B 0 1    2 C C ¼ 0: B D1 D1 1 C B B B h2 w3 P 1 ð1q1 Þ  2 h2 ðD2  P 1 q1 Þ P 1 ð1q1 Þ C C B B CC   B B CC D1 B þB h2 D12 ðD2  P 1 q1 Þw3 P 1 ð1q CC 1Þ B B CC @ @ AA  2 2 2 D1 1 1 1 þ 2 h2 D2 ðD2  P 1 q1 Þ P 1 ð1q1 Þ þ 2 h2 w3 w4 From which (19) is obtained. First-order condition with respect to b1 yields: oK 1 ðT ; b1 ; b2 Þ p1 b1 ðh1 w2 þ c p1 w2 Þ  h1 w2 w1 ¼ 0 ¼ þ ob1 T T from which (20) follows. First-order condition with respect to b2 yields: 0 1 p2 þ ð pb2 w4Tþh2 w4 Þ b2 oK 1 ðT ; b1 ; b2 Þ @ T h    i A ¼ 0 ¼ D1 D1 1 1 ob2 h2 D12 w3 þ P 1 ð1q w  ðD  P q Þ þ 2 1 1 ðP 2 D2 Þ 3 D2 P 1 ð1q1 Þ 1Þ from which (21) is obtained. To check second-order conditions, we obtain the following Hessian matrix H ðT ; b1 ; b2 Þ: 1 3 2 0 2A1 þ 2A2     p p 1 2 1 7 6 13 @ þb21 w2 ðh1 þ c p1 Þ A T12 6T T2 w2 b1 ðh1 þ c p1 Þ w4 b2 ðh2 þ c p2 Þ 7 2 7 6 p2 Þ 7 6  þb2 w4 ðh2 þ c  7 6 p1 7: 6 1 1 7 6 T2 ð w ðc p þ h Þ Þ 0 1 1 2 T 7 6 c w b ðh þ p Þ 1 2 1 1 7 6   5 4 p 2 1 1 ðw4 ðh2 þ c p2 ÞÞ 0 T T2 w4 b2 ðh2 þ c p2 Þ

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809

We have constructed the model by incorporating pi P 0, but we cannot establish the convexity of the cost function when pi > 0. When pi are set to zero we see that the Hessian matrix is positive semi-definite as required. The condition that all leading principal minors are strictly positive is sufficient for H ðT ; b1 ; b2 Þ to be positive semi-definite and hence K 1 ðT ; b1 ; b2 Þ to be convex. The leading principal minors are derived as follows:  1  H 1 ¼ 3 2A1 þ 2A2 þ b21 w2 ðh1 þ c p1 Þ þ b22 w4 ðh2 þ c p2 Þ > 0: T For the second principal sub-matrix: 2 detðH 2 Þ ¼ 4 w2 ðh1 þ c p1 Þw4 ðh2 þ c p2 ÞðA1 þ A2 Þ > 0: T And finally: 2 p1 Þðh2 þ c p2 Þw2w4 > 0: detðH 3 Þ ¼ 5 ðA1 þ A2 Þðh1 þ c T Since all parameters are non-negative by assumption and also all the wi P 0, detðH 3 Þ P 0 for pi ¼ 0. Hence, all leading principal minors are strictly positive and K 1 ðT ; b1 ; b2 Þ is convex. h A.4. Total cost function with backorders for Case 2 Total cost function, TC 2 ðT ; b1 ; b2 Þ, for Case 2 is given as 1 0 2 2 D1 1 1 T þ h A1 þ c1 ð1q p 1 w2 ðT w1  b1 Þ þ p1 b1 þ 2 c 1 w2 b1 2 Þ 1 C B q1 C B þ A2 þ c2 ðD2  D1 ð1q ÞT 1Þ C B ! 2 C B 2 1 1 ðT w3 b2 Þ C: w ðT w  b  D S Þ þ TC 2 ðT ; b1 ; b2 Þ ¼ B 2 2 1 3 2 P 2 D2 C B þ h2 2 5 C B 2 1 C B þðT w3  b2  D2 S 1 ÞS 1 þ 2 D2 S 1 A @   2 b 2 2 p2 w5 ðb2  D2 S 2 Þ þ 2b2 S 2  D2 S 22 þ P 2 D þ p2 b2 þ 12 c 2 A.5. Proof of Theorem 5 Proof. The long-run average cost function is: K 2 ðT ; b1 ; b2 Þ ¼ TC 2 ðT ; b1 ; b2 Þ=T . The average cost function is obtained as 3 21 A þ T1 A2 T 1 7 6þ 1p b þ 1 p b 7 6 T 1 1 T 2 2 7 6 11 2 2 7 6 þ D2 ðc p S þ h S Þw 2 2 2 1 6 7 6 T 2 7 6 þ 1 b ðc 7 6 T 2 p2 S 2  h2 S 1 Þw6 7 6 11 1 1 2 K 2 ðT ; b1 ; b2 Þ ¼ 6 þ w2 b2 ðh1 þ c 7: p1 Þ þ T 2 b2 ðh2 þ c p2 Þw7 1 7 6 2T 2 3 7 6 q D1 1  h c1 ð1q 7 6 1 w2 w1 b1 þ c2 ðD2  D1 ð1q ÞÞ Þ 1 1 6 þ4   57 7 6 b2 7 6 h2 w3 w5 b2 þ w5 D2 S 1 þ P 2 D2  S 1 5 4 1

þ T 2 h1 w2 w21 þ 12 h2 w23 w7 The first-order condition with respect to T yields 0 1 1  T 2 A1  T12 A2  T12 p1 b1  T12 p2 b2 B 1 1 C  T 2 2 D2 ðc p2 S 22 þ h2 S 21 Þw6  T12 b2 ðc p2 S 2  h2 S 1 Þw6 C oK 2 ðT ; b1 ; b2 Þ B C ¼ 0: ¼B B  1 1 w b2 ðh þ c C 1 1 2 @T p2 Þw7 @ T 2 2 2 1 1 p1 Þ  T 2 2 b2 ðh2 þ c A þ 12 ðh1 w2 w21 þ h2 w23 w7 Þ

¨ ner, T. Bilgic¸ / European Journal of Operational Research 188 (2008) 793–810 S. O

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From which we can obtain (27). The first-order condition with respect to b1 yields: oK 2 ðT ; b1 ; b2 Þ 1 1 1 w b1 ðh1 þ c ¼ p1 þ p1 Þ  h1 w2 w1 ¼ 0 @b1 T T 2 2 from which (28) follows. The first-order condition with respect to b2 yields: 01 1 p T 2 oK 2 ðT ; b1 ; b2 Þ B 1 C p2 S 2  h2 S 1 Þw6 ¼ @ þ T ðc A¼0 @b2 1 1 1 þ T 2 b2 ðh2 þ c p2 Þw7  h2 w3 P 2 D2 from which we have (29). To show that the optimal set of ðT  ; b1 ; b2 Þ is the global minimum of the given problem, the second- order conditions have to be checked. The result is analogous to that used in the proof of Theorem 1 and is omitted.  Thus, under the given assumptions the optimal set of ðT  ; b 1 ; b2 Þ as given in (27)–(29) yield the global minimum of the given problem. h References Alle, A., Pinto, J., Papageorgiou, L., 2004. The economic lot scheduling problem under performance decay. Industrial & Engineering Chemistry Research 43 (20), 6463–6475. Bitran, G., Dasu, S., 1992. Ordering policies in an environment of stochastic yields and substitutable demand. Operations Research 40 (5), 999–1017. Bitran, G., Gilbert, S., 1994. Co-production processes with random yields in the semi conductor industry. Operations Research 42 (3), 476–491. Bomberger, E.E., 1966. A dynamic approach to a lot size scheduling problem. Management Science 12 (11). Cooke, D.L., Rohleder, T.R., Silver, E.A., 2004. Finding effective schedules for the economic lot scheduling problem: A simple mixed integer programming approach. International Journal of Production Research 42 (1), 21–36. Daya, M., Hariga, M., 2000. Economic lot scheduling problem with imperfect production processes. Journal of Operational Research Society 51, 875–881. Delporte, C.M., Thomas, L.J., 1977. Lot sizing and sequencing for n products on one facility. Management Science 23 (1), 1070. Deuermeyer, B.L., Pierskalla, W., 1978. A by-production production system with an alternative. Management Science 24, 1373–1383. Doll, C.L., Whybark, D., 1973. An iterative procedure for the single machine multi-product lot scheduling problem. Management Science 20 (1), 50–55. Elmagrabhy, S.E., 1978. The economic lot scheduling problem (ELSP) review and extensions. Management Science 24 (6), 578–598. Gallego, G., Roundy, R., 1992. The economic lot scheduling problem with finite backorder costs. Naval Research Logistics 39, 729–739. Hsu, W.L., 1983. On the general feasibility test of scheduling lot sizes for several products on one machine. Management Science 29 (1), 93–105. Jones, P., Inman, R., 1989. When is the economic lot scheduling problem easy? IIE Transactions 21 (1), 11–20. Lee, H., 1992. Lot sizing to reduce capacity utilization in a production process with defective items, process correction and rework. Management Science 38, 1314–1328. Lee, H., Rosenblatt, M., 1985. Optimal inspection and ordering policies for products with imperfect quality. IIE Transactions 17 (3), 284– 289. Lin, G.C., Kroll, D.E., Lin, C., 2006. Determining a common production cycle time for an economic lot scheduling problem with deteriorating items. European Journal of Operational Research 173 (2), 669–682. Ou, J., Wein, L., 1995. Dynamic scheduling of a production/inventory system with by-products and random yields. Management Science 41 (6), 1000–1017. Porteus, E., 1986. Optimal lot-sizing, process quality improvement and setup reduction. Operations Research 34, 137–144. Silver, E., Schweitzer, P., 1983. Mathematical pitfalls in the one machine multi-product economic lot scheduling problem. Operations Research 31 (1), 401–405. Vemuganti, A., 1977. Production runs for multiple products: The two product heuristic. Management Science 23 (1), 1321–1327. Yano, C., Lee, H., 1995. Lot sizing with random yields: A review. Operations Research 43 (2), 311–335. Zipkin, P., 1991. Computing optimal lot sizes in the economic lot scheduling problem. Operations Research 39 (1), 56–63.