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European Journal of Operational Research 126 (2000) 371±385

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Theory and Methodology

Input control in a batch production system with lead times, due dates and random yields Yunzeng Wang a

a,*

, Yigal Gerchak

b,1

Department of Operations Research and Operations Management, Weatherhead School of Management, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7235, USA b Department of Management Sciences, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received 1 September 1998; accepted 1 April 1999

Abstract The presence of random yields in many manufacturing processes can considerably complicate production planning and control. We investigate a batch production system with due dates, where the yield of each batch is random and the production lead time is longer than the time interval between starting consecutive batches. To satisfy an order with a given due date, several input batches could be initiated. But the realized yields of batches still in process are unknown when the next batch size needs to be determined. We formulate this general problem as a dynamic program with the objective of minimizing the total expected discounted costs. For a simple version of the model with linear cost parameters and one work-in-process batch, we show that the structure of the optimal input control policy is of a single critical level type for the work-in-process batch size. We prove that, for given outstanding demand, this critical level becomes larger as the number of input opportunities becomes smaller. Furthermore, if the discount factor equals to 1, this critical level is shown to be strictly greater than the outstanding demand. Production often starts earlier than necessary in order to utilize yield realizations of initial batches for judicious choice of later batches, and to achieve diversi®cation over time. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Lot sizing; Random yield; Production control; Due dates

1. Introduction Consider an imperfect production system, which processes batches sequentially. At each review epoch (beginning of a review period) a batch of items is input into the system for processing. The processing time,

*

Corresponding author. Tel.: +1-216-368-3811; fax: +1-216-368-4776. E-mail addresses: [email protected] (Y. Wang), [email protected] (Y. Gerchak). 1 This research was supported by the Natural Science and Engineering Research Council of Canada. Helpful comments by referees are acknowledged. 0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 2 9 5 - 7

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or production lead time, for a batch is longer than the review period. Thus, at any time, several work-inprocess batches at di€erent processing stages will be present in the system. Production processes of this type are quite common. They can be generically viewed as a ``conveyor'', which carries the batches through the system, as shown in Fig. 1(a), where Ik is the size of batch k. Quality inspection takes place at the end of the line, and the net yield of good units is random. Now suppose that an order is received, and it is to be satis®ed within a due date. Then, the number of input batches designated to satisfy the order is limited, due to production lead times. The problem is to choose the input quantities for each of these consecutive input batches in order to balance the potential costs of over production and shortage penalty at the due date. What really makes this problem challenging, both theoretically and practically, is the fact that the yields of batches that are in process are not known when the size of the next input batch needs to be selected. Fig. 1(b) provides an example to illustrate the temporal dynamics of the system, where the production of a batch takes two review periods. Demand due date is four review periods away. As a consequence, only three batches can be input to meet the due date. The batches are labeled in the reverse order of their input sequence. At the point when the size of the last batch I1 is to be determined, the ®rst two input batches I3 and I2 are still in process and their yields (number of good items), namely, YI3 and YI2 , respectively, are not yet known. Our model shares some common features with several branches of the literature on lot sizing with random yields (Yano and Lee, 1995). If specialized to the case of no work-in-process (i.e., up-to-date information about the realized yields of all previous batches), our physical scenario becomes similar to that of multiple-lot-sizing (``reject-allowance'') production-to-order (e.g., Beja, 1977; Sepehri et al., 1986; GrosfeldNir and Gerchak, 1996 and references therein). Within that literature, Klein (1966) and Sepehri et al. (1986) address the case of limited number of production runs, while White (1965), Grosfeld-Nir and Gerchak (1990) and Wein (1992) consider stochastically proportional yields. But, the economic trade-o€ in this branch of literature is between setup and variable production costs, while our focus is on trading-o€ production and overage costs against penalties for not completing the order by its due date. The several ``stages'' of work-in-process in our setting are reminiscent of multi-stage production models with random yields (Lee and Yano, 1988; Barad and Braha, 1996; Grosfeld-Nir and Ronen, 1993; Grosfeld-Nir and Gerchak, 1998). Yet, in these models batches are inspected after every processing stage, and thus the whole decision structure, as well as the economic trade-o€s, are quite di€erent. Although ours is a single-product setting, the fact that lot sizing decisions have to be made before the results of previous (or simultaneous) decisions are known is somewhat reminiscent of lot sizing problems in

Fig. 1. The batch production system.

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373

assembly systems with random yields (Singh et al., 1990; Gerchak et al., 1994; Gurnani et al., 1996) as well as control problems with random lead times (Chu et al., 1993; Proth et al., 1997). Due-date-based models in production management typically ask when to start some activities, and in what order, so as to minimize tardiness and earliness penalties. Our random yield (rather than random processing time) setting raises lot sizing issues, rarely discussed in conjunction with due dates. Our model minimizes the total expected discounted costs of production, overage and penalties for the portion of an order not satis®ed by the due date. No explicit earliness penalties for units completed before the due date are included. We ®rst formulate a very general model with an arbitrary number of work-inprocess batches, general cost functions and capacity constraints. We then present a detailed analysis for a simpli®ed version of the general model. In this simpli®ed model, we assume that all the cost functions are linear and that there is only one work-in-process batch at any decision point. We show that the optimal policy is of a single critical level type. That is, a new input batch is started if and only if the size of the workin-process batch is less than a critical level. We prove that, for given outstanding demand, this critical level becomes larger as the number of input opportunities becomes smaller. Furthermore, in the absence of discounting, this critical level for each input batch is shown to be strictly greater than the outstanding demand. The reasons for starting production earlier than needed to meet the due date is that information about realized yields of early batches helps in judicious choice of latter batch sizes, and the multiple trials provide yield diversi®cation over time, similar to what has been observed by Henig and Gerchak (1990) in a periodic-review setting with random yields.

2. Problem de®nition and formulation The production system under consideration can be described as follows. At every review epoch one ``batch'' of items is input into the system for processing. The processing/production lead time of each batch takes a ®xed time equal to K review periods, independent of the batch size. As a consequence, there are always up-to K batches in process just before any additional batch is fed into the system at the beginning of a review period. Note that those batches are at di€erent processing stages. Inspection, through which defects (bad items) will be picked out, takes place at the last ``processing'' stage; the yield (# of good items) from any batch is random. Speci®cally, we model the random yield associated with a batch of size x by the stochastically proportional (multiplicative) yield model: Yx ˆ x
U ; where, U is the random fraction of good items, which is assumed to be independent of x. The random multiplier U has a density g(á) on [0, 1], cumulative distribution G(á) and a mean l; and Yx the resulting yield (good items). We assume that the multipliers (U's) corresponding to di€erent batches are i.i.d. This common-cause yield model has been used extensively by various researchers (e.g., Lee and Yano, 1988; Henig and Gerchak, 1990; Wang and Gerchak, 1996; see also Yano and Lee (1995) and references therein). Suppose we are working towards ®lling a single order with a due date of H …H > K† periods away. Since the last batch should be fed at least K periods (the production lead time) before this due date, the maximal number of batches we can process to meet the order will be N ˆ H ÿ K: At each of these N input epochs, we know the following: 1. The outstanding demand to be satis®ed, denoted by D, which equals the initial demand minus the total number of good items from all of previous completed and inspected batches. 2. The number of items in each of the K batches still in process, denoted by I1 ; I2 ; . . . ; IK . (The subscripts are in reverse order of the sequence by which those batches have been fed into the system; so the next batch whose production will be completed will be of size IK .)

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3. How many possible input chances are left within the due date; denote this number by n. Then, taking into consideration of the pattern of yield randomness and the relevant costs, we have to choose the number of items for the current input batch (if any). When there are n possible input chances left, we call the corresponding problem an ``n-period decision problem'', or simply an ``n-period problem''. We consider three types of costs: (1) production costs c(x) for a batch of size x (regardless of realized yield), which may include setup costs; (2) an underage (shortage) cost p(á), which applies to the portion of the order not met by the due date and (3) overage costs h(á), which apply to ®nished good units. h(á) could be negative corresponding to a salvage value. For simplicity, we assume that there are no explicit holding costs for good items produced in early batches. The problem of minimizing the total expected discounted costs can be formulated as a dynamic program. Let fn …D; I1 ; I2 ; . . . ; IK † denote the total expected discounted costs for an n-period problem given the starting state D (the outstanding demand) and I1 ; I2 ; . . . ; IK (the sizes of the K batches in process), when an optimal production control rule is used at each of the subsequent decision opportunities. Then, fn …D; I1 ; I2 ; . . . ; IK † satis®es the functional equation   Z 1 fnÿ1 …D ÿ IK u; x; I1 ; . . . ; IKÿ1 †g…u† du ; fn …D; I1 ; I2 ; . . . ; IK † ˆ min c…x† ‡ a 06x6B

0

for n ˆ 1; . . . ; N , where B is the production capacity (if limited) for each batch and a the discount factor per period. Note that the dynamics modeled here is that, when period n ) 1 arrives, every work-in-process batch advances by one step, with IK being completed and x becoming the ®rst work-in-process batch. Since after the last input batch (i.e., the Nth input epoch) only the outputs from the most recent K batches are still to be realized, the value of f0 …D; I1 ; I2 ; . . . ; IK † can be calculated by the following recursion: Z 1 fÿ1 …D ÿ IK u; 0; I1 ; . . . ; IKÿ1 †g…u† du; f0 …D; I1 ; I2 ; . . . ; IK † ˆ a 0

Z fÿi …D; 0; . . . ; 0; Ii‡1 ; . . . ; IK † ˆ a

0

1

fÿ…i‡1† …D ÿ IK u; 0; . . . ; 0; Ii‡1 ; . . . ; IKÿ1 †g…u† du

for i ˆ 1; 2; . . . ; K ÿ 2 and Z fÿ…Kÿ1† …D; 0; . . . ; 0; IK † ˆ

1 D=IK

Z h…IK u ÿ D†g…u† du ‡

D=IK 0

p…D ÿ IK u†g…u† du:

While formulating this very general model is conceptually useful, it turns out to be too complex to analyze. Instead, in the following we analyze a simple version of it, assuming that: 1. There is only one work-in-process batch at any decision making epoch (i.e., K ˆ 1), i.e., the production lead time equals one review period. Denote the size of this batch by I. 2. All cost functions are linear: c ± unit production cost (in particular, setup costs are assumed to equal zero); h ± unit overage cost; p ± unit shortage cost. These are common assumptions in random yield models. 3. There is no production capacity limit. (Note: most of the ``reject allowance'' literature also made this assumption). Since it contains all the essential features of the original problem, we feel that if we can solve this simpli®ed problem, we will then gain useful insights into the original complex problem which we have de®ned above (it might be only a matter of more complex mathematical manipulations).

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The simpli®ed problem can be written as follows:   Z 1 fnÿ1 …D ÿ Iu; x†g…u† du fn …D; I† ˆ min cx ‡ a xP0

0

for n ˆ 1; 2; . . . ; N

375

…1†

and Z f0 …D; I† ˆ h

1

D=I

Z …Iu ÿ D†g…u† du ‡ p

0

D=I

…D ÿ Iu†g…u† du:

…2†

Note D=I can be either less or greater than 1. Regarding the cost parameters, we assume that c < alp (i.e., one would rather input a unit into the process at the last input chance instead of incurring a unit of shortage penalty), and c > ÿalh (that is, one cannot bene®t from salvage value alone). As a consequence, we have h ‡ p > 0. De®ne Z 1 fnÿ1 …D ÿ Iu; x†g…u† du for n ˆ 1; 2; . . . N : …3† Vn …x; D; I† ˆ cx ‡ a 0

Note that, for a given outstanding order D and a work-in-process I, Vn (x, D, I) is a function of the current batch size x only. At each decision point, we wish to minimize Vn (x, D, I) by choosing the batch size x. Since I is actually the input batch of the previous period, we always have I P 0. But D may be negative at some later periods, since the total yield (# of good items) from all previous output batches may already exceed the initial order. The reader should note the following notation that will be used in our analysis: ofn …
;
†=oD and ofn …
;
†=oI always indicate the partial derivatives of fn …
;
† with respect to its ®rst and second arguments, respectively. The following properties regarding f0 …D; I† will be of use: Proposition 1. (i) f0 …D; I† is convex in …D; I†. o2 f0 …D; I† o2 f0 …D; I† ˆ < 0: (ii) oDoI oIoD of0 …D; I† of0 …D; I† ˆ ÿh and ˆ lh: (iii) If D < 0; then oD oI of0 …D; I† of0 …D; I† ˆ p and ˆ ÿlp: (iv) If D > I; then oD oI Proof. See Appendix A.



3. Analysis of one-period problem In this section we provide a complete analysis for a one-period (i.e., only one decision epoch left) decision problem. We show that the optimal policy for choosing the batch size x is a single critical level type with respect to the work-in-process batch size I. Note that this one-period problem looks similar to the classic Newsvendor problem, with the work-in-process batch here roughly corresponding to the outstanding replenishment order in the Newsvendor problem. The key di€erence is that the yield of the work-in-process batch (as well as that of the current input batch) here is random. As a consequence, the optimal policy is not

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an order-up-to type as in the Newsvendor problem. The yield randomness of the work-in-process batch also di€erentiates our problem here from the previous random yield literature (e.g., Henig and Gerchak, 1990; Lee and Yano, 1988; Zipkin, 1994, Section 8.4.8), where the yields of all earlier batches are assumed to be known at the current decision epoch. Now, in choosing the batch size x for a one-period problem, the goal, from (3), is to minimize the function Z 1 f0 …D ÿ Iu; x†g…u† du: …4† V1 …x; D; I† ˆ cx ‡ a 0

We have the following property: Proposition 2. V1 …x; D; I† is convex in …x; D; I†. Proof. We know, from Part (i) of Proposition 1, that f0 …D; I† is convex in (D, I). The arguments of f0 …D ÿ Il; x† are linear in (x, D, I). By Theorem 5.7 of Rockafellar (1970), f0 …D ÿ Il; x† is thus convex in (x, D, I). Now, since V1 …x; D; I† is a linear combination of convex functions, it is itself convex.  Furthermore, it follows from (4) that Z 1 oV1 …x; D; I† ˆc‡a ‰of0 …D ÿ Iu; x†=oIŠg…u† du ox 0 and o2 V1 …x; D; I† ˆ ÿa oxoI

Z

1 0

‰o2 f0 …D ÿ Iu; x†=oIoDŠug…u† du > 0;

…5†

…6†

where, the inequality follows from Part (ii) of Proposition 1. Eq. (6), in conjunction with Proposition 2, implies that the derivative of V1 …x; D; I† with respect to x, i.e., oV1 …x; D; I†=ox, is increasing in (x, I). The above simple properties of the function V1 …x; D; I† allow us to establish the single critical level nature of optimal policy as follows. Let I1 …D† solve oV1 …0; D; I†=ox ˆ 0. That is, I1 …D† is given, from (5), by Z 1 ‰of0 …D ÿ I1 …D†u; 0†=oIŠg…u† du ˆ 0: …7† c‡a 0

Since V1 …x; D; I† is convex and oV1 …x; D; I†=ox is increasing in (x, I), it follows that, in order to minimize V1 …x; D; I†, the optimal batch size is x1 ˆ 0 if I P I1 …D†, and it is x1 > 0 if I < I1 …D†. That is, when the outstanding order is D, I1 …D† is the singe critical level for the work-in-process batch size I, which determines whether the current batch size x should be zero or not. When the work-in-process batch size is below its critical level, i.e., I < I1 …D†, due to the convexity of V1 …x; D; I†, the corresponding optimal batch size x1  x1 …D; I† > 0 is given by equating (5) to zero (the ®rst order condition). That is Z 1 fof0 ‰D ÿ Iu; x1 …D; I†Š=oIgg…u† du ˆ 0: …8† c‡a 0

In summary, the optimal decision rule for a one-period lot-sizing problem is as follows:  x1 …D; I†; if I < I1 …D†;  x1 ˆ 0; if I P I1 …D†: I1 …D† and x1 …D; I† solve Eqs. (7) and (8), respectively.

…9†

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377

The following Proposition characterizes the directions in which the critical level I1 …D† varies with the outstanding order D, and the optimal batch size x1 …D; I† with the outstanding order D and the work-inprocess batch size I. These properties are rather intuitive. Proposition 3. (i) I1 …D† is increasing in D. (ii) x1 …D; I† is increasing in D and decreasing in I. Proof. See Appendix B.



Corresponding to the optimal policy in (9), the value function (the total expected discounted costs) of the one-period problem is given by  V1 ‰x1 …D; I†; D; IŠ if I < I1 …D†; …10† f1 …D; I† ˆ minV1 …x; D; I† ˆ if I P I1 …D†: V1 …0; D; I† xP0 We conclude our analysis of the one-period problem by proving the following properties regarding f1 …D; I†, which will be useful in our analysis of a two-period problem in Section 4. Proposition 4. …i† f1 …D; I† is convex in …D; I†: of1 …D; I† ˆ alh for D < 0: …ii† oI

…11†

Proof. Part (i): Since V1 …x; D; I† is convex in (x, D, I) (Proposition 2), the convexity of f1 …D; I† in (D, I) follows from Theorem 5.7 of Rockafellar (1970). Part (ii): When the outstanding order is negative (i.e., D < 0), the optimal batch size is obviously zero (i.e., x1 ˆ 0). As a consequence, we have f1 …D; I† ˆ V1 …0; D; I†. From Eq. (4) and part (iii) of Proposition 1, we can then obtain the required result.  Note that the second property is intuitive.

4. Analysis of two-period problem In this section, we ®rst show that the structural properties obtained for the one-period problem also hold for a two-period problem. We then derive some monotonicity properties regarding the critical levels for the work-in-process batch sizes as the number of review periods (i.e., input opportunities) increases. The analysis of a two-period problem will carry over to an n-period problem. From (3), the objective function for a two-period problem is given by Z V2 …x; D; I† ˆ cx ‡ a

0

1

f1 …D ÿ Iu; x†g…u† du:

…12†

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Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

Proposition 5. V2 …x; D; I† is convex in …x; D; I†. Proof. Using part (i) of Proposition 4, the proof will be similar to that of Proposition 2.



To show that a ``single-critical-level'' type policy is optimal for the two-period problem, we will need the following assumption concerning the dependence of f1 …D; I† on (D, I): o2 f1 …D; I† o2 f1 …D; I† ˆ < 0: oDoI oIoD

…13†

Although we proved a similar property for f0 …D; I†, as given in Part (ii) of Proposition 1, it turns out to be too complex to prove the inequality part of (13) for f1 …D; I† directly in the case I < I1 …D†. Nevertheless, we believe intuitively that this condition should hold: While D is the demand (withdrawal) from the system, I tends to increase the supply; hence, the ``e€ects'' of D and I to the system should be in opposite directions, as expressed in (13). From (12), we have Z 1 oV2 …x; D; I† ˆc‡a ‰of1 …D ÿ Iu; x†=oIŠg…u† du …14† ox 0 and o2 V2 …x; D; I† ˆ ÿa oxoI

Z

1 0

‰o2 f1 …D ÿ Iu; x†=oIoDŠug…u† du > 0;

…15†

where, the inequality follows from (13). Thus, oV2 …x; D; I†=ox is increasing in (x, I). With the above properties, we can establish the optimality of a single critical level type policy for a twoperiod problem. Let I2 …D† solve oV2 …0; D; I†=ox ˆ 0, i.e., Z 1 ‰of1 …D ÿ I2 …D†u; 0†=oIŠg…u† du ˆ 0: …16† c‡a 0

Then, since oV2 …x; D; I†=ox is increasing in (x, I), oV2 …0; D; I†=ox > 0 for I > I2 …D†. This, in conjunction with the convexity of V2 …x; D; I†, implies that the optimal batch size for the two-period problem is zero, i.e., x2 ˆ 0 if the work-in-process batch size is greater than or equal to the critical level; otherwise, x2 > 0. When I < I2 …D†, due to the convexity of V2 …x; D; I†, the corresponding optimal input batch size x2  x2 …D; I† > 0 is obtained by equating (14) to zero, i.e., Z 1 fof1 ‰D ÿ Iu; x2 …D; I†Š=oIgg…u† du ˆ 0: …17† c‡a 0

In conclusion, we have shown that the optimal decision rule for a two-period lot-sizing problem is again of a ``single-critical-level'' type:  x2 …D; I† if I < I2 …D†; …18† x2 ˆ 0 if I P I2 …D†: One can easily show that I2 …D† is increasing in D and that x2 …D; I† is increasing in D and decreasing in I. These properties, which are similar to those in Proposition 3 for a one-period problem, are again rather intuitive. The two-step value function is

Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

 f2 …D; I† ˆ minV2 …x; D; I† ˆ xP0

V2 ‰x2 …D; I†; D; IŠ if I < I2 …D†; if I P I2 …D†: V2 …0; D; I†

379

…19†

Proposition 6. f2 …D; I† is convex in …D; I†. Proof. Using Proposition 5, the proof will be similar to that of Proposition 4.  Finally, we derive and discuss some important properties regarding the critical levels for problems with di€erent review periods. These properties are summarized in the next Proposition: Proposition 7. (i) I1 …D† > I2 …D† for all 0 < a 6 1. (ii) In …D† > D for a ˆ 1; n ˆ 1; 2. Proof. See Appendix C.



We note that the analysis of the two-period problem can actually be extended to an n-period problem. As a consequence, the above Proposition can be extended to (i)0 I1 …D† > I2 …D† >


> IN …D† for 0 < a 6 1 and (ii)0 In …D† > D for a ˆ 1; n ˆ 1; 2; . . . ; N , respectively. A possible intuitive explanation for the second result might be as follows. Since the realized yield of any batch will be less than the batch size with probability of one, at any input chance, as long as the size of the work-in-process batch is less than or equal to the outstanding demand D, one should feed some items in the current input batch. But, when the value of a becomes suciently small, it may then not be worthwhile to input anything, especially at the earlier input chances, due to the discount e€ect; so In …D† > D is not necessarily true for any value of a. But it turns out to be true for a ˆ 1. The intuition for the property I1 …D† > I2 …D† >


> IN …D† is as follows. First, when more future input chances are available, one would rather input relatively fewer items and wait to see if the yield of the work-in-process items could actually satisfy D, and thus reduce the risk of larger overage costs; we call this the ``information e€ect'' ± information about earlier input batches' yields could be of assistance to the future input decision. Second, when there are more input chances left, the risk of overage and underage could be diversi®ed among (or shared by) more input batches; we call this the ``risk diversi®cation e€ect''. This also explains why it is worthwhile to start production earlier than needed to meet the due date. The bene®t of such ``diversi®cation over time'' when yields are random was also observed in the context of periodic review models by Henig and Gerchak (1990, Theorem 7 and subsequent discussion); see also Zipkin (1994, Section 8.4.8).

5. A numerical example We provide an example to illustrate how the critical levels at di€erent review periods are computed. To that end, we assume that the random yield factor U takes the uniform distribution on ‰0; 1Š. That is, g…u† ˆ 1 for 0 6 u 6 1 and l ˆ 1=2. A simple closed-form formula for computing the critical level of an one-period problem, namely, I1 …D†, can be derived from Eq. (C.1) in Appendix C as I1 …D† ˆ

a…h ‡ p†
D: 2c ‡ ah

…20†

380

Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

From (C.8), we obtain the constant M as s 2c ‡ ah Mˆ : a…h ‡ p† Substituting M into (C.2), we obtain a closed-form formula for computing the critical level of a two-period problem as s a2 …h ‡ p† 2c ‡ ah
D: …21† I2 …D† ˆ 2c ‡ a2 h a…h ‡ p† For some concrete results and comparison, we let a ˆ 0:8; c ˆ 10; h ˆ 5; p ˆ 30

and

D ˆ 100:

Then, we have I1 …D† ˆ 117 and I2 …D† ˆ 86, respectively. If we let a ˆ 1 and keep all the other parameters unchanged, we then have I1 …D† ˆ 140 and I2 …D† ˆ 118. Note that I1 …D† > I2 …D† in both cases and that I2 …D† ˆ 118 > D ˆ 100 when a ˆ 1. 6. Concluding remarks and future research The single-critical-level (reorder point) but non-order-up-to type policy shown to be optimal in our model is similar to that of the periodic review models with stochastically proportional random yield (e.g., Henig and Gerchak, 1990). More interesting properties of the optimal policy obtained here are that I1 …D† > I2 …D† >


> IN …D†, and that, in the absence of discounting, In …D† > D. In …D† > D implies that the total input quantity (work-in-process I plus the input x for the current batch) will be greater than the outstanding demand D. Most of the reject allowance literature have assumed that the input lot is always greater than the outstanding demand. While this seems rather intuitive and was proven to be true in the case of Binomial yield by Beja (1977), Grosfeld-Nir and Gerchak (1996) provide examples which show that it is not true in general. Several possible extensions to our current model are of interest both theoretically and practically. Like most of the reject allowance literature, we have assumed that the demand is deterministic. This may not always be the case in practice. We believe that a single critical level type policy will still be optimal when the demand is random. To incorporate setup costs into the current model and provide an analytical solution would be a dicult problem. This being so, some heuristics may need to be devised.

Appendix A. Proof of Proposition 1 From Eq. (2), we have, after some algebra, that of0 …D; I† ˆ ÿh ‡ …h ‡ p†G…D=I†; oD

…A:1†

o2 f0 …D; I† ˆ …h ‡ p†g…D=I†=I > 0; oD2

…A:2†

Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

of0 …D; I† ˆ lh ÿ …h ‡ p† oI

Z 0

D=I

ug…u† du;

381

…A:3†

o2 f0 …D; I† ˆ …h ‡ p†D2 g…D=I†=I 3 > 0; oI 2

…A:4†

o2 f0 …D; I† o2 f0 …D; I† ˆ ˆ ÿ…h ‡ p†Dg…D=I†=I 2 < 0; oDoI oIoD

…A:5†

o2 f0 …D; I† o2 f0 …D; I† o2 f0 …D; I† o2 f0 …D; I†
ˆ 0:
ÿ oD2 oI 2 oDoI oIoD

…A:6†

Thus, Part (i) of the Proposition follows from (A.2), (A.4) and (A.6), and Part (ii) follows from (A.4). Parts (iii) and (iv) will follow from a direct check of (A.1) and (A.3). 

Appendix B. Proof of Proposition 3 Part (i): From (7), it follows, after some algebra, that R1 2 fo f0 ‰D ÿ I1 …D†u; 0Š=oIoDgg…u† du dI1 …D† ˆ R 10 > 0; dD fo2 f0 ‰D ÿ I1 …D†u; 0Š=oIoDgug…u† du 0

where the inequality follows from Part (ii) of Proposition 1. Part (ii): From (8), we have that R1 2 fo f0 ‰D ÿ I1 …D†u; 0Š=oIoDgg…u† du ox1 …D; I† ˆ ÿ R0 1 > 0; oD fo2 f0 ‰D ÿ I1 …D†u; 0Š=oI 2 gg…u† du 0 and R1 2 fo f0 ‰D ÿ I1 …D†u; 0Š=oIoDgug…u† du ox1 …D; I† ˆ 0R 1 < 0; oI fo2 f0 ‰D ÿ I1 …D†u; 0Š=oI 2 gg…u† du 0 where the inequalities follow from Parts (ii) and (i) of Proposition 1 (which implies that o2 f0 ‰D ÿ I1 …D†u; 0Š=oI 2 > 0). Hence, x1 …D; I† is increasing in D and decreasing in I. 

Appendix C. Proof of Proposition 7 Part (i): By Parts (iii) and (iv) of Proposition 1, the left-hand side of (7) can be written as

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Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

Z

oV1 ‰0; D; I1 …D; I†Š ˆc‡a ox Z ‡a

D=I1 …D†

0 1

D=I1 …D†

Z

ˆc‡a

fof0 ‰D ÿ I1 …D†u; 0Š=oIgg…u† du

D=I1 …D† 0

fof0 ‰D ÿ I1 …D†u; 0Š=oIgg…u† du

Z …ÿlp†g…u† du ‡ a

1 D=I1 …D†

…lh†g…u†du

ˆ c ÿ alf…h ‡ p†G‰D=I1 …D†Š ÿ hg: So, I1 …D† is determined by the equation c ÿ alf…h ‡ p†G‰D=I1 …D†Š ÿ hg ˆ 0:

…C:1†

Eq. (16), by which I2 …D† is determined, can be written as (detailed proof follows) c ÿ a2 lf…h ‡ p†G…M†G‰D=I2 …D†Š ÿ hg ˆ 0;

…C:2†

where, M is a given constant with 0 < M < 1. Comparing (C.1) with (C.2), it is clear that each of the two extra factors a and G…M† in (C.2), both of which are between 0 and 1, will force I2 …D† to become less than I1 …D†. That is, in general we will have I1 …D† > I2 …D†. Now, we show the equivalence of (16) and (C.2). That is, we need to prove that oV2 ‰0; D; I2 …D†Š ˆ c ÿ a2 lf…h ‡ p†G…M†G‰D=I2 …D†Š ÿ hg: ox

…C:3†

First, when D > 0 and I ˆ 0, we have x1 …D; I† ˆ x1 …D; 0† > 0. From (8), x1 …D; 0† solves Z 1 fof0 ‰D; x1 …D; 0†Š=oIgg…u† du ˆ 0; c‡a 0

that is, c ‡ afof0 ‰D; x1 …D; 0†Š=oIg ˆ 0:

…C:4†

From (2), we have of0 …D; I† ˆ lh ÿ …h ‡ p† oI

Z 0

D=I

ug…u† du;

…C:5†

of0 …D; I† ˆ ÿh ‡ …h ‡ p†G…D=I†: oD Substituting (C.5) into (C.4), we get Z D=x1 …D; 0† ug…u† du ˆ 0: c ‡ alh ÿ a…h ‡ p†

…C:7†

De®ne D=x1 …D; 0†  M in (C.7) for all D > 0. So, M is a constant determined by Z M ug…u† du ˆ 0: c ‡ alh ÿ a…h ‡ p†

…C:8†

0

0

We can easily verify that 0 < M < 1, in order to satisfy (C.8).

…C:6†

Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

383

Second, for I < I1 …D†, we have from (10) that f1 …D; I† ˆ V1 ‰x1 …D; I†; D; IŠ. It follows that of1 …D; I† oV1 ‰x1 …D; I†; D; IŠ oV1 ‰x1 …D; I†; D; IŠ ox1 …D; I† ˆ ‡
oI oI ox oI oV1 ‰x1 …D; I†; D; IŠ ˆ oI Z 1 fof0 ‰D ÿ Iu; x1 …D; I†Š=oDgug…u† du; ˆ ÿa 0

where, the second equality follows from the optimality of x1 …D; I†, and the third from (4). When D > 0 and I ˆ 0, this reduces to of1 …D; 0† ˆ ÿa oI

Z

1

0

fof0 ‰D; x1 …D; 0†Š=oDgug…u† du

ˆ ÿalfof0 ‰D; x1 …D; 0†Š=oDg

…C:9†

ˆ ÿalfÿh ‡ …h ‡ p†G‰D=x1 …D; 0†Šg ˆ ÿal‰ÿh ‡ …h ‡ p†G…M†Š; where, the third equality follows from (C.6) and M is de®ned in (C.8). Third, using (C.9) and (11) for of1 …D; 0†=oI when D > 0 and D < 0, respectively, we can write, from (14), that oV2 ‰0; D; I2 …D†Š ˆc‡a ox

Z Z

ˆc‡a

0

D=I2 …D†

D=I2 …D† 0

Z

ˆc‡a Z ‡a

fof1 ‰D ÿ I2 …D†u; 0Š=oIgg…u† du

0

Z ‡a

1

0

fof1 ‰D ÿ I2 …D†u; 0Š=oIgg…u† du

D=I2 …D† 0

D=I2 …D†

fof1 ‰D ÿ I2 …D†u; 0Š=oIgg…u† du

fÿal‰ÿh ‡ …h ‡ p†G…M†Šgg…u† du

…alh†g…u† du

ˆ c ÿ a2 lf…h ‡ p†G…M†G‰D=I2 …D†Š ÿ hg; which gives us (C.3). We thus completed the proof of Part (i). Part (ii): For a ˆ 1, we ®rst show that I1 …D† > D and then I2 …D† > D. Assume I1 …D† < D. With a ˆ 1, the left-hand side of (C.1) would reduce to c ÿ lp < 0 ˆ right-hand side (where the inequality follows by our assumption c ÿ alp < 0 about the values of the cost parameters). Hence, we must have I1 …D† > D in order to satisfy (C.1), since oV1 …x; D; I†=ox is increasing in I. When a ˆ 1, the next argument shows that the left-hand side of (C.2) becomes negative if I2 …D† < D, and as a consequence, we must also have I2 …D† > D. If I2 …D† < D, simply substituting a ˆ 1 and G‰D=I2 …D†Š ˆ 1 into (C.2), we have

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Y. Wang, Y. Gerchak / European Journal of Operational Research 126 (2000) 371±385

Z Left-hand ˆ c ‡ lh ÿ l…h ‡ p†G…M† ˆ …h ‡ p† Z ˆ …h ‡ p†

M 0

Z ug…u† du ÿ l

0

M

0

M

ug…u† du ÿ l…h ‡ p†G…M† 

g…u† du

< 0;

RM where the second equality is due to c ‡ lh ˆ …h ‡ p† 0 ug…u† du from (C.8); and the inequality follows from Lemma 1 (see below). We thus completed the proof.  Lemma 1. If U is a random variable on [0, 1] with a density g(á) and mean l, then for any given value of M, 0 < M < 1, Z M Z M ug…u† du ÿ l g…u† du < 0: 0

0

Proof. Consider the following two cases: (a) For 0 < M < l, Z M Z M Z Z M ug…u† du ÿ l g…u† du < l ug…u† du ÿ l 0

0

(b) For l < M < 1, let Z Z M ug…u† du ÿ l F …M† ˆ 0

0

M 0

0

M

g…u† du ˆ 0:

g…u† du:

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