Multiple lot-sizing decisions with an interrupted geometric yield and variable production time

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Multiple lot-sizing decisions with an interrupted geometric yield and variable

production time

Hsi-Mei Hsu

*

_{, Tai-Sheng Su, Muh-Cherng Wu, Liang-Chuan Huang}

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 30010, Taiwan

a r t i c l e

i n f o

Article history:

Received 13 October 2006

Received in revised form 12 December 2008 Accepted 19 January 2009

Available online 25 January 2009 Keywords:

Lot-sizing

Interrupted geometric distribution Dynamic programming

Production lead-time Production/inventory system

a b s t r a c t

This study examines a multiple lot-sizing problem for a single-stage production system with an inter-rupted geometric distribution, which is distinguished in involving variable production lead-time. In a ﬁnite number of setups, this study determined the optimal lot-size for each period that minimizes total expected cost. The following cost items are considered in optimum lot-sizing decisions: setup cost, variable production cost, inventory holding cost, and shortage cost. A dynamic programming model is formulated in which the duration between current time and due date is a stage variable, and remaining demand and work-in-process status are state variables. This study then presents an algorithm for solving the dynamic programming problem. Additionally, this study examines how total expected costs of opti-mal lot-sizing decisions vary when parameters are changed. Numerical results show that the optimum lot-size as a function of demand is not always monotonic.

1. Introduction

Multiple lot-sizing production-to-order (MLPO) problems have been studied for several decades (Bowman, 1955). Such problems typically arise from variations in production yield. Consider a pro-duction system with an uncertain process yield. To fulﬁll a partic-ular customer demand, lots may need to be released several times to minimize total expected costs. The MLPO problem is to deter-mine the optimal lot-size for each possible lot release.

This study describes and formulates a single-stage MLPO prob-lem with one salient feature—uncertain production lead-time. According toYano (1987), this feature may arise due to many fac-tors such as unreliable vendors, unreliable transportation time, job queuing, machine breakdowns, and rework. Uncertain lead-time characteristic has seldom been considered in MLPO studies; although it has been examined in production control studies (Hsu, Wee, & Teng, 2007). In this study, we assume production lead-time is a random variable; the probability for one period is p and that for two periods is 1 p.

In the MLPO problem, process yield follows an interrupted geo-metric (IG) distribution. The delivery agreement includes due dates; that is, customers will not accept products after delivery due dates, and salvage values of products are negligible. In con-trast, ﬁnished goods produced ahead of the due date become

inventory and incur holding costs. The following cost items are in-cluded: setup cost, variable production cost, inventory holding cost, and shortage cost.

An example of the MLPO problem in this study is a process of drawing special steel coils. The manufacturing process has two operations: pickling and wire drawing. The pickling operation removes rust from steel coils. The processing time required for pick-ling a steel coil varies. In practice, a steel coil undergoes one or two pickling operations depending on the duration the coil has been in air. The drawing operation reduces the size of the input coil. Draw-ing speed is very fast. All coils in a lot are inspected when the whole lot is complete. The drawing operation involves a die that is worn gradually over time. When this die is excessively worn, the output does not meet speciﬁcations. This implies that the integrated draw-ing process follows an IG distribution, and production lead-time for a lot from release to output takes one or two periods. Special steels are customized products that in most cases cannot be sold to other customers. Thus, we assume product salvage value is negligible.

This study develops a dynamic programming (DP) approach to solve the MLPO problem. Several lemmas are proposed to reduce the DP problem solution space. Numerical experiments show that the optimum lot-size, as a function of demand, is not necessarily monotonic. This study experimentally investigated how total ex-pected costs of optimal lot-sizing decisions vary when various parameters change.

The remainder of this paper is organized as follows. A literature review is given inSection 2.Section 3presents the MLPO problem as a DP model by including a simple example to facilitate under-standing the formulation. Lemmas for reducing the DP solution

* Corresponding author. Tel.: +886 3 5731761; fax: +886 3 5722 392. E-mail addresses: [email protected] (H.-M. Hsu), tyson.iem92g@nctu. edu.tw(T.-S. Su),[email protected](M.-C. Wu),[email protected]

(L.-C. Huang).

Contents lists available atScienceDirect

Computers & Industrial Engineering

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space are presented inSection 4. An algorithm for solving the DP is presented inSection 5. Numerical examples are given inSection 6. Conclusions are provided inSection 7.

2. Related literature

Grosfeld-Nir and Gerchak (2004) and Yano and Lee (1995) com-prehensively surveyed studies of MLPO problem. Such studies can be categorized as: single-stage and multiple-stage. This study is in the category of single-stage MLPO problems; thus, recent studies in this category are reviewed.

Recent single-stage MLPO studies can be analyzed from multi-ple perspectives. The ﬁrst perspective is associated with customer demand and delivery requirements. Customer demand may be stochastic (Gerchak & Grosfeld-Nir, 1998) or deterministic. Delivery requirements can be based on due dates or quantities. In a tity-based agreement (also called rigid-demand delivery), the quan-tity ordered must delivered in full; that is, partial delivery is unacceptable. In due-date-based agreements (also called non-rigid demand delivery), customers will not accept products after the due date. Prior studies are either based on rigid-demand (e.g.,Anily, 1995; Anily, Beja, & Mendel, 2002; Beja, 1977; Zhang & Guu, 1998), or non-rigid demand (e.g., Guu & Zhang, 2003; Pentico, 1988; Sepehri, Silver, & New, 1986; Wang & Gerchak, 2000).

The second perspective is associated with production character-istics such as process yield, lead-time, and quality classiﬁcations. Previous studies assumed process yield is governed by a probabil-ity distribution, which includes the discrete uniform (Anily, 1995), the binomial distribution (Beja, 1977; Pentico, 1988; Sepehri et al., 1986), the interrupted geometric (Anily et al., 2002; Guu & Zhang, 2003; Zhang & Guu, 1998), the general distribution (Zhang & Guu, 1997), and the stochastically proportional (Grosfeld-Nir & Gerchak, 1990; Wang & Gerchak, 2000). In terms of lead-time, few studies (Wang & Gerchak, 2000) addressed an MLPO problem in which production lead-time is longer than the time epochs between any two lot releases. Most reseachers assumed production out-comes have only two possible states, either acceptable or unaccept-able quality, while a few other studies (Gerchak & Grosfeld-Nir,

1999) examined scenarios that may have three or more

out-comes—for example, high quality, medium quality and unaccept-able quality.

The third perspective is associated with cost items and objec-tive functions for the MLPO decision making. The most widely addressed cost items include setup cost, variable production cost, inventory cost, and shortage cost. A few researchers also consid-ered inspection cost (Grosfeld-Nir, Gerchak, & He, 2000) and

dis-posal cost (Wang & Gerchak, 2000). For the objective function, most researchers attempted to minimize total expected cost, while a few considered the impact of risk caused by cost variance ( Gros-feld-Nir & Gerchak, 1996).

The fourth perspective is associated with the solution approach. Most formulations of MLPO problems include recursive formulas and have been widely interpreted as DP problems. Therefore, DP has been widely used to solve MLPO problems; however, such a solution approach may be very demanding computationally. Some researchers proposed lemmas to reduce the solution space (Anily, 1995; Beja, 1977; Zhang & Guu, 1998); some others attempted to develop near–optimal heuristic rules (Pentico, 1988; Sepehri et al., 1986); and a few others approximately model the DP prob-lem using a relatively simpler non-DP probprob-lem for cases with ex-tremely large/small demand quantities (Anily et al., 2002).

The four perspectives highlight the various complex scenarios that can occur in single-stage MLPO problems. Some researchers investigated multiple-stage MLPO problems, in which additional complexity may arise due to inclusion of lot-sizing decisions made at the start of each stage. For example, a production system with two stages needs a lot-sizing decision for the ﬁrst stage. Releasing all output items of the ﬁrst stage immediately to the next stage may not be an optimal decision. A lot-sizing decision at the start of the second stage is needed. Example studies that addressed

the multiple-stage MLPO problem include Grosfeld-Nir (2005)

and Grosfeld-Nir and Robinson (1995).

Compared to those in literature, the singe-stage MLPO problem in this study is unique in that it includes one salient feature—uncer-tain production lead-time. This feature has rarely been considered in either single-stage or multiple-stage MLPO studies.

3. Modeling

To model the MLPO problem, the notation is ﬁrst presented, fol-lowed by a description of the IG distribution. A simple example is then given to explain the idea of the formulation. Finally, the cost function of the MLPO problem is modeled using a recursive for-mula, and its boundary conditions (BCs) are deﬁned.

3.1. IG distribution

As process yield is governed by an IG distribution, the produc-tion system manufactures each unit in a one-by-one manner and operates in two possible states in-control or of-control. The out-put unit is non-defective when the system is in-control, and is defective when the system is out-of-control. The process can Notation

D quantity required by a customer

T number of periods in the decision time horizon

t index of time, t = 0 is the due date, t = 0, 1, 2, . . . , T

a

setup cost incurred at each lot input,

a

> 0 b variable production cost per unit, b > 0

kt lot-size released at t

Wt a binary variable indicating the demand of a setup,

Wt¼ 0 if kt¼ 0

1 if kt>0

Dt remaining demand at t (number of demand units still

not fulﬁlled at t)

h inventory holding cost per unit per period

($/unit-period), h > 0

m shortage cost per unit, m > 0

p probability of producing a lot in one period

1 p probability of producing a lot in two periods

h probability that the production system is in-control

Ykt a random variable for the number of output units for lot kt

Rt(kt+1) number of work-in-process (WIP) at t,

Rtðktþ1Þ ¼ 0_ktþ1if the realized production time for ktþ1_otherwise is one period;

st= (Dt, Rt(kt+1)) the production system status at t, also called

state t

Ct(st, kt) total expected cost incurred after t

CtðstÞ ¼ Min

06kt61fCtðst;ktÞg minimum total expected cost incurred after t

Nt(st) optimal lot-size at state st; that is,

Min 06kt61

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switch from an in-control state to an out-of-control state, but not vice versa. This irreversible characteristic leads to the naming of ‘‘interrupted” geometric distribution. The IG distribution can be interpreted as follows. To produce exactly ykt non-defective units, the system must be in-control for the ﬁrst yktunits and out-of-con-trol at the ðyktþ 1Þ

th

unit; the probability is ð1 hÞhy_kt_{. In contrast,}

the probability of producing ktnon-defective units is hkt because

the system must be in-control for each unit produced. The IG dis-tribution is as follows: PðYkt ¼ yktÞ ¼ ð1 hÞhy_kt _y kt¼ 0; 1; 2; . . . ; kt 1; hkt _y kt¼ kt: ( 3.2. Simple example

Consider an order demand D with T = 3 as the current time (Fig. 1). Three lot-sizing decisions must be made at t = 3, 2, and 1, respectively. At t = 3, consider the state s3= (D, 0), where lot k3

is released. If lot k3 is completed at t = 2, then D2¼ D3 yk3, R2(k3) = 0, and s2¼ ðD3 yk3;0Þ. If lot k3is not completed at t = 2, then D2= D3, R2(k3) = k3, and s2= (D3, k3).

At t = 2, consider the state s2¼ ðD3 yk3;0Þ, where lot k2 is released. If lot k2 is completed at t = 1, then D1¼ D2 yk2, R1(k2) = 0, and s1¼ ðD2 yk2;0Þ. If lot k2 is not completed at t=1, then D1= D2, R1(k2) = k2, and s1= (D2, k2)

At t = 2, consider the state s2= (D3, k3), where lot k2is released. If

lot k2 is completed at t = 1, then D1¼ D2 yk3 yk2. That is, two lots, k3and k2, are now completed, where the realized production

time of lot k3is two periods and that of lot k2is one period.

Nota-bly, R1(k2) = 0 because all released lots (k3and k2) are now

com-pleted. Therefore, s1¼ ðD2 yk3 yk2;0Þ. If lot k2is not completed at t = 1, then D1¼ D2 yk3, R1(k2) = k2, and s1¼ ðD2 yk3;k2Þ. The other portions in Fig. 1 about the production system status can be likewise derived by following the above procedure.

3.3. Cost function formulation

Fig. 2shows the general representation of a lot-sizing decision made at st= (Dt, Rt(kt+1)). Cost function Ct(st, kt) in the intermediate

stage, when t P 1, can be formulated as follows:

Ctðst;ktÞ ¼ H1þ p X Rtðktþ1Þ y_{Rt ðktþ1Þ}¼0 Xkt y_kt¼0 pðyRtðktþ1ÞÞ pðyktÞ ðH2þ H3Þ þ ð1 pÞ X Rtðktþ1Þ y_{Rt ðktþ1Þ}¼0 pðyRtðktþ1ÞÞ ðH4þ H5Þ ð1Þ where H1¼

a

Wtþ bkt; H2¼ hðt 1ÞðyRtðktþ1Þþ yktÞ; H3¼ C_t1ðst1¼ ðDt yRtðktþ1Þ ykt;0ÞÞ; H4¼ hðt 1ÞyRtðktþ1Þ; H5¼ Ct1ðst1¼ ðDt yRtðktþ1Þ;ktÞÞ:

where H1is the aggregated production cost for lot kt, including both

setup and variable production costs. At st= (Dt, Rt(kt+1)) with a lot kt

released, the possible outcomes of st1can be represented in two

) 0 , ( 3 D s = k3 2 k ) , ( 2 2 1 D k s= 1 k ) 0 , ( 1 ₁ 0 D yk s = − ) , ( 1 1 0 D k s = ) 0 , ( 1 2 1 0 D yk yk s = − − ) , ( 1 1 0 D y2 k s = − k ) 0 , ( 2 3 2 1 D yk yk s = − − ) 0 , ( 1 1 0 D yk s = − ) , ( 1 1 0 D k s = ) 0 , ( 1 2 1 0 D yk yk s ₌ ₋ ₋ ) , ( ₁ ₁ 0 D y2 k s ₌ ₋ _k ) , ( 3 3 2 D k s = ) , ( 2 2 1 D y3 k s = − k 1 k 3 = t t=2 t=1 t=0 p p − 1 p p − 1 p − 1 p p p p p p − 1 p − 1 p − 1 p − 1 1 k 1 k 2 k ) 0 , ( 3 3 2 D yk s = − ) 0 , ( 2 2 1 D yk s = −

Fig. 1. A simple example illustrating the decision structure.

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(

( ) 1 t Rk 1 t t

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cases, which lead to the derivation of the second and third terms in (1)(Fig. 2).

Case 1. st1¼ ðDt yRtðktþ1Þ ykt;0Þ, with probability p. The num-ber of total output units is yRtðktþ1Þþ ykt, with a joint probability pðyRtðktþ1ÞÞ pðyktÞ. These output units are produced at t-1 and incur an inventory holding cost of H2. The expected inventory holding

cost can then be expressed as PRtðktþ1Þ y_{Rt ðktþ1Þ}¼0

Pkt

y_kt¼0pðyRtðktþ1ÞÞ pðyktÞ H2. The termPRyt_{Rt ðktþ1Þ}ðktþ1Þ¼0

Pkt

y_kt¼0pðyRtðktþ1ÞÞ pðyktÞ H3is the minimum total expected cost incurred after t 1.

Case 2. st1¼ ðDt yRtðktþ1Þ;ktÞ with probability 1 p. The number of total output units is yRtðktþ1Þ, with an occurrence probability pðyRtðktþ1ÞÞ. These output units are produced at t 1 and incur an inventory holding cost H4. The expected inventory holding cost

can then be represented as PRtðktþ1Þ

y_{Rt ðktþ1Þ}¼0pðyRtðktþ1ÞÞ H4. The term PRtðktþ1Þ

y_{Rt ðktþ1Þ}¼0pðyRtðktþ1ÞÞ H5denotes the minimum total expected cost incurred after t 1.

3.4. Boundary conditions

As a recursive formula,(1)has two BCs. The ﬁrst BC is intended to address costs incurred at t while Dt= 0; that is, st= (0, Rt(kt+1)).

Since demand now has been fulﬁlled, no lot needs to be released. Therefore, we can conclude:

C

tðst¼ ð0; Rtðktþ1ÞÞÞ ¼ 0: ð2Þ The second BC addresses the costs incurred at t = 0 with a status s0= (D0, R0(k1)). The WIP R0(k1) is produced after t = 0 and cannot

be used to fulﬁll the customer demand. Shortage cost for unful-ﬁlled demand D0> 0 is mD0. At t = 0, no lot is released and the cost

incurred is C 0ðs0¼ ðD0;R0ðk1ÞÞÞ ¼ mD0 if D0>0; 0 if D0¼ 0: ð3Þ

In summary,(1) is a recursive formula for determining costs prior to the due date,(2)is a BC for cost in the situation in which demand has already been satisﬁed, and(3)is a BC for cost in the case in which demand has not been satisﬁed at the due date. 4. Solution space reduction

The recursive formula in(1), as well as its two BCs, deﬁne a DP problem, where Nt(st) is to be found. To reduce the solution space, Lemma 1 is proposed to deﬁne an upper bound for Nt(st), with Proposition 1as a prerequisite to its proof.

Proposition 1. Given t P 1, DtP1, Rt(kt+1) P Dt, st= (Dt, Rt(kt+1)), and s0 t¼ ðDt;Rtðktþ1Þ þ 1Þ, then Ctðs0tÞ > C tðstÞ. Proof.

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kt; Ctðs0t;ktÞ Ctðst;ktÞ ¼ hðt 1ÞðE½YRtðktþ1Þþ1 E½YRtðktþ1ÞÞ ¼ hðt 1ÞðhRtðktþ1Þþ1_{Þ > 0:} That is, Ctðs0t;ktÞ > Ctðst;ktÞ, "kt. Let kt¼ Ntðs0tÞ. Then Ctðs0t;Ntðs0tÞÞ > Ctðst;Ntðs0tÞÞ. By deﬁnition, C tðs0tÞ ¼ Min kt fCtðs0t;ktÞg ¼ Ctðs0t;Ntðs0tÞÞ; CtðstÞ ¼ Min kt fCtðst;ktÞg ¼ Ctðst;NtðstÞÞ: Therefore, C tðstÞ ¼ Ctðst;NtðstÞÞ 6 Ctðst;Ntðs0tÞÞ < Ctðs0t;Ntðs0tÞÞ ¼ C tðs0tÞ . h

This proposition implies that while the WIP is greater than the remaining demand, changing a state by including one more unit in WIP typically increases cost. It is intuitively rational when consid-ering a case in which the remaining demand is 12 units. With an IG distribution, we infer that all the states with 12 units or more in WIP lead to the same probability for meeting remaining demand. However, for a state with additional quantity in WIP, holding cost of ﬁnished goods increases.

Lemma 1. Given t P 1, DtP1, and st= (Dt, Rt(kt+1)), then

Nt(st) 6 Dt.

Proof. If Nt(st) = 0, trivially, one can obtain Nt(st) = 0 < 1 6 Dt. If

Nt(st) P 1, "kt> Dt, Ctðst;ktÞ Ctðst;DtÞ ¼ bðkt DtÞ þ phðt 1ÞðE½Ykt E½YDtÞ þ ð1 pÞ X Rtðktþ1Þ y_{Rt ðktþ1Þ}¼0 pðyRtðktþ1ÞÞ ½C t1ðst1¼ ðDt yRtðktþ1Þ;ktÞÞ Ct1ðst1¼ ðDt yRtðktþ1Þ;DtÞÞ[bðkt DtÞ þ phðt 1ÞðE½Ykt

E½YDtÞ > 0ðby Proposition 1Þ

That is, Ct(st, kt) > Ct(st, Dt) for any kt> Dt. This implies that Nt(st) 6 Dt.

h

This lemma implies that the optimal lot size should always be less than or equal to remaining demand. It is helpful to reducing the solution space of the dynamic program. This lemma is intui-tively rational. Likewise, consider a case in which the remaining demand is 12 units. With an IG distribution, releasing a lot with at least 12 units would lead to the same probability of meeting remaining demand. Thus, at most 12 units should be released in this case.

Lemmas 2 and 3 are intended to quickly compute Nt(st) and

C

tðstÞ for cases of st= (1, Rt(kt+1)). InLemmas 2 and 3, thresholds

a and b are derived to determine whether to release a lot at st= (1, 0) and st= (1, Rt(kt+1) > 0), respectively.

Lemma 2. For st= (1, Rt(kt+1)), where Rt(kt+1) = 0,

if C_t1ðst1¼ ð1; 0ÞÞ 6 a; then NtðstÞ ¼ 0 and C tðstÞ ¼ Ct1ðst1¼ ð1; 0ÞÞ; if C t1ðst1¼ ð1; 0ÞÞ > a; then NtðstÞ ¼ 1 and CtðstÞ ¼

a

þ b þ phðt 1Þh þ pð1 hÞC_t1ðst1¼ ð1; 0ÞÞ þ ð1 pÞC_t1ðst1¼ ð1; 1ÞÞ; where a ¼ ½

a

þ b þ phðt 1Þh þ ð1 pÞCt1ðst1¼ ð1; 1ÞÞ=½1 pð1 hÞ.

Proof. If kt= 0, then Ctðst;0Þ ¼ Ct1ðst1¼ ð1; 0ÞÞ. If kt= 1, then

Ctðst;1Þ ¼

a

þ b þ phðt 1Þh þ pð1 hÞCt1ðst1¼ ð1; 0ÞÞ þ ð1 pÞ C t1ðst1¼ ð1; 1ÞÞ. Then, Ctðst;0Þ Ctðst;1Þ ¼ C_t1ðst1¼ ð1; 0ÞÞ ½

a

þ b þ phðt 1Þh þ pð1 hÞC_t1ðst1¼ ð1; 0ÞÞ þ ð1 pÞCt1ðst1¼ ð1;1ÞÞ ¼ Ct1ðst1¼ ð1; 0ÞÞ ½1 pð1 hÞ ½

a

þ b þ phðt 1Þh þ ð1 pÞCt1ðst1¼ ð1;1ÞÞ: Let a ¼ ½

a

þ b þ phðt 1Þh þ ð1 pÞCt1ðst1¼ ð1; 1ÞÞ=½1 pð1 hÞ Then, Ctðst;0Þ Ctðst;1Þ ¼ ½Ct1ðst1¼ ð1;0ÞÞ a ½1 pð1 hÞ

There-fore, we conclude that

if C_t1ðst1¼ ð1; 0ÞÞ 6 a; then NtðstÞ ¼ 0;

if C_t1ðst1¼ ð1; 0ÞÞ > a; then NtðstÞ ¼ 1: And C

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Lemma 3. For st= (1, Rt(kt+1)), where Rt(kt+1) > 0, if C_t1ðst1¼ ð1; 0ÞÞ 6 b; then NtðstÞ ¼ 0 and CtðstÞ ¼ hðt 1ÞE½YRtðktþ1Þ þ ð1 hÞC t1ðst1¼ ð1; 0ÞÞ; if C_t1ðst1¼ ð1; 0ÞÞ > b; then NtðstÞ ¼ 1 and C tðstÞ ¼

a

þ b þ hðt 1ÞE½YRtðktþ1Þ þ phðt 1Þh þ pð1 hÞ 2_C t1 ðst1¼ ð1; 0ÞÞ þ ð1 pÞð1 hÞCt1ðst1¼ ð1; 1ÞÞ; where b ¼ ½

a

þbþphðt 1Þhþð1pÞð1hÞCt1ðst1¼ ð1;1ÞÞ=fð1hÞ ½1pð1hÞg.

Proof. If kt= 0, then Ctðst;0Þ ¼ hðt 1ÞE½YRtðktþ1Þ þ ð1 hÞC

t1 ðst1¼ ð1; 0ÞÞ. If kt= 1, we have Ctðst;1Þ ¼

a

þ b þ hðt 1ÞE½YRtðktþ1Þ þ phðt 1Þh þ pð1 hÞ2C t1ðst1¼ ð1; 0ÞÞ þ ð1 pÞð1 hÞCt1ðst1¼ ð1; 1ÞÞ: Let b ¼ ½

a

þbþphðt 1Þhþð1pÞð1hÞC t1ðst1¼ ð1;1ÞÞ=fð1hÞ½1 pð1hÞg Then, Ctðst;0ÞCtðst;1Þ ¼ ½Ct1ðst1¼ ð1;0ÞÞbfð1hÞ½1

pð1hÞg. This implies that

if C_t1ðst1¼ ð1; 0ÞÞ 6 b; then NtðstÞ ¼ 0;

if C_t1ðst1¼ ð1; 0ÞÞ > b; then NtðstÞ ¼ 1: Thus, C

tðstÞ for each case can be accordingly computed. h

In summary,Proposition 1is a prerequisite ofLemma 1.Lemma 1is used to deﬁne the upper bound for releasing a lot, which helps reduce the solution space of the DP. BothLemmas 2 and 3are used to accelerate decision making when Dt= 1.

5. Dynamic programming algorithm

Based on the above lemmas, we propose an algorithm for com-puting NT(sT), beginning with sT= (D, 0).

Algorithm Computing_Optimal_Lot_Size (sT= (D, 0))

Step 1: Based on the ﬁrst BC, compute Nt(st) and CtðstÞ at

st= (0, Rt(kt+1)).

Step 2: Based on the second BC, compute N0(s0) and C0ðs0Þ at

s0= (D0, R0(k1)).

Step 3: Based on Lemma 2, compute Nt(st) and CtðstÞ at

st= (1, Rt(kt+1) = 0) for 1 6 t 6 T.

Fig. 3. The set of states considered in Steps 1 and 2.

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Step 4: Based on Lemma 3, compute Nt(st) and CtðstÞ at

st= (1, Rt(kt+1) > 0) for 1 6 t 6 T.

Step 5: Based on(1)andLemma 1, compute Nt(st) and CtðstÞ at

st= (2 6 Dt6D 1, Rt(kt+1) P 0) for 1 6 t 6 T.

Step 6: Based on(1)andLemma 1, compute NT(sT) and CtðstÞ at

sT= (D, 0).

The DP algorithm is utilized to calculate the cost function of each state in the DP decision tree, whose number of states could be quite huge. The complexity of the DP decision tree can be understood by examining the three component variables of a state st, which involves t, Dt, and Rt(kt+1). Step 1 determines Nt(st) and

C

tðstÞ for plane Dt= 0 based on the ﬁrst BC, and Step 2 is for the

plane t = 0 based on the second BC (Fig. 3). With the results ob-tained from Steps 1 and 2, Step 3 together with Step 4 determine Nt(st) and CtðstÞ for plane Dt= 1 (Fig. 4) based on Lemmas 2 and

3. The results for the remaining planes are computed by Steps 5 and 6 (Fig. 5).

6. Numerical examples

The properties of the multiple lot-sizing problems are examined using numerical examples. First, this work examines how the value

of decision parameters, T, D, p, h,

a

, b, h, and m, affect total expected cost. Second, this work examines whether NT(sT) against the order

size D is monotonically increasing. 6.1. Properties of decision parameters

We could readily justify that the total expected cost decreases as the values of cost parameters,

a

, b, h, and m decrease, while total expected cost decreases as the value of h increases. However, the relationships between total expected cost and parameters p, T, and D are not explicit and must be examined by numerical tests.

To examine the relationship between p and total expected cost, 700 cases are used, which are designed by setting D = 30, m = 200,

a

= 100 and b = 1, and varying the other parameter values as fol-lows: T = 3, 4, 6, and 10; p = 0, 0.1, 0.3, 0.5, 0.7, 0.9, and 1; h = 0.5, 0.6, 0.7, 0.8, and 0.9; and h = 0, 1, 3, 5, and 10.

Experimental results imply that total expected cost decreases with p when the value of p is sufﬁciently large (Fig. 6). This implies that a company that with shorter production lead-time tends to in-cur less cost. Suppose a motivational mechanism is established by sharing with workers a certain percentage (say, 20%) of the cost-sav-ing amount. The proposed model can then be used to determine the percentage that should be shared with workers for a particular p.

Fig. 5. The set of states considered in Steps 5 and 6.

1200 1600 2000 2400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p Cost h= 0 h= 1 h= 3 h= 5 h= 10

(7)

To examine the impacts of T and D on total expected cost, we used 700 cases which are designed by setting

a

= 100, b = 1 and m = 200, and varying the other parameter values as follows: T = 2, 4, 6, 8, and 10; D = 10, 20, 30, 40, and 50; p = 0, 0.1, 0.3, 0.5, 0.7, 0.9, and 1; h = 0.025, 2.5; and h = 0.6, 0.9.

Total expected cost decreases with T (Fig. 7), implying that the production of rush orders (with a small T value) increases cost. To ensure a constant contribution margin, say, 30%, for any lead-time commitment, one must adopt a price discrimination policy. That is, as lead-time commitment decreases the unit price charged in-creases. Thus, the proposed model is helpful in determining the pricing policy in terms of lead-time commitment.

Total expected cost increases as D increases in a convex manner (Fig. 8). InFig. 8, a scenario is considered in which unit price for any order size is constant. Based on revenue and cost curves, proﬁt, as a function of order size, does not increase monotonically (Fig. 8). This highlights the need for computing an optimal order size, which is one application of the proposed MLPO model.

6.2. Counter-examples of monotonic property

To justify whether NT(sT) is increasing monotonically with

re-spect to D, this work uses two scenarios, p=0.7 and p=1.0. The other parameters are T = 6,

a

= 50, b = 1, h = 1, m = 200, and h = 0.95. The proposed model is used to compute NT(sT) for 1 6 D 6 100.Fig. 9

presents computational results, revealing that NT(sT) as a function

of D is not necessarily monotonic, further supporting the

impor-tance of applying the proposed model to the lot-sizing decision. The size of NT(sT) for p = 0.7 is in general larger than that for

p = 1.0 (Fig. 9). Thus, the optimal lot-size for this scenario with var-iable lead-times was larger than that with a constant lead-time. 7. Conclusions

This study addresses a new single-stage MLPO problem, which is distinguished by its inclusion of one salient feature—production lead time is uncertain with two possible outcomes. That is, produc-tion lead time is either one or two periods. Such a problem has ap-peared in various production processes, such as when the drawing of steel coils; however, it has scarcely been studied in literature. This study formulates the MLPO problem as a dynamic problem and examines its properties via numerical experiments.

Some properties of decision variables (optimal lot-sizes) are summarized as follow. First, the optimal lot-size at any period is less than or equal to remaining demand, as proved inLemma 1. Second, optimal lot-size as a function of demand is not necessarily monotonic. Third, optimal lot-size with variable lead-time tends to be larger than that with a ﬁxed lead-time.

Properties of decision parameters T and p are also summarized. Total cost appears to decreases with T, implying that the unit sell-ing price can be lowered when customers accept an extended lead-time. While the value ofp is large enough, the higher is p, the lower the total expected cost tends to be. This implies that a production system would be more cost-competitive if its production lead-time

T

Fig. 7. Total expected cost as a function of T (T = 2, 4, 6, 8, 10, D = 50, p = 0.5, h = 0.9,a= 100, b = 1, h = 0.025, and m = 200).

D

Optimal order size

$

(8)

could become shorter, in terms of probability. With the proposed DP model, this study determined the lowest total expected cost for any production scenario, and in turn determined the appropri-ate quoted price.

If salvage costs for after-due products are not negligible, some lemmas in this work may not be valid. Therefore, one possible extension is to develop an MLPO model that includes substantial salvage costs. Another extension is to investigate the MLPO prob-lem with more than two possible outcomes in production lead-time. The proposed approach appears to be applicable to such an extension; however, formulating and solving a relatively much more complex DP problem is challenging. Additionally, some other extensions include investigating different probability distributions for modeling process yield and a scenario of a multiple-stage pro-duction system.

Acknowledgements

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for partially supporting this re-search under Contract No. NSC-93-2213-E-009-105. The anony-mous reviewers are thanked for their assistance in improving the quality of this paper. Ted Knoy is appreciated for his editorial assistance.

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