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Optimal multi-stage logistic and inventory policies with production bottleneck in a serial supply chain
ARTICLE IN PRESS Int. J. Production Economics 124 (2010) 408–413
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Optimal multi-stage logistic and inventory policies with production bottleneck in a serial supply chain Yu-Cheng Hsiao a,, Yi Lin b,c, Yun-Kuei Huang b a
Department of Logistics, Takming University of Science and Technology, Taiwan Department of Business Administration, Takming University of Science and Technology, Taiwan c Graduate Institute of Industrial and Business Management, National Taipei University of Technology, Taiwan b
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 October 2007 Accepted 23 November 2009 Available online 16 December 2009
Bottleneck appears in a serial supply chain if the minimum production rate at all stages is smaller than the demand rate. Operation manager must focus on keeping the bottleneck stage fully utilized and forcing the other stages to produce in synch with the bottleneck. This study applies the lot size division method, the recursive tightening method, and the drum-buffer-rope strategy. A pull and reverse pull algorithm is designed to solve the multi-stage logistic and inventory problem with a production bottleneck in a serial supply chain. A numerical example is included to illustrate the algorithm procedures. & 2009 Elsevier B.V. All rights reserved.
Keywords: Inventory Supply chain Bottleneck Lot size division Recursive tightening
1. Introduction In the 1980s, manufacturing organizations were required to become more flexible and responsive, which involved modifying existing products and processes. Organizations even had to develop new products to meet ever-changing customer needs. In the 1990s, managers realized that material and service inputs from suppliers significantly impacted organization performances in meeting customer needs (Handfield and Nichols, 1999). Recently, continuing advances in web communications and transportation technologies have accelerated the evolution of supply chains and the techniques for managing them. Companies are recognizing the importance of incorporating supply chain strategy into their overall planning (Lummus et al., 1998). Firms generally hold excess production capacity. This strategy allows them to respond sufficiently quickly to unexpected increases in demand, and also enables rapid delivery to the customers without overtime costs or production disruptions (Martinich, 1997). However, bottlenecks may occur in the supply chain if the minimal production rate of all stages is below the demand rate. External factors that affect the demand for firm products are sometimes beyond the control of management. A booming economy may increase demand (Krajewski and Ritzman, 1996). For example, in 1990 and 1993, Compaq Computer
Corresponding author. Tel.: + 886 2 26582507.
E-mail addresses: [email protected] (Y.-C. Hsiao), [email protected] (Y. Lin), [email protected] (Y.-K. Huang). 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.12.002
introduced new computer models for which demand outstripped production capacity (Burrows, 1994). On the other hand, equipment breakdowns and defects caused by machine wear are two major sources of lost production. For instance, an earthquake seriously damaged power systems and production facilities in Taiwan on September 21, 1999. The production capacities of many industries declined after the earthquake, especially those in the semiconductor industry. When production lags demand, enterprises may expand production capacity to prevent lost sales and simultaneously may markup prices to maximize profit. However, enterprises generally overlook reducing production costs, particularly logistic and inventory costs. In typical models for determining the economic production quantity (EPQ), the supplier produces and transports a single lot product to the customer. However, in order to decrease inventory level, partial lots (batches) can be transported to the customer. Szendrovits (1975) reduced both manufacturing cycle time and total costs with equal sized batches over all stages for a given order quantity. Transportation costs were treated as sunk costs. Goyal (1976) noted that Szendrovits’ model (1975), which involved fixed cost per transport in all stages could use the search procedure to determine the economic production quantity and optimal batches number. Assuming equal batch size at any particular stage and variation of batch number with stage, Goyal (1977a) and Szendrovits and Drezner (1980) determined the economic batch quantity and the optimal batch number at every stage with constant lot size. Also, Goyal (1977b) determined the optimal production quantity for a two-stage production system with unequal batch sizes that follow increasing or decreasing
ARTICLE IN PRESS Y.-C. Hsiao et al. / Int. J. Production Economics 124 (2010) 408–413
geometric series. Moreover, Goyal and Szendrovits (1986) presented a constant lot size model with equal and unequal sized batch shipments between production stages, and solved the model heuristically. Additionally, Vercellis (1999) proposed multi-plant production planning in capacitated self-configuring two-stage serial systems. The resulting mixed {0, 1} linear programming model was solved using LP-based heuristic algorithms. Furthermore, Cachon and Zipkin (1999) examined a twostage serial supply chain with fixed transportation time and no ordering costs. Most of the literature concerns only two stages (buyer and supplier). However, all stages of the supply chain system must be taken into consideration. Bogaschewsky et al. (2001) developed a model for a multi-stage production and inventory system and established a heuristic method for identifying an upper bound for the total cost of the optimal integer batch number solution and a lower bound for the corresponding production lot size. Subsequently, a scanning process was used to optimize the values of these bounds. In the serial supply chain, a uniform lot size was produced through all stages with a single setup and without interruption at each stage. Partial lots, called batches, may be transported to the next stage on completion. The number of unequal sized batches may differ among stages. Hsiao (2008) solved the one-warehouse multi-retailer problem by the order interval division (OID) and recursive tightening (RT) methods. The proposed solution procedure has a polynomial complexity of O(n log n) where n denotes the number of retailers involved in the problem. The Bogaschewsky et al. (2001) model assumed that the production rate at every stage exceeds the product demand rate. Actually, the supply chain may contain a bottleneck. Goldratt (1988) developed synchronous production, which was based on the theory of constraints, and recommended the drum-bufferrope method to solve the scheduling problem for a production system with a bottleneck. The key principle of the theory of constraints is that effective production management requires focusing on the constraining resources, namely the bottleneck. In synchronous production, the drum is the bottleneck and the mechanism controlling the pace of production. The drum pulls production from earlier stages and pushes it to subsequent stages. In a JIT pull system and a MRP push system, the drum is the final product demand and the master production schedule, respectively. A constraint is anything that limits an organization, operation, or a system from maximizing its output or meeting its stated objectives. Constraints may be physical such as insufficient labor or plant capacity or non-physical such as poor scheduling or lack of motivation. A bottleneck is also a constraint but its common usage in the supply chain is to describe a situation when the downstream operation has insufficient capacity to accept the upstream load. The capacity of a plant is governed by the physical space, the labor force, financial resources, materials, and machines. In the short-term operating environment, physical space is not normally variable. Plant expansion is considered long-term capacity planning. In the supply chain, one should concentrate on a smooth and regular flow of material through the system rather than attempting to balance the capacity (Goldratt, 1990; Waller, 2003). Operation manager and supply chain coordinator should face the production capacity constraints frequently. With continuing advances in production technologies and shorter product lifecycles, increasing numbers of companies, in industries as diverse as personal computers, toys, and even agricultural chemicals, are being forced to deal with markdowns. To protect against decline in inventory value, upstream firms only build inventory on receiving orders from downstream firms.
409
This study investigates a modification of the model of Bogaschewsky et al. (2001). In the multi-stage supply chain, the production rate of at least one stage should be smaller than the demand rate—that is, a bottleneck exists in the supply chain. This investigation adopts the drum-buffer-rope method to keep the bottleneck stage fully utilized and synchronize production during the other stages with the bottleneck. Since the demand and production rate are determined, no buffer is needed to protect the bottleneck against material shortages. Two different uniform lot sizes are produced: one that is pulled through the upstream stages from the bottleneck, and another that is pushed through the downstream stages from the bottleneck. Partial lots, or batches, can be transported to the next stage upon completion. Unequal batch sizes are allowed at each stage, as are different numbers of batches among stages. These unequal batch sizes at each stage follow increasing or decreasing geometric series (Goyal, 1977b). An integer nonlinear programming model is presented that considers setup costs, inventory holding costs and transportation costs for the whole system, and this model is divided into pull and push sub-models. Modify the order interval division and recursive tightening methods of Hsiao (2008), then the lot size division (LSD) and recursive tightening (RT) methods are developed. Based on the LSD and RT methods, the pull and reverse pull method is established to solve the pull and the push sub-models. A numerical example is presented to illustrate the procedures of the pull and reverse pull algorithm.
2. Notations and assumptions The notations used in this study are listed as follows: Qu Qd mj D Pj hj Sj Fj [Pj] + [Pj] Rj
upstream lot size pulled through the upstream stages from the bottleneck downstream lot size pushed through the downstream stages from the bottleneck number of batches at stage j constant product demand rate (units per unit time) constant production rate at stage j (units per unit time) holding cost of stage j per unit per unit time setup cost of stage j fixed transportation cost between stages j and (j+ 1) ($ per batch) the greater production rate of stages j and (j+ 1), or, max(Pj, Pj + 1) the smaller production rate of stages j and (j + 1), or, min(Pj, Pj + 1) production rate ratio of stages j and (j + 1), i.e., [Pj] + /[Pj]
The 11 assumptions of the model in this study are listed below. 1. The product demand rate is a known constant in the planning period. 2. The product units are infinitely divisible. 3. The supply chain comprises n serial manufacturing stages. Stage (n + 1) represents end-customer demand; therefore, Pn + 1 =D. 4. The production rate of stage k is the smallest of all stages, i.e., Pj 4Pk, for j= 1, 2,y,k 1, k+ 1,y,n. 5. The production rate of at least one stage is smaller than the demand rate; that is, a bottleneck exists in the supply chain. In the planning period, the supply chain can only produce Pk units of end products for customers. Consequently, there are shortages and lost sales at the demand stage for the end customers. 6. No ordering cost was involved. 7. Backlog and interruption are not permitted at every produc-
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8.
9.
10. 11.
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tion stage. Only a single setup is performed for a production lot at each production stage. Both upstream and downstream lot size are smaller than the production rate of stage k in the planning period, i.e., Qu oPk and Qd oPk; otherwise, the planning period should be extended. Upstream and downstream lot sizes are uniform: upstream lot size is pulled through the upstream stages from the bottleneck and downstream lot size is pushed through the downstream stages from the bottleneck; but partial lots, or batches, can be transported to the next stage upon completion. Consequently, batch size and batch number may differ across stages. Unequal batch sizes at each stage follow increasing or decreasing geometric series (Goyal, 1977b). Fixed batch transportation costs and transportation time between adjacent stages are constant, so variable transportation costs and holding costs during transportation are treated as sunk costs.
3. Integer nonlinear programming model
under the following conditions: Pi 4D
for
i A S1
ð8Þ
Pj oD
for
j A S2 ; S2 a |
ð9Þ
Pj 4 Pk
for
j ¼ 1; 2; . . . ; k1; kþ 1; . . . ; n
S 1 [ S 2 ¼ f1; 2; . . . ; ng
ð10Þ ð11Þ
The first and second parts of the objective function (1) represent the total costs of the upstream and downstream stages, respectively. Eqs. (2) and (3) state that both upstream and downstream lot size are smaller than the production rate of stage k in the planning period. Eqs. (9) and (10) demonstrate that stage k is the bottleneck stage. Meanwhile, Eqs. (4)–(7) together define the decision variables. When the objective function and decision variables are gathered, the original model can be divided into two sub-models: the upstream sub-model and the downstream sub-model. The upstream sub-model can be expressed as ( " # ) k1 hj Qu2 ðRj 1Þ hj Qu2 Pk X 1 1 þSj þ Fj mj þ Minimize Qu j ¼ 1 ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j ð12Þ
Bogaschewsky’s model (2001) considers a serial supply chain in which a uniform lot size was produced through all stages with a single setup and without interruption at each stage. The number of unequal sized batches may differ among stages. The production rate at every stage exceeds the product demand rate. This study examines a serial supply chain in which the production rate of stage k is the smallest of all stages, and the production rate of at least one stage is smaller than the demand rate; that is, stage k is the bottleneck stage. Let S1 and S2 be the sets of the production stages’ number. If Pi 4D, then iAS1. If Pj oD, then jAS2. Since the production rate of at least one stage is smaller than the demand rate and stage k is the bottleneck stage, the set S2 has at least one element, k, and S2 a|. Goldratt’s (1988) synchronous production concept and the drum-buffer-rope method are adopted to maintain full bottleneck stage utilization. Uniform upstream lot size Qu and downstream lot size Qd are produced: upstream lot size Qu is pulled through the upstream stages from the bottleneck and downstream lot size Qd is pushed through the downstream stages from the bottleneck. During the planning period, the supply chain can only produce Pk units of end products for customers. A serial supply chain with bottleneck problem thus can be formulated as the following integer nonlinear programming model: ( " # ) k1 hj Qu2 ðRj 1Þ hj Qu2 Pk X 1 1 þ S þ þ F m Minimize j j j Qu j ¼ 1 ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j n P X þ k Qd j ¼ k
(
hj Qd2 ðRj 1Þ m
ðRj j 1Þ½Pj þ
" # ) hj Qd2 1 1 þ S þ þF m j j j 2 ½Pj ½Pj þ
ð1Þ
subject to Pk 4Qu
ð2Þ
Pk 4Qd
ð3Þ
Qu 40
ð4Þ
Qd 40
ð5Þ
mj Z 1
for
j ¼ 1; 2; . . . ; n
for
j ¼ 1; 2; . . . ; n
Pk 4 Qu
ð13Þ
Qu 4 0
ð14Þ
Pj 4 Pk
for
j ¼ 1; 2; . . . ; k1
ð15Þ
mj Z 1
for
j ¼ 1; 2; . . . ; k1
ð16Þ
and mj are integers
for
j ¼ 1; 2; . . . ; k1
ð17Þ
Meanwhile, the downstream sub-model can be expressed as ( " # ) n hj Qd2 ðRj 1Þ hj Qd2 Pk X 1 1 Minimize þ Sj þ Fj mj þ Qd j ¼ k ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j
ð18Þ subject to Pk 4 Qd
ð19Þ
Qd 4 0
ð20Þ
Pj 4 Pk
for
j ¼ kþ 1; k þ 2; . . . ; n; n þ 1
ð21Þ
mj Z 1
for
j ¼ k; k þ1; k þ 2; . . . ; n
ð22Þ
and mj are integers
for
j ¼ k; k þ1; k þ 2; . . . ; n
ð23Þ
Inspecting the objective functions and the decision variables reveals no correlation between the upstream and downstream sub-models; in other words, both sub-models are mutually independent. The optimal solutions of two sub-models could be obtained, respectively, in this study. Therefore, this study concluded that: the minimal cost of the original model is the sum of the minimal costs of two sub-models and the optimal solution of the original model is the union of the optimal solutions of two sub-models.
ð6Þ
4. The methods
ð7Þ
Let M represent the batch vector and let TCu(Qu, Mu) and TCd(Qd, Md) be the objective function of the upstream sub-model
and mj are integers
subject to
ARTICLE IN PRESS Y.-C. Hsiao et al. / Int. J. Production Economics 124 (2010) 408–413
and the downstream sub-model, respectively. Setting qTCu(Qu, Mu)/qQu and qTCd(Qd, Md)/qQd to zero, we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " #)ffi , u k1 k1 X uX hj ðRj 1Þ hj 1 1 t Qu ðMu Þ ¼ ðSj þ mj Fj Þ þ 2 ½Pj ½Pj þ ððRj Þmj 1Þ½Pj þ j¼1 j¼1 ð24Þ and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " #) , u n n X uX hj ðRj 1Þ hj 1 1 Qd ðMd Þ ¼ t ðSj þ mj Fj Þ þ 2 ½Pj ½Pj þ ððRj Þmj 1Þ½Pj þ j¼k j¼k
ð25Þ
4.1. The lot size division method Modify the order interval division method of Hsiao (2008), the s be the critical lot size division (LSD) method is developed. Let Qj;m j lot size of the stage j, we have s hj ðQj;m Þ2 ðRj 1Þ j mj
ððRj Þ 1Þ½Pj
þ
¼
s hj ðQj;m Þ2 ðRj 1Þ j
ððRj Þmj þ 1 1Þ½Pj þ
þFj
s , then Solve for Qj;m j vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Fj ½Pj þ u s ! ¼u Qj;m j u 1 1 th ðR 1Þ j j ðRj Þmj 1 ðRj Þmj þ 1 1
ð26Þ
ð27Þ
Lemma 1. For stage j in an n-stage serial supply chain, if the lot size s Q is greater than the critical lot size Qj;m , then (mj +1) is chosen as the j number of batches; otherwise, mj is selected as the number of batches. s , For j= 1, 2,y,n, use Eq. (27) to find all critical lot sizes Qj;m j s s namely, Qj;1 , Qj;2 and so on, till the resulting critical lot size exceeds D. Merge these critical lot sizes found for all stages to form an ordered critical lot size vector, lot size ranges RG and batch vectors M. The procedures of the lot size division method are similar to the procedures of the order interval division method in Section 3 of Hsiao’s paper (2008).
411
4.3. The pull and reverse pull methods Goldratt (1988) designed synchronous production, which is based on the theory of constraints, and also recommended the drum-buffer-rope method for solving the scheduling problem for the production system containing a bottleneck. In synchronous production, the drum is both the bottleneck and the mechanism controlling the pace of production. The production schedule is developed around the bottleneck, which pulls production from earlier stages and pushes it to the subsequent stages. In a JIT pull system and a MRP push system, the drum is both the final product demand and the master production schedule. We have developed the LSD and RT methods to solve the logistic and inventory problem of a serial supply chain that is a JIT pull system. If stage k (the bottleneck stage) is treated as the final product demand stage, the upstream sub-model becomes a kstage serial supply chain in which production rate at every stage exceeds the product demand rate. The LSD and RT methods can be easily applied to the upstream sub-model to solve the optimal upstream lot size Qu and the batch number associated with each stage. The LSD and RT methods cannot be directly applied to the downstream sub-model because the downstream sub-model is an MRP push system. Therefore the reverse pull method is designed to solve the MRP push system. First, pseudo-reverse procedures transfer a push system into a pseudo-pull system. The pseudoreverse procedures include three steps: (1) pseudo-reversing the logistic direction of a serial supply chain system; (2) letting the demand stage be the pseudo-earliest stage; and (3) letting the earliest stage be the pseudo-demand stage. After executing the pseudo-reverse procedures, the downstream sub-model was converted to a pseudo-pull model. The LSD and RT methods can be directly applied to this pseudo-pull model to solve the optimal solutions and obtain the minimal cost. The pseudo-reverse procedures then are executed again, and the pseudo-pull model backs into the downstream sub-model. The minimal cost of the downstream sub-model is the same as the minimal cost of the pseudo-pull model. Moreover, the pseudoreversed optimal solutions of the pseudo-pull model are the optimal solutions of the downstream sub-model. A special case does exist. If the earliest stage is the bottleneck stage, then the original problem is simply an MRP push system. This problem can be solved using the reverse pull method only.
4.2. The recursive tightening method Modify the recursive tightening method of Hsiao (2008), two theorems can be rewritten as follows: Theorem 1. For an n-stage serial supply chain, if the lot size Qx(Mi) does not lie within its associated lot size range RGi, but rather lies in the lot size range RGj, where x= u, d and ioj, then TCx(Qx(Mi), Mi)4TCx(Qx(Mi), Mj). Consequently, none of the batch vectors Mi, Mi + 1,y, or Mj 1 is the global optimal batch vector, and none of Qx(Mi), Qx(Mi + 1),y, or Qx(Mj 1) is the global optimal lot size. Theorem 2. For an n-stage serial supply chain, if the lot size Qx(Mi) does not lie within its associated lot size range RGi, but rather lies in the lot size range RGj, where x= u, d and i4j, then TCx(Qx(Mi), Mi)4TCx(Qx(Mi), Mj). Consequently, none of the batch vectors Mi, Mi + 1,y, or Mj 1 is the global optimal batch vector, and none of Qx(Mi), Qx(Mi + 1),y, or Qx(Mj 1) is the global optimal lot size. The upward and downward recursive tightening procedures are similar to the procedures of the recursive tightening method in Section 4 of Hsiao’s paper (2008).
5. Algorithm The following procedures integrate the pull and reverse pull methods to determine the optimal solutions for the problem being studied. Step 1. Step 2. Step 3.
Step 4.
Let Pmin =D and k=n + 1. For j = 1, 2, 3,y,n, do If Pj oPmin, then let Pmin =Pj and k=j. If k=n + 1 (i.e. without bottleneck), then Use the LSD and RT methods to solve the problem and stop. Execute the pseudo-reverse procedures to convert the downstream sub-problem (from stage k to the demand stage) to be a pseudo-pull problem. (a) Pseudo-reverse the logistic direction of the downstream sub-problem. (b) Let the demand stage of the original problem be the pseudo-earliest stage of the pseudo-pull problem.
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Step 5. Step 6.
Step 7.
Step 8.
Step 9.
(c) Let stage k of the original problem be the pseudo-demand stage of the pseudo-pull problem. Use the LSD and RT methods to solve the pseudopull problem. Execute the pseudo-reverse procedures to reverse the logistic direction of the solution of the pseudo-pull problem. If k= 1 (i.e. the earliest stage is the bottleneck stage), then the optimal solutions are obtained and stop. Let stage k of the original problem denote the final product demand stage of the upstream subproblem. Use the LSD and RT methods to solve the upstream sub-problem. The optimal solution of the original model is the combination of the optimal solutions of the two sub-models. The minimal cost of the original model is the sum of the minimal costs of the two sub-models.
i
Lot size region RGi
m1, m2 Mi
Q(Mi)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
(0, 167.3] [167.3, 182.6] [182.6, 293.8] [293.8, 324.9] [324.9, 424.1] [424.1, 478.2] [478.2, 562.5] [562.5, 650.6] [650.6, 712.2] [712.2, 849.6] [849.6, 876.6] [876.6, 1059.1] [1059.1, 1083.5] [1083.5, 1263.4] [1263.4, 1362.0] [1362.0, 1494.0] [1494.0, 1696.7]
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8,
351.6
a
1 2 2 3 3 4 4 5 5 6 6 7 8 8 9 9 10
Total cost of upstream
653.6
866.3 951.3 998.7a 1039.5 1067.0
423.57
1160.4
The optimal lot size of upstream.
Table 3 Results of the psedo-downstream sub-problem using LSD and RT methods.
6. Numerical example The inventory problem of a serial supply chain involves four manufacturing stages. The 5th stage is the demand stage, during which the demand rate is 2150 units per month. Table 1 lists the problem data. The production rate of stage 3 is the smallest of all stages and is also smaller than the demand rate. Stage 3 is the bottleneck of the supply chain. Therefore, the problem can be divided into two sub-problems: one is the upstream sub-problem containing stages 1, 2 and 3; the other is the downstream subproblem containing stages 3, 4 and the demand stage. After performing the LSD and RT methods, the lot size ranges RGi and the associated batch vector Mi of the upstream subproblem and the downstream sub-problem are listed in Tables 2 and 3, respectively. In Table 2, there are 17 lot size ranges (RGi). Use Eq. (24) to calculate the lot size of RG1, we have Q(M1) =351.6 which lies in RG5. Applying recursive tightening method, the upward recursive tightening procedures are expressed as follows: (1) (2) (3) (4)
Table 2 Results of the upstream sub-problem using LSD and RT methods.
Q(M5)= 653.6 which falls in RG9. Q(M9)= 866.3 which lies in RG11. Q(M11)= 951.3 which falls in RG12. Q(M12)= 998.7 which lies in RG12.
i
Lot size region RGi
m3, m4 Mi
Q(Mi)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(0, 160.2] [167.3, 182.6] [182.6, 277.6] [277.6, 324.9] [324.9, 392.8] [392.8, 478.2] [478.2, 507.4] [507.4, 622.0] [622.0, 650.6] [650.6, 736.7] [736.7, 849.6] [849.6, 851.7] [851.7, 967.1] [967.1, 1082.9] [1082.9, 1083.5] [1083.5, 1199.2] [1199.2, 1316.2] [1316.2, 1362.0] [1362.0, 1433.8] [1433.8, 1552.1]
1, 1 1, 2 2, 2 2, 3 3, 3 3, 4 4, 4 4, 5 4, 6 5, 6 5, 7 6, 7 6, 8 6, 9 6,10 7,10 7,11 7,12 8,12 8,13
195.6
a b
The same procedures are performed for the downward recursive tightening. Only the lot size Q(M12) falls in its own lot size range RG12. According to the RT method, Qu =Q(M12)= 998.7 and (m1, m2)= M12 =(6, 7) are the optimal solutions to the upstream sub-problem. Moreover, the optimal total cost of the upstream sub-problem is 423.57.
Table 1 The data of the numerical example.
321.2 374.5 437.1 489.8 545.1 596.2b 641.3b 694.5b 738.7a
Total cost of downstream
513.36 509.92 505.40 503.61
827.9
948.2
1090.7
The optimal lot size of downstream. The lot sizes located in their own lot size ranges.
Table 3 contains four downstream lot sizes located in their own lot size ranges. The total costs for each downstream lot size then can be determined and the smallest one selected. Qd =738.7 and (m3, m4)= (5, 7) denote the optimal solutions to the downstream sub-problem. The optimal total cost of the downstream sub-problem is 503.61. The optimal solutions of two sub-models are combined as follows: (m1, m2, m3, m4)=(6, 7, 5, 7). Finally, the total cost of the overall problem is 927.18. Table 4 lists the batch sizes of the four stages.
7. Conclusions
Number of stage j
Production rate Pj (units/ month)
Setup cost Sj ($)
Fixed transportation cost Fj ($/trip)
Holding cost hj ($/units month)
1 2 3 4
3000 2000 1500 2250
45 30 20 25
4 6 8 7
0.60 0.75 0.90 1.20
This study proposes an integer nonlinear programming model for a multi-stage logistic and inventory problem in a serial supply chain with a production bottleneck where the unequal batch sizes at each stage follow increasing or decreasing geometric series. The drum-buffer-rope method is adopted to keep the bottleneck stage fully utilized and to force the other stages to produce in synch with the bottleneck. Moreover, the reverse pull method converts a
ARTICLE IN PRESS Y.-C. Hsiao et al. / Int. J. Production Economics 124 (2010) 408–413
References
Table 4 The batch sizes of four stages. The batch sizes
Batch
Stage
1 2 3 4 5 6 7
413
1
2
3
4
48.1 72.1 108.1 162.2 243.3 364.9
51.3 68.4 91.2 121.5 162.1 216.1 288.1
266.7 177.8 118.5 79.0 52.2
91.7 96.0 100.4 105.1 110.0 115.1 120.4
push system into a pseudo-pull system. Therefore, the LSD and RT methods can be directly applied to this pseudo-pull model to obtain the minimal cost and the optimal solutions. Additionally, the pseudo-reversed optimal solutions of the pseudo-pull model are allowed to be the optimal solutions of the downstream submodel. The minimal cost of the original model is the sum of the minimal costs of the two sub-models and the optimal solution of the original model is the union of the optimal solutions of the two sub-models.
Acknowledgment The author would like to thank the National Science Council (NSC) of Taiwan for financially supporting this research under Contract no. NSC 91-2416-H-147-002.
Bogaschewsky, R.W., Buscher, U.D., Lindner, G., 2001. Optimizing multi-stage production with constant lot size and varying number of unequal sized batches. Omega 29, 183–191. Burrows, P., 1994. Compaq stretches for the crown. Business Week July 11, 140–142. Cachon, G.P., Zipkin, P.H., 1999. Competitive and cooperative inventory policies in a two-stage supply chain. Management Science 45, 936–953. Goldratt, E., 1988. Computerized shop floor scheduling. International Journal of Production Research 26, 443–455. Goldratt, E., 1990. Theory of Constraints. North River Press, Massachusetts. Goyal, S.K., 1976. Note on: manufacturing cycle time determination for a multistage economic production quantity model. Management Science 23, 332–333. Goyal, S.K., 1977a. Economic batch quantity in a multi-stage production system. International Journal of Production Research 16, 267–273. Goyal, S.K., 1977b. Determination of optimum production quantity for a two-stage production system. Operation Research Quarterly 28, 865–870. Goyal, S.K., Szendrovits, A.Z., 1986. A constant lot size model with equal and unequal sized batch shipments between production stages. Engineering Costs and Production Economics 10, 203–210. Handfield, R.B., Nichols Jr., E.L., 1999. Introduction to Supply Chain Management. Prentice-Hall, Englewood Cliffs, NJ. Hsiao, Y.C., 2008. Optimal single-cycle polices for the one-warehouse multiretailer inventory/distribution system. International Journal of Production Economics 114, 219–229. Krajewski, L.J., Ritzman, L.P., 1996. Operations Management: Strategy and Analysis. Addison Wesley, New York. Lummus, R.R., Vokurka, R.J., Alber, K.L., 1998. Strategic supply chain planning. Production and Inventory Management Journal 39, 49–58. Martinich, S.J., 1997. Production and Operations Management. Wiley, New York. Szendrovits, A.Z., 1975. Manufacturing cycle time determination for a multi-stage economic production quantity model. Management Science 22, 298–308. Szendrovits, A.Z., Drezner, Z., 1980. Optimizing multi-stage production with constant lot size and varying numbers of batches. Omega 8, 623–629. Vercellis, C., 1999. Multi-plant production planning in capacitated self-configuring two-stage serial systems. European Journal of Operational Research 119, 451–460. Waller, D.L., 2003. Operations Management: A Supply Chain Approach, second ed. Thomson Business Press, London.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Optimal multi-stage logistic and inventory policies with production bottleneck in a serial supply chain Yu-Cheng Hsiao a,, Yi Lin b,c, Yun-Kuei Huang b a
Department of Logistics, Takming University of Science and Technology, Taiwan Department of Business Administration, Takming University of Science and Technology, Taiwan c Graduate Institute of Industrial and Business Management, National Taipei University of Technology, Taiwan b
a r t i c l e in f o
a b s t r a c t
Article history: Received 23 October 2007 Accepted 23 November 2009 Available online 16 December 2009
Bottleneck appears in a serial supply chain if the minimum production rate at all stages is smaller than the demand rate. Operation manager must focus on keeping the bottleneck stage fully utilized and forcing the other stages to produce in synch with the bottleneck. This study applies the lot size division method, the recursive tightening method, and the drum-buffer-rope strategy. A pull and reverse pull algorithm is designed to solve the multi-stage logistic and inventory problem with a production bottleneck in a serial supply chain. A numerical example is included to illustrate the algorithm procedures. & 2009 Elsevier B.V. All rights reserved.
Keywords: Inventory Supply chain Bottleneck Lot size division Recursive tightening
1. Introduction In the 1980s, manufacturing organizations were required to become more flexible and responsive, which involved modifying existing products and processes. Organizations even had to develop new products to meet ever-changing customer needs. In the 1990s, managers realized that material and service inputs from suppliers significantly impacted organization performances in meeting customer needs (Handfield and Nichols, 1999). Recently, continuing advances in web communications and transportation technologies have accelerated the evolution of supply chains and the techniques for managing them. Companies are recognizing the importance of incorporating supply chain strategy into their overall planning (Lummus et al., 1998). Firms generally hold excess production capacity. This strategy allows them to respond sufficiently quickly to unexpected increases in demand, and also enables rapid delivery to the customers without overtime costs or production disruptions (Martinich, 1997). However, bottlenecks may occur in the supply chain if the minimal production rate of all stages is below the demand rate. External factors that affect the demand for firm products are sometimes beyond the control of management. A booming economy may increase demand (Krajewski and Ritzman, 1996). For example, in 1990 and 1993, Compaq Computer
Corresponding author. Tel.: + 886 2 26582507.
E-mail addresses: [email protected] (Y.-C. Hsiao), [email protected] (Y. Lin), [email protected] (Y.-K. Huang). 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.12.002
introduced new computer models for which demand outstripped production capacity (Burrows, 1994). On the other hand, equipment breakdowns and defects caused by machine wear are two major sources of lost production. For instance, an earthquake seriously damaged power systems and production facilities in Taiwan on September 21, 1999. The production capacities of many industries declined after the earthquake, especially those in the semiconductor industry. When production lags demand, enterprises may expand production capacity to prevent lost sales and simultaneously may markup prices to maximize profit. However, enterprises generally overlook reducing production costs, particularly logistic and inventory costs. In typical models for determining the economic production quantity (EPQ), the supplier produces and transports a single lot product to the customer. However, in order to decrease inventory level, partial lots (batches) can be transported to the customer. Szendrovits (1975) reduced both manufacturing cycle time and total costs with equal sized batches over all stages for a given order quantity. Transportation costs were treated as sunk costs. Goyal (1976) noted that Szendrovits’ model (1975), which involved fixed cost per transport in all stages could use the search procedure to determine the economic production quantity and optimal batches number. Assuming equal batch size at any particular stage and variation of batch number with stage, Goyal (1977a) and Szendrovits and Drezner (1980) determined the economic batch quantity and the optimal batch number at every stage with constant lot size. Also, Goyal (1977b) determined the optimal production quantity for a two-stage production system with unequal batch sizes that follow increasing or decreasing
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geometric series. Moreover, Goyal and Szendrovits (1986) presented a constant lot size model with equal and unequal sized batch shipments between production stages, and solved the model heuristically. Additionally, Vercellis (1999) proposed multi-plant production planning in capacitated self-configuring two-stage serial systems. The resulting mixed {0, 1} linear programming model was solved using LP-based heuristic algorithms. Furthermore, Cachon and Zipkin (1999) examined a twostage serial supply chain with fixed transportation time and no ordering costs. Most of the literature concerns only two stages (buyer and supplier). However, all stages of the supply chain system must be taken into consideration. Bogaschewsky et al. (2001) developed a model for a multi-stage production and inventory system and established a heuristic method for identifying an upper bound for the total cost of the optimal integer batch number solution and a lower bound for the corresponding production lot size. Subsequently, a scanning process was used to optimize the values of these bounds. In the serial supply chain, a uniform lot size was produced through all stages with a single setup and without interruption at each stage. Partial lots, called batches, may be transported to the next stage on completion. The number of unequal sized batches may differ among stages. Hsiao (2008) solved the one-warehouse multi-retailer problem by the order interval division (OID) and recursive tightening (RT) methods. The proposed solution procedure has a polynomial complexity of O(n log n) where n denotes the number of retailers involved in the problem. The Bogaschewsky et al. (2001) model assumed that the production rate at every stage exceeds the product demand rate. Actually, the supply chain may contain a bottleneck. Goldratt (1988) developed synchronous production, which was based on the theory of constraints, and recommended the drum-bufferrope method to solve the scheduling problem for a production system with a bottleneck. The key principle of the theory of constraints is that effective production management requires focusing on the constraining resources, namely the bottleneck. In synchronous production, the drum is the bottleneck and the mechanism controlling the pace of production. The drum pulls production from earlier stages and pushes it to subsequent stages. In a JIT pull system and a MRP push system, the drum is the final product demand and the master production schedule, respectively. A constraint is anything that limits an organization, operation, or a system from maximizing its output or meeting its stated objectives. Constraints may be physical such as insufficient labor or plant capacity or non-physical such as poor scheduling or lack of motivation. A bottleneck is also a constraint but its common usage in the supply chain is to describe a situation when the downstream operation has insufficient capacity to accept the upstream load. The capacity of a plant is governed by the physical space, the labor force, financial resources, materials, and machines. In the short-term operating environment, physical space is not normally variable. Plant expansion is considered long-term capacity planning. In the supply chain, one should concentrate on a smooth and regular flow of material through the system rather than attempting to balance the capacity (Goldratt, 1990; Waller, 2003). Operation manager and supply chain coordinator should face the production capacity constraints frequently. With continuing advances in production technologies and shorter product lifecycles, increasing numbers of companies, in industries as diverse as personal computers, toys, and even agricultural chemicals, are being forced to deal with markdowns. To protect against decline in inventory value, upstream firms only build inventory on receiving orders from downstream firms.
409
This study investigates a modification of the model of Bogaschewsky et al. (2001). In the multi-stage supply chain, the production rate of at least one stage should be smaller than the demand rate—that is, a bottleneck exists in the supply chain. This investigation adopts the drum-buffer-rope method to keep the bottleneck stage fully utilized and synchronize production during the other stages with the bottleneck. Since the demand and production rate are determined, no buffer is needed to protect the bottleneck against material shortages. Two different uniform lot sizes are produced: one that is pulled through the upstream stages from the bottleneck, and another that is pushed through the downstream stages from the bottleneck. Partial lots, or batches, can be transported to the next stage upon completion. Unequal batch sizes are allowed at each stage, as are different numbers of batches among stages. These unequal batch sizes at each stage follow increasing or decreasing geometric series (Goyal, 1977b). An integer nonlinear programming model is presented that considers setup costs, inventory holding costs and transportation costs for the whole system, and this model is divided into pull and push sub-models. Modify the order interval division and recursive tightening methods of Hsiao (2008), then the lot size division (LSD) and recursive tightening (RT) methods are developed. Based on the LSD and RT methods, the pull and reverse pull method is established to solve the pull and the push sub-models. A numerical example is presented to illustrate the procedures of the pull and reverse pull algorithm.
2. Notations and assumptions The notations used in this study are listed as follows: Qu Qd mj D Pj hj Sj Fj [Pj] + [Pj] Rj
upstream lot size pulled through the upstream stages from the bottleneck downstream lot size pushed through the downstream stages from the bottleneck number of batches at stage j constant product demand rate (units per unit time) constant production rate at stage j (units per unit time) holding cost of stage j per unit per unit time setup cost of stage j fixed transportation cost between stages j and (j+ 1) ($ per batch) the greater production rate of stages j and (j+ 1), or, max(Pj, Pj + 1) the smaller production rate of stages j and (j + 1), or, min(Pj, Pj + 1) production rate ratio of stages j and (j + 1), i.e., [Pj] + /[Pj]
The 11 assumptions of the model in this study are listed below. 1. The product demand rate is a known constant in the planning period. 2. The product units are infinitely divisible. 3. The supply chain comprises n serial manufacturing stages. Stage (n + 1) represents end-customer demand; therefore, Pn + 1 =D. 4. The production rate of stage k is the smallest of all stages, i.e., Pj 4Pk, for j= 1, 2,y,k 1, k+ 1,y,n. 5. The production rate of at least one stage is smaller than the demand rate; that is, a bottleneck exists in the supply chain. In the planning period, the supply chain can only produce Pk units of end products for customers. Consequently, there are shortages and lost sales at the demand stage for the end customers. 6. No ordering cost was involved. 7. Backlog and interruption are not permitted at every produc-
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8.
9.
10. 11.
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tion stage. Only a single setup is performed for a production lot at each production stage. Both upstream and downstream lot size are smaller than the production rate of stage k in the planning period, i.e., Qu oPk and Qd oPk; otherwise, the planning period should be extended. Upstream and downstream lot sizes are uniform: upstream lot size is pulled through the upstream stages from the bottleneck and downstream lot size is pushed through the downstream stages from the bottleneck; but partial lots, or batches, can be transported to the next stage upon completion. Consequently, batch size and batch number may differ across stages. Unequal batch sizes at each stage follow increasing or decreasing geometric series (Goyal, 1977b). Fixed batch transportation costs and transportation time between adjacent stages are constant, so variable transportation costs and holding costs during transportation are treated as sunk costs.
3. Integer nonlinear programming model
under the following conditions: Pi 4D
for
i A S1
ð8Þ
Pj oD
for
j A S2 ; S2 a |
ð9Þ
Pj 4 Pk
for
j ¼ 1; 2; . . . ; k1; kþ 1; . . . ; n
S 1 [ S 2 ¼ f1; 2; . . . ; ng
ð10Þ ð11Þ
The first and second parts of the objective function (1) represent the total costs of the upstream and downstream stages, respectively. Eqs. (2) and (3) state that both upstream and downstream lot size are smaller than the production rate of stage k in the planning period. Eqs. (9) and (10) demonstrate that stage k is the bottleneck stage. Meanwhile, Eqs. (4)–(7) together define the decision variables. When the objective function and decision variables are gathered, the original model can be divided into two sub-models: the upstream sub-model and the downstream sub-model. The upstream sub-model can be expressed as ( " # ) k1 hj Qu2 ðRj 1Þ hj Qu2 Pk X 1 1 þSj þ Fj mj þ Minimize Qu j ¼ 1 ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j ð12Þ
Bogaschewsky’s model (2001) considers a serial supply chain in which a uniform lot size was produced through all stages with a single setup and without interruption at each stage. The number of unequal sized batches may differ among stages. The production rate at every stage exceeds the product demand rate. This study examines a serial supply chain in which the production rate of stage k is the smallest of all stages, and the production rate of at least one stage is smaller than the demand rate; that is, stage k is the bottleneck stage. Let S1 and S2 be the sets of the production stages’ number. If Pi 4D, then iAS1. If Pj oD, then jAS2. Since the production rate of at least one stage is smaller than the demand rate and stage k is the bottleneck stage, the set S2 has at least one element, k, and S2 a|. Goldratt’s (1988) synchronous production concept and the drum-buffer-rope method are adopted to maintain full bottleneck stage utilization. Uniform upstream lot size Qu and downstream lot size Qd are produced: upstream lot size Qu is pulled through the upstream stages from the bottleneck and downstream lot size Qd is pushed through the downstream stages from the bottleneck. During the planning period, the supply chain can only produce Pk units of end products for customers. A serial supply chain with bottleneck problem thus can be formulated as the following integer nonlinear programming model: ( " # ) k1 hj Qu2 ðRj 1Þ hj Qu2 Pk X 1 1 þ S þ þ F m Minimize j j j Qu j ¼ 1 ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j n P X þ k Qd j ¼ k
(
hj Qd2 ðRj 1Þ m
ðRj j 1Þ½Pj þ
" # ) hj Qd2 1 1 þ S þ þF m j j j 2 ½Pj ½Pj þ
ð1Þ
subject to Pk 4Qu
ð2Þ
Pk 4Qd
ð3Þ
Qu 40
ð4Þ
Qd 40
ð5Þ
mj Z 1
for
j ¼ 1; 2; . . . ; n
for
j ¼ 1; 2; . . . ; n
Pk 4 Qu
ð13Þ
Qu 4 0
ð14Þ
Pj 4 Pk
for
j ¼ 1; 2; . . . ; k1
ð15Þ
mj Z 1
for
j ¼ 1; 2; . . . ; k1
ð16Þ
and mj are integers
for
j ¼ 1; 2; . . . ; k1
ð17Þ
Meanwhile, the downstream sub-model can be expressed as ( " # ) n hj Qd2 ðRj 1Þ hj Qd2 Pk X 1 1 Minimize þ Sj þ Fj mj þ Qd j ¼ k ðRmj 1Þ½Pj þ 2 ½Pj ½Pj þ j
ð18Þ subject to Pk 4 Qd
ð19Þ
Qd 4 0
ð20Þ
Pj 4 Pk
for
j ¼ kþ 1; k þ 2; . . . ; n; n þ 1
ð21Þ
mj Z 1
for
j ¼ k; k þ1; k þ 2; . . . ; n
ð22Þ
and mj are integers
for
j ¼ k; k þ1; k þ 2; . . . ; n
ð23Þ
Inspecting the objective functions and the decision variables reveals no correlation between the upstream and downstream sub-models; in other words, both sub-models are mutually independent. The optimal solutions of two sub-models could be obtained, respectively, in this study. Therefore, this study concluded that: the minimal cost of the original model is the sum of the minimal costs of two sub-models and the optimal solution of the original model is the union of the optimal solutions of two sub-models.
ð6Þ
4. The methods
ð7Þ
Let M represent the batch vector and let TCu(Qu, Mu) and TCd(Qd, Md) be the objective function of the upstream sub-model
and mj are integers
subject to
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and the downstream sub-model, respectively. Setting qTCu(Qu, Mu)/qQu and qTCd(Qd, Md)/qQd to zero, we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " #)ffi , u k1 k1 X uX hj ðRj 1Þ hj 1 1 t Qu ðMu Þ ¼ ðSj þ mj Fj Þ þ 2 ½Pj ½Pj þ ððRj Þmj 1Þ½Pj þ j¼1 j¼1 ð24Þ and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( " #) , u n n X uX hj ðRj 1Þ hj 1 1 Qd ðMd Þ ¼ t ðSj þ mj Fj Þ þ 2 ½Pj ½Pj þ ððRj Þmj 1Þ½Pj þ j¼k j¼k
ð25Þ
4.1. The lot size division method Modify the order interval division method of Hsiao (2008), the s be the critical lot size division (LSD) method is developed. Let Qj;m j lot size of the stage j, we have s hj ðQj;m Þ2 ðRj 1Þ j mj
ððRj Þ 1Þ½Pj
þ
¼
s hj ðQj;m Þ2 ðRj 1Þ j
ððRj Þmj þ 1 1Þ½Pj þ
þFj
s , then Solve for Qj;m j vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Fj ½Pj þ u s ! ¼u Qj;m j u 1 1 th ðR 1Þ j j ðRj Þmj 1 ðRj Þmj þ 1 1
ð26Þ
ð27Þ
Lemma 1. For stage j in an n-stage serial supply chain, if the lot size s Q is greater than the critical lot size Qj;m , then (mj +1) is chosen as the j number of batches; otherwise, mj is selected as the number of batches. s , For j= 1, 2,y,n, use Eq. (27) to find all critical lot sizes Qj;m j s s namely, Qj;1 , Qj;2 and so on, till the resulting critical lot size exceeds D. Merge these critical lot sizes found for all stages to form an ordered critical lot size vector, lot size ranges RG and batch vectors M. The procedures of the lot size division method are similar to the procedures of the order interval division method in Section 3 of Hsiao’s paper (2008).
411
4.3. The pull and reverse pull methods Goldratt (1988) designed synchronous production, which is based on the theory of constraints, and also recommended the drum-buffer-rope method for solving the scheduling problem for the production system containing a bottleneck. In synchronous production, the drum is both the bottleneck and the mechanism controlling the pace of production. The production schedule is developed around the bottleneck, which pulls production from earlier stages and pushes it to the subsequent stages. In a JIT pull system and a MRP push system, the drum is both the final product demand and the master production schedule. We have developed the LSD and RT methods to solve the logistic and inventory problem of a serial supply chain that is a JIT pull system. If stage k (the bottleneck stage) is treated as the final product demand stage, the upstream sub-model becomes a kstage serial supply chain in which production rate at every stage exceeds the product demand rate. The LSD and RT methods can be easily applied to the upstream sub-model to solve the optimal upstream lot size Qu and the batch number associated with each stage. The LSD and RT methods cannot be directly applied to the downstream sub-model because the downstream sub-model is an MRP push system. Therefore the reverse pull method is designed to solve the MRP push system. First, pseudo-reverse procedures transfer a push system into a pseudo-pull system. The pseudoreverse procedures include three steps: (1) pseudo-reversing the logistic direction of a serial supply chain system; (2) letting the demand stage be the pseudo-earliest stage; and (3) letting the earliest stage be the pseudo-demand stage. After executing the pseudo-reverse procedures, the downstream sub-model was converted to a pseudo-pull model. The LSD and RT methods can be directly applied to this pseudo-pull model to solve the optimal solutions and obtain the minimal cost. The pseudo-reverse procedures then are executed again, and the pseudo-pull model backs into the downstream sub-model. The minimal cost of the downstream sub-model is the same as the minimal cost of the pseudo-pull model. Moreover, the pseudoreversed optimal solutions of the pseudo-pull model are the optimal solutions of the downstream sub-model. A special case does exist. If the earliest stage is the bottleneck stage, then the original problem is simply an MRP push system. This problem can be solved using the reverse pull method only.
4.2. The recursive tightening method Modify the recursive tightening method of Hsiao (2008), two theorems can be rewritten as follows: Theorem 1. For an n-stage serial supply chain, if the lot size Qx(Mi) does not lie within its associated lot size range RGi, but rather lies in the lot size range RGj, where x= u, d and ioj, then TCx(Qx(Mi), Mi)4TCx(Qx(Mi), Mj). Consequently, none of the batch vectors Mi, Mi + 1,y, or Mj 1 is the global optimal batch vector, and none of Qx(Mi), Qx(Mi + 1),y, or Qx(Mj 1) is the global optimal lot size. Theorem 2. For an n-stage serial supply chain, if the lot size Qx(Mi) does not lie within its associated lot size range RGi, but rather lies in the lot size range RGj, where x= u, d and i4j, then TCx(Qx(Mi), Mi)4TCx(Qx(Mi), Mj). Consequently, none of the batch vectors Mi, Mi + 1,y, or Mj 1 is the global optimal batch vector, and none of Qx(Mi), Qx(Mi + 1),y, or Qx(Mj 1) is the global optimal lot size. The upward and downward recursive tightening procedures are similar to the procedures of the recursive tightening method in Section 4 of Hsiao’s paper (2008).
5. Algorithm The following procedures integrate the pull and reverse pull methods to determine the optimal solutions for the problem being studied. Step 1. Step 2. Step 3.
Step 4.
Let Pmin =D and k=n + 1. For j = 1, 2, 3,y,n, do If Pj oPmin, then let Pmin =Pj and k=j. If k=n + 1 (i.e. without bottleneck), then Use the LSD and RT methods to solve the problem and stop. Execute the pseudo-reverse procedures to convert the downstream sub-problem (from stage k to the demand stage) to be a pseudo-pull problem. (a) Pseudo-reverse the logistic direction of the downstream sub-problem. (b) Let the demand stage of the original problem be the pseudo-earliest stage of the pseudo-pull problem.
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Step 5. Step 6.
Step 7.
Step 8.
Step 9.
(c) Let stage k of the original problem be the pseudo-demand stage of the pseudo-pull problem. Use the LSD and RT methods to solve the pseudopull problem. Execute the pseudo-reverse procedures to reverse the logistic direction of the solution of the pseudo-pull problem. If k= 1 (i.e. the earliest stage is the bottleneck stage), then the optimal solutions are obtained and stop. Let stage k of the original problem denote the final product demand stage of the upstream subproblem. Use the LSD and RT methods to solve the upstream sub-problem. The optimal solution of the original model is the combination of the optimal solutions of the two sub-models. The minimal cost of the original model is the sum of the minimal costs of the two sub-models.
i
Lot size region RGi
m1, m2 Mi
Q(Mi)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
(0, 167.3] [167.3, 182.6] [182.6, 293.8] [293.8, 324.9] [324.9, 424.1] [424.1, 478.2] [478.2, 562.5] [562.5, 650.6] [650.6, 712.2] [712.2, 849.6] [849.6, 876.6] [876.6, 1059.1] [1059.1, 1083.5] [1083.5, 1263.4] [1263.4, 1362.0] [1362.0, 1494.0] [1494.0, 1696.7]
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8,
351.6
a
1 2 2 3 3 4 4 5 5 6 6 7 8 8 9 9 10
Total cost of upstream
653.6
866.3 951.3 998.7a 1039.5 1067.0
423.57
1160.4
The optimal lot size of upstream.
Table 3 Results of the psedo-downstream sub-problem using LSD and RT methods.
6. Numerical example The inventory problem of a serial supply chain involves four manufacturing stages. The 5th stage is the demand stage, during which the demand rate is 2150 units per month. Table 1 lists the problem data. The production rate of stage 3 is the smallest of all stages and is also smaller than the demand rate. Stage 3 is the bottleneck of the supply chain. Therefore, the problem can be divided into two sub-problems: one is the upstream sub-problem containing stages 1, 2 and 3; the other is the downstream subproblem containing stages 3, 4 and the demand stage. After performing the LSD and RT methods, the lot size ranges RGi and the associated batch vector Mi of the upstream subproblem and the downstream sub-problem are listed in Tables 2 and 3, respectively. In Table 2, there are 17 lot size ranges (RGi). Use Eq. (24) to calculate the lot size of RG1, we have Q(M1) =351.6 which lies in RG5. Applying recursive tightening method, the upward recursive tightening procedures are expressed as follows: (1) (2) (3) (4)
Table 2 Results of the upstream sub-problem using LSD and RT methods.
Q(M5)= 653.6 which falls in RG9. Q(M9)= 866.3 which lies in RG11. Q(M11)= 951.3 which falls in RG12. Q(M12)= 998.7 which lies in RG12.
i
Lot size region RGi
m3, m4 Mi
Q(Mi)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(0, 160.2] [167.3, 182.6] [182.6, 277.6] [277.6, 324.9] [324.9, 392.8] [392.8, 478.2] [478.2, 507.4] [507.4, 622.0] [622.0, 650.6] [650.6, 736.7] [736.7, 849.6] [849.6, 851.7] [851.7, 967.1] [967.1, 1082.9] [1082.9, 1083.5] [1083.5, 1199.2] [1199.2, 1316.2] [1316.2, 1362.0] [1362.0, 1433.8] [1433.8, 1552.1]
1, 1 1, 2 2, 2 2, 3 3, 3 3, 4 4, 4 4, 5 4, 6 5, 6 5, 7 6, 7 6, 8 6, 9 6,10 7,10 7,11 7,12 8,12 8,13
195.6
a b
The same procedures are performed for the downward recursive tightening. Only the lot size Q(M12) falls in its own lot size range RG12. According to the RT method, Qu =Q(M12)= 998.7 and (m1, m2)= M12 =(6, 7) are the optimal solutions to the upstream sub-problem. Moreover, the optimal total cost of the upstream sub-problem is 423.57.
Table 1 The data of the numerical example.
321.2 374.5 437.1 489.8 545.1 596.2b 641.3b 694.5b 738.7a
Total cost of downstream
513.36 509.92 505.40 503.61
827.9
948.2
1090.7
The optimal lot size of downstream. The lot sizes located in their own lot size ranges.
Table 3 contains four downstream lot sizes located in their own lot size ranges. The total costs for each downstream lot size then can be determined and the smallest one selected. Qd =738.7 and (m3, m4)= (5, 7) denote the optimal solutions to the downstream sub-problem. The optimal total cost of the downstream sub-problem is 503.61. The optimal solutions of two sub-models are combined as follows: (m1, m2, m3, m4)=(6, 7, 5, 7). Finally, the total cost of the overall problem is 927.18. Table 4 lists the batch sizes of the four stages.
7. Conclusions
Number of stage j
Production rate Pj (units/ month)
Setup cost Sj ($)
Fixed transportation cost Fj ($/trip)
Holding cost hj ($/units month)
1 2 3 4
3000 2000 1500 2250
45 30 20 25
4 6 8 7
0.60 0.75 0.90 1.20
This study proposes an integer nonlinear programming model for a multi-stage logistic and inventory problem in a serial supply chain with a production bottleneck where the unequal batch sizes at each stage follow increasing or decreasing geometric series. The drum-buffer-rope method is adopted to keep the bottleneck stage fully utilized and to force the other stages to produce in synch with the bottleneck. Moreover, the reverse pull method converts a
ARTICLE IN PRESS Y.-C. Hsiao et al. / Int. J. Production Economics 124 (2010) 408–413
References
Table 4 The batch sizes of four stages. The batch sizes
Batch
Stage
1 2 3 4 5 6 7
413
1
2
3
4
48.1 72.1 108.1 162.2 243.3 364.9
51.3 68.4 91.2 121.5 162.1 216.1 288.1
266.7 177.8 118.5 79.0 52.2
91.7 96.0 100.4 105.1 110.0 115.1 120.4
push system into a pseudo-pull system. Therefore, the LSD and RT methods can be directly applied to this pseudo-pull model to obtain the minimal cost and the optimal solutions. Additionally, the pseudo-reversed optimal solutions of the pseudo-pull model are allowed to be the optimal solutions of the downstream submodel. The minimal cost of the original model is the sum of the minimal costs of the two sub-models and the optimal solution of the original model is the union of the optimal solutions of the two sub-models.
Acknowledgment The author would like to thank the National Science Council (NSC) of Taiwan for financially supporting this research under Contract no. NSC 91-2416-H-147-002.
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