queueing analysis on a single-station make-to-stock/make-to-order inventory-production system

Applied Mathematical Modelling 34 (2010) 978–991

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Applied Mathematical Modelling

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Queueing analysis on a single-station make-to-stock/make-to-orderinventory-production system

Kuo-Hwa Chang *, Yang-Shu LuDepartment of Industrial and Systems Engineering, Chung Yuan Christian University, 200, Chung Pei Rd., Chung-Li 32023, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 9 May 2008Received in revised form 13 July 2009Accepted 23 July 2009Available online 30 July 2009

Keywords:Queueing systemMake-to-orderMake-to-stockBase-stock policyInventory-queue

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.07.009

* Corresponding author. Tel.: +886 2654416; fax:E-mail address: [email protected] (K.-H. Cha

a b s t r a c t

We consider a one-station production system that produces standard products for ordinarydemands and custom products for specific demands. In this system, the workstation hastwo manufacturing modes. In mode 1, it produces standard products and, in mode 2, it pro-duces custom products by performing the additional alternating works on one existing fin-ished standard product. Base-stock control policy is applied to control the production ofstandard products. The fill rate of the ordinary demand and the on-time-delivery-rate ofthe specific demand are considered as the measures of the qualities of service. By assumingan Markovian system, qualities of service under base-stock policy are obtained; further-more, the optimal base-stock level can be obtained numerically under the requirementson the qualities of services.

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1. Introduction

In the supply chain system, there are different sources of uncertainties such as arrival times of the demands, processingtime, transportation time, reliability of the machines and the capability of the operator. Customers treat the response time ofthe order as one of the service performance measure. Because of the new business model of Internet/telephone ordering andquick response service requirement, make-to-order (MTO) business model is increasing its significance rapidly. MTO produc-tion is usually utilized to deal with the specific demand for a non-standard and custom products, however, make-to-stock(MTS) production is used for the ordinary demand for standard products, which are delivered directly from inventory (stock).That is, in MTS production, products are stocked in advance, while in MTO production, a product only starts to be producedwhen an order of demand is received. Therefore, one of the main challenge for the production system management is how todeal with those randomly arriving demands in order to have the immediate satisfaction to the ordinary demand and shortresponse time to the specific demand.

Assembly manufacturing plays a very important role in the global supply chain of consumer products, such as laptopcomputers. Assemblers, in addition to fulfill the ordinary demands for the standard products by adopting MTS production,are often asked to take care of the specific demands for the custom products and to adopt MTO production. In usual cases,ordinary demands should be satisfied immediately, however, there is a time window for the specific demand. To the assem-bler, it is not profitable to maintain a solo MTO production line exclusively for the specific demand. In some case, customproducts share almost all the parts of the standard products and can be produced by alternating the existing standard prod-ucts with some additional works, therefore, the assembler usually consider embedding the MTO lines into the main streamMTS lines, which becomes a hybrid production system. In this system, stocking a large amount of finished standard productscan increase the percentage of satisfied ordinary demand and shorten the response time of the specific demand, however, it

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also incurs higher inventory cost. Therefore, inventory control to a production system is an important issue for the manage-ment. One simple but efficient way to resolve this issue is to have some initial stock, called base-stock, and the control mech-anism is operated as an inventory-queue.

In our study, we assume the custom products can be made by alternating the existing standard ones with additionalworks. We consider a single-station hybrid MTO/MTS production system (see Fig. 1) with random ordinary and specific de-mands, in which, the workstation has two manufacturing modes. In mode 1, it produces standard products for the ordinarydemands (MTS orders). The finished standard product also serves as semi-product used to fulfill the specific demands. Inmode 2, the workstation performs the additional alternating works on the standard product according to specific demands(MTO orders). There is a base-stock level for the finished standard products. An ordinary demand will take a standard prod-uct (if there is any) away directly from finished-product inventory upon its arrival, otherwise it will leave the system. Eachspecific demand will also take one standard product and will wait for the processing of the corresponding additional work atworkstation in mode 2 on the first-come-first-served basis. In this hybrid system, producing the custom products for specificdemands has higher priority than producing the standard product. An arriving specific demand will stop undergoing mode 1production process for a standard product and the workstation will switch to mode 2 until there are no specific demandswaiting. Furthermore, the system will reserve as many as finished standard products available for the specific demands insystem. Since base-stock policy is used, every satisfied ordinary demand will send a production order to workstation forthe a standard product at the same time it receives a finished-product. Every arriving specific demand will also send a pro-duction order to workstation for the a standard product upon their arrivals.

Among the researches on hybrid systems, Soman et al. [1] review the studies on the hybrid MTO–MTS production andmention that such systems can often be seen in the food industry. Krishnamurthy et al. [2] use simulation to analyze aMTO/MTS hybrid system in which base-stock controlled MTS production system supplies finished product to multipleMTO production systems. They also compare the performance of MRP and Kanban for a multi-stage, multi-product manu-facturing system. Adan and ven der Wal [3] present two single-station systems. The first system deals with ordinary andspecific demands with base-stock control. Production is preempted by the specific demand. The second system deals withthe specific demands with base-stock control for the semi-finished products. Productions are in two phases. The first phaseis to produce semifinished products and the second phase is to perform the further work on the semi-finished products instock according to specific demands. Nguyen [4] considers a single-station hybrid production system for multiple MTS ordersand multiple MTO orders. MTS orders are satisfied from the inventory controlled by base-stock policy and they are lost ifthere are no inventory. He models it as a mixed queueing network and approximate the performances under heavy trafficconditions by using the corresponding limiting theorem. Federguruen and Katalan [5] consider a single-station system thatproduces some MTS products and one MTO product. For the MTS products the base-stock policies with general periodic se-quence are considered. By using M/G/1 model with vacations, the impacts of various priority rules for the MTO products arestudied. Carr and Duenyas [6] consider a single-station hybrid production system for the MTS orders and MTO orders. TheMTS orders are satisfied from the finished-product inventory. There is no backorder for the MTS order and unsatisfied MTSorders are lost. They apply admission control on the MTO orders and sequencing on jobs at the workstation. They use Markovdecision process to find an optimal policy to maximize the average profit rate and obtain the corresponding switchingcurves. Arreola-Risa and DeCroix [7] consider a single-station system that produce multiple products with base-stock

Fig. 1. The hybrid MTO/MTS system dealing with the ordinary demands and specific demands.

980 K.-H. Chang, Y.-S. Lu / Applied Mathematical Modelling 34 (2010) 978–991

inventory policies. They study the optimality conditions to decide which products are make-to-stock and which are make-to-order (with base-stock level zero) in order to have the minimum average cost per unit and average cost rate per unit,respectively. Rajagopalan [8] also considers a single-station system for the MTS order and MTO order. The inventory controlpolicy for the MTS products is a ðq; rÞ policy. Production orders for both MTS and MTO items are served on a first-come-first-served (FCFS) basis.

In the system, the percentage of the satisfied ordinary demands, called the fill rate, and the percentage of response timesof specific demands that are less than a predetermined lead time, called the on-time-delivery-rate (OTDR), are considered asthe measures of the qualities of service. Our system is analyzed by modeling it as inventory-queue system. By assuming theMarkovian property, the limiting probabilities are obtained by using matrix geometric method (MGM) and their correspond-ing performances under base-stock control policy are obtained. Based on these analyses, we can determine the optimalbase-stock level numerically under the requirements on the qualities of service; furthermore, we can determine the optimalbase-stock level according to some cost structure.

The remainder of this paper is organized as follows. In Section 2, we present the Markovian model for our hybrid system.We derive the limiting probabilities for the system, and, based on which, we further derive the fill rate and OTDR as well asother performance indexes. In Section 3, we present some numerical examples. We validate our results and observe the lim-iting behaviors of fill rates and OTDR as base-stock level gets large. We also study the optimal base-stock level numericallyunder some cost structure. We conclude our study in Section 4.

2. Inventory-queue model

In this system (as shown in Fig. 2), we assume that the ordinary demands and specific demands arrive according to Pois-son processes with respective rate k1 and k2. The production times of workstation under mode 1 and mode 2 are assumed tobe exponentially distributed with respective rates l1 and l2. Let S be the base-stock level for the finished standard products.Note that, in our system, the specific demand has higher priority than the ordinary one in that the workstation will fulfill thespecial demand first if there is any with preemption on the production of the standard products. The system will also reserveas many finished standard products available for the specific demands in system as follows. Let X be the number of waitingspecific demands in system and let Y be the number of finished standard products in inventory or under mode 2 processing.If X 6 Y , the system will reserve Y � X finished standard products for these X specific demands; if X > Y , then the system willreserve all the Y finished standard products. After satisfying the first Y specific demands, the production will be changed tomode 1 for producing one standard product for the ðY þ 1Þst specific demand and then mode 2 on the same finished standardproduct to satisfy the ðY þ 1Þst specific demand and the same process will apply to the ðY þ 2Þnd specific demand and so onuntil all the specific demands outstanding in system are satisfied.

The system can be described by a Markov process with states ðn;mÞ, where n is the number of specific demands in thesystem(waiting or under processing); m is the number of finished standard products. Let l be the number of production or-ders for producing standard products, then we have lþm� S ¼ n. We denote our state space by X ¼ fn P 0; S P m P 0g.Since our system always reserves as many finished standard products available for the outstanding specific demands, an

Fig. 2. The hybrid MTO/MTS system dealing with the ordinary demands and specific demands.


arriving ordinary demand will be lost if it finds state ðn;mÞ where n P m. The state space with transition rates is depicted inFig. 3.

Since it is a loss system for the ordinary demands, this system is stable under the condition k2=l1 þ k2=l2 < 1 for anyfinite S. The corresponding balance equations for the limiting probabilities Pðn;mÞ are as follows.

Pðn;mÞðk1 þ k2Þ ¼ Pðn;m� 1Þl1; n ¼ 0; m ¼ S;

Pðn;mÞðk1 þ k2 þ l2Þ ¼ Pðn;m� 1Þk1 þ Pðn;m� 1Þl1 þ Pðnþ 1;mþ 1Þl2; n ¼ 0; 1 6 m 6 S� 1;Pðn;mÞðk2 þ l1Þ ¼ Pðn;mþ 1Þk1 þ Pðnþ 1;mþ 1Þl2; n ¼ 0; m ¼ 0;Pðn;mÞðk1 þ k2 þ l2Þ ¼ Pðn� 1;mÞk2;1 6 n 6 S� 1; m ¼ S;

Pðn;mÞðk1 þ k2 þ l2Þ ¼ Pðn;mþ 1Þk1 þ Pðn� 1;mÞk2 þ Pðnþ 1;mþ 1Þl2;1 6 n 6 S� 2; 2 6 m 6 S� n;

Pðn;mÞðk2 þ l2Þ ¼ Pðn� 1;mÞk1; n P S; m ¼ S;

Pðn;mÞðk2 þ l2Þ ¼ Pðn;mþ 1Þk1 þ Pðnþ 1;mþ 1Þl2 þ Pðn� 1;mÞk2 þ Pðn;m� 1Þl1; n ¼ 1; m ¼ 1;Pðn;mÞðk2 þ l2Þ ¼ Pðn� 1;mÞk2 þ Pðn;mþ 1Þk1 þ Pðnþ 1;mþ 1Þl2; 2 6 n ¼ m 6 S� 1;Pðn;mÞðk2 þ l1Þ ¼ Pðn� 1;mÞk2 þ Pðnþ 1;mþ 1Þl2; n P 1; m ¼ 0;Pðn;mÞðk2 þ l2Þ ¼ Pðn� 1;mÞk2 þ Pðnþ 1;mþ 1Þl2 þ Pðn;m� 1Þl1; n P 2; m ¼ 1;Pðn;mÞðk2 þ l2Þ ¼ Pðn� 1;mÞk2 þ Pðnþ 1;mþ 1Þl2;n P mþ 1; 2 6 m 6 S� 1:

We have the corresponding generator matrix written in following block form:

Q ¼

D AC E1 A

C E2. .

.

C . ..

A. .

.ES�1 A

C B . ..

C . ..

. ..

26666666666666666664

37777777777777777775

;

Fig. 3. State transition rates diagram.


where A;B;C;D;E1; . . . ;ES�1 are the corresponding submatrices for the transition rates in generator matrix and they areshown in Appendix. We define the vectors of limiting probabilities

Pi ¼ fPðSþ i� 1; SÞ; PðSþ i� 1; S� 1Þ; . . . ; PðSþ i� 1;0Þg;

for i P 1 and P00;P01;P02; . . . ;P0;S�1, where

P0i ¼ Pði; SÞ; Pði; S� 1Þ; . . . ; Pði;0Þf g;

for 0 6 i 6 S� 1.In order to solve PQ ¼ 0, the balance equations can be expressed as follows:

P00Dþ P01C ¼ 0;P00Aþ P01E1 þ P02C ¼ 0;P01Aþ P02E2 þ P03C ¼ 0;

..

.

P0S�1Aþ P1ES�1 þ P2C ¼ 0;PiAþ Piþ1Bþ Piþ2C ¼ 0; i P 1:

ð1Þ

We define a constant rate matrix R such that

Pi ¼ Pi�1R ¼ P1Ri�1: ð2Þ

Eq. (1) then can be simplified as the follows:

Aþ RBþ R2C ¼ 0:

R in Eq. (2) can be solved numerically. Then Pi; i > 1 can be obtained if P1 is known. However, P1 and other P0i;0 6 i 6 S� 1can be solved by the following linear equations.

P00;P01; � � � ;P0;S�1;P1½ �;D A 0 0 0 e

C E1 A ...

0 e

0 C E2. .

. ... ..

. ...

..

.0 C A 0 ..

.

..

. ...

0 ES�1 A� e0 0 0 C ðBþ RCÞ� ðI� RÞ�1e

2666666666664

3777777777775

¼ 0;1½ �;ð3Þ

where each vector e is a ðSþ 1Þ � 1 column of 1s and A� and ðBþ RCÞ� are the matrixes without the last column of A andðBþ RCÞ, respectively. On the right-hand side, 0 is a 1� SðSþ 2Þ row of 0s and 1 is the constant. Note that the last equationin linear system (3) corresponds to the law of probabilities.

From here we can obtain those important performance indexes of the system described as follow. Let Pf be fill rate of theordinary demand; LF be the average number of finished standard products, LPO be the average number of production ordersfor the standard products, LSD be the average number of specific demands in system, and WSD be the expected response timeof a specific demand, respectively, then

Pf ¼XS�1

n¼0

XS

m¼nþ1

Pðn;mÞ;

LF ¼X1n¼0

XS

m¼1

mPðn;mÞ;

LPO ¼X1n¼0

XS

m¼0

½ðS�mþ nÞ�Pðn;mÞ;

LSD ¼X1n¼1

XS

m¼0

nPðn;mÞ;

WSD ¼LSD

k2:


Define R be the response time of a specific demand. When a specific demand arrives and state ðn;mÞ;m > n, is seen by thisdemand, the response time of this demand will be the sum of nþ 1 processing times of mode 2. If m 6 n, then this demandhas to wait m mode 2 processings for the first m specific demands already in system and n�mþ 1 mode 1 and mode 2 pro-cesses for the remaining specific demands including itself. By PASTA property, we have

Table 1Compar

S

102030

Table 2Compar

S

102030

Table 3Compar

S

102030

Table 4Compar

S

102040

P R 6 uð Þ ¼XS

m¼1

Xm�1

n¼0

P Ynþ1 6 uð ÞP n;mð Þ þXS

m¼0

X1n¼m

P Xn�mþ1 þ Ynþ1 6 uð ÞPðn;mÞ;

where Xk is a gamma random variable with parameters k and l1 and Yk is a gamma random variable with parameters k andl2. If u is the required lead time for the specific demand, then OTDR is PðR 6 uÞ.

3. Numerical results

Let q1 ¼ k1=l1 and q2 ¼ k2=l1 þ k2=l2. Note that, for every finite S, the stability condition is q2 < 1. We first compare thenumerical results with the simulated ones on various q1 and q2. The results are shown in Tables 1–6. The numbers in column‘Num.’ are obtained by our analysis and those in the column ‘Sim.’ are the simulation results.

The numerical results and simulation results are very close to each other. In the cases corresponding to Table 3, 5 and 6,q1 þ q2 P 1. Let ke be the effective arrival rates for the ordinary demands, then Pf ¼ ke=k1. Since our system is a loss system

ison results on various base-stock levels for case of q1 ¼ 0:2 and q2 ¼ 0:15.

Pf LF LPO LSD WSD OTDR

Num. Sim. Num. Sim. Num. Sim. Num. Sim. Num. Sim. Num. Sim.

0.999 1 9.485 9.489 0.540 0.538 0.026 0.025 2.051 2.048 0.623 0.6191 1 19.485 19.489 0.541 0.540 0.026 0.025 2.051 2.049 0.623 0.6211 1 29.485 29.492 0.541 0.540 0.026 0.026 2.510 2.049 0.623 0.621




0.981 0.983 7.852 7.848 2.256 2.248 0.108 0.107 2.571 2.547 0.589 0.5840.999 0.998 17.685 17.642 2.407 2.399 0.092 0.091 2.195 2.167 0.599 0.5930.999 1 27.675 27.655 2.416 2.414 0.091 0.091 2.183 2.166 0.599 0.594




0.560 0.559 3.063 3.059 9.071 9.031 2.135 2.122 35.503 30.314 0.329 0.3290.681 0.678 6.461 6.431 15.134 15.151 1.590 1.612 22.776 23.028 0.397 0.3980.732 0.731 9.485 9.466 21.910 21.878 1.368 1.338 19.544 19.114 0.425 0.424




0.919 0.910 5.467 5.460 4.570 4.574 0.038 0.038 3.007 3.040 0.574 0.5740.968 0.969 11.807 11.776 8.223 8.202 0.030 0.033 2.429 2.640 0.604 0.6030.991 0.991 26.666 26.664 13.361 13.342 0.027 0.027 2.152 2.153 0.623 0.621

Table 5Comparison results on various base-stock levels for case of q1 ¼ 0:8 and q2 ¼ 0:5.

S Pf LF LPO LSD WSD OTDR


10 0.602 0.606 2.229 2.224 8.204 8.201 0.432 0.438 10.297 10.429 0.371 0.37220 0.619 0.618 2.708 2.738 17.710 17.729 0.418 0.419 9.957 9.976 0.380 0.38130 0.620 0.626 2.772 2.780 27.646 27.643 0.417 0.417 9.934 9.917 0.381 0.381

Table 6Comparison results on various base-stock levels for case of q1 ¼ 0:8 and q2 ¼ 0:84.

S Pf LF LPO LSD WSD OTDR


10 0.199 0.197 0.644 0.643 13.106 13.054 3.750 3.760 53.570 53.714 0.122 0.12120 0.200 0.209 0.662 0.664 23.088 23.077 3.751 3.759 53.581 53.700 0.123 0.12330 0.200 0.205 0.663 0.665 33.089 32.005 3.752 3.758 53.600 53.686 0.123 0.123


to the ordinary demands, for the case of q1 þ q2 > 1, the remaining utilization of workstation that can be used for the ordin-ary demand is 1� q2. Therefore, when S is getting large, ordinary demands can only be served with the utilization 1� q2 atmost (note that the specific demands will always be served). Based on this, we can make the following inference as the Sincreases to infinity: for ordinary demands, Pf will converge to the maximum fill rate ð1� q2Þ/q1, and ke will converge toPf k1. The maximum fill rate for the cases corresponding to Table 3, 5 and 6 are 0.8, 0.620 and 0.2, respectively, and thenumerical results demonstrate these convergences. On the other hand, when q1 þ q2 < 1; ke will converge to k1 and Pf willconverge to 1 as S tends to infinity. However, the response time of the specific demand will be the waiting time in system of aM/M/1 queue with arrival rate k2 and service rate l2. That is, the response time will be exponentially distributed with ratel2 � k2, while the fill rates will converge to 1. The OTDR obtained by the exponential waiting times for the cases correspond-ing to Table 1, 2 and 4 are 0.623, 0.599 and 0.623, respectively, and the numerical results also demonstrate theseconvergences.

Figs. 4 and 5 show the convergences of fill rate and OTDR on S as k1 and k2 increase, respectively. In either figure, when S issmall, fill rates are always greater than OTDRs because of our reservation mechanism. However, for the case ofq ¼ q1 þ q2 < 1, fill rate will be greater than OTDR as S gets large because fill rate will converge to 1 while OTDR willnot. We also observe that, for small S, the increase rate (slope of the curves) of OTDR is greater than the corresponding fillrate. It means that the improvement on OTDR is more significant than that on fill rate when we increase the base-stock levelstarting from a small level.

We further consider the following example which is adopted from a real case. For assembling a laptop computer, thereare two parallel pre-assembly processes, the pre-assembly of the printed circuit board(PCB) and the pre-assembly of thedisplay, cases and other the peripherals such as the hard drive, DVD drive, etc. At the following main assembly station, anoperator then assembles PCB and those peripherals into a finished standard laptop computer. Special demands for customlaptop computers usually ask for adding more memory, replacing standard hard drive by a larger one or upgrading thedisplay. The assembly of custom computer will also be performed at the same station by alternating the existing finishedstandard laptop computer. The custom computer should be ready in 2 h at this station after the request has been re-ceived. The time to perform the main assembly takes around 40–45 min and the alternation around 30–35 min. Com-bined demands arrive to this station (operator) one per hour in average and special demands count one-fifth of them.To simulate this system, we use triangle distribution [40, 45] and triangle distribution [30, 35] to simulate the mainassembly time and alternation time, respectively. In our Markovian model, we assume mode 1 processing time to beexponentially distributed with mean 42.5 min and mode 2 with mean 32.5 min. The comparisons between results fromthe simulated real system and the analytical model on fill rates and OTDRs are shown in Table 7 and the correspondingfigures are presented by Figs. 6 and 7. Also note that the data denoted by ‘Num.’ are obtained by our analytical modeland those by ‘Sim.’ are the simulation results. The results from our model are lower but close to the simulation resultsand the gaps decrease when base-stock level increases. This means that if high fill rate and OTDR are required and weneed to set a higher base-stock level, our model can deliver a relatively accurate approximation on the required base-stock level. From our example, suppose both the fill rate and OTDR are required at least 90%, then based on our modelwe can set base-stock level to 9, however, the actual quality of services may be slightly higher.

Since we assume the custom products can be made by alternating the existing standard ones with additional works andthe corresponding processing should be shorter than the one for standard product, in the following examples, we assumethat l1 < l2.

Fig. 4. Convergences of fill rate and OTDR on S as k1 increases.


Fig. 5. Convergences of fill rate and OTDR on S as k2 increases.


Table 7Comparison results on fill rate Pf and OTDR.

S Pf OTDR

Num. Sim. Num. Sim.

3 0.7173 0.7917 0.8369 0.97416 0.8596 0.9254 0.8978 0.99219 0.9192 0.9666 0.9232 0.9967

12 0.9503 0.9820 0.9365 0.997015 0.9683 0.9923 0.9441 0.999018 0.9794 0.9949 0.9488 0.999021 0.9864 0.9979 0.9518 0.9990

Fig. 6. Comparisons on fill rate.

Fig. 7. Comparisons on OTDR.


Table 8Fill rates and OTDR on various S (k1 ¼ 5; k2 ¼ 2;l1 ¼ 10 and l2 ¼ 20).

S Pf OTDR S Pf OTDR

3 0.7623 0.8554 12 0.9762 0.96104 0.8271 0.8874 13 0.9809 0.96335 0.8706 0.9089 14 0.9847 0.96516 0.9012 0.9240 15 0.9877 0.96667 0.9235 0.9350 16 0.9901 0.96788 0.9402 0.9432 17 0.9920 0.96879 0.9529 0.9494 18 0.9935 0.9695

10 0.9626 0.9542 19 0.9948 0.970111 0.9702 0.9580 20 0.9958 0.9706

Table 9TCs on various base-stock levels (k1 ¼ 5; k2 ¼ 2;l1 ¼ 10;l2 ¼ 20;C1 ¼ $20;C2 ¼ $30 C3 ¼ $0:7 and C4 ¼ $1).

S TC S TC

3 35.098 14 14.5844 27.479 15 14.9205 22.627 16 15.3336 19.432 17 15.8037 17.306 18 16.3188 15.906 19 16.8729 15.017 20 17.455

10 14.500 21 18.06111 14.256 22 18.68512 14.218 23 19.32413 14.340 24 19.975

Fig. 8. TCs on various base-stock levels(k1 ¼ 5; k2 ¼ 2;l1 ¼ 10;l2 ¼ 20;C1 ¼ $20; C2 ¼ $30C3 ¼ $0:7 and C4 ¼ $1).


Consider a system with k1 ¼ 5; k2 ¼ 2;l1 ¼ 10 and l2 ¼ 20. Suppose that the fill rate is required to, at least, 0.9 and theon-time-delivery-rate (with the required lead time 0.2) must be at least 0.95. We first try to find the base-stock levels thatthese qualities of services are satisfied. The results on various base-stock levels are shown in Table 8. We can see that thesequalities of services are satisfied if S is greater than or equal to 10.


Now, we apply some cost structure by defining the following costs. Let C1 denote the penalty cost for each unsatisfiedordinary demand; Let C2 denote the penalty cost for each unsatisfied specific demand; Let C3 denote the inventory cost rateper each finished standard product in stock; Let C4 denote the waiting cost rate per work-in-process. Then, the total cost rate,TC, is expressed as

TC ¼ 1� Pf

� �k1C1 þ PðR > uÞk2C2 þ LFC3 þ LPOC4;

where u is the required maximal lead time for the specific demand.We consider the same case of k1 ¼ 5; k2 ¼ 2;l1 ¼ 10 and l2 ¼ 20 with C1 ¼ $20;C2 ¼ $30;C3 ¼ $0:7 and C4 ¼ $1. Suppose

the qualities of services are that the fill rate must be at least 0.9 and the on-time-delivery-rate (with the required lead time0.2) must be at least 0.95. The TCs on various base-stock level S are shown in Table 9 and the corresponding figure is in Fig. 8.From Table 8, we have known that the feasible base-stock levels are those greater than or equal to 10. Among these feasiblelevels, we then obtain the optimal base-stock level at S ¼ 12 with the minimum total cost rate of 14.218.

4. Conclusion

We study a practical production system consistent of MTO and MTS productions and dealing with two random demands:ordinary demands and specific demands. We are interested in the fill rate of the ordinary demand and the on-time-delivery-rate of the specific demand. We analyze these systems by using inventory-queue model and solve the limiting probabilitiesby using matrix geometry method. Some numerical examples are provided. Based on the numerical results, more insightsabout the system are found such as limiting behaviors of the fill rate and on-time-delivery-rate as base-stock levels getslarge. These insights inspire us to study the limiting behavior of the system when the base stock level increases to infinityin the future. Under the cost structure, we also solve the optimal base-stock level numerically. In future study, we may con-sider a multistation system that produces the standard product with the single final process performing the additional workfor the custom product.

Acknowledgments

This work was supported by the National Science Council of Taiwan, ROC, under the grant contract NSC-97-2221-E-033-036.

Appendix A

2 3
A ¼
am;m

. ..

am;m

64 75ðSþ1ÞðSþ1Þ

am;m ¼ k2 0 6 m 6 S;

B ¼

bm;m

bm;m

. ..

bm;m

bmþ1;m bm;m

26666664

37777775ðSþ1ÞðSþ1Þ

bm;m ¼�ðk2 þ l2Þ; 0 6 m 6 S� 1�ðk2 þ l1Þ; m ¼ S

�

bmþ1;m ¼ l1; m ¼ S� 1;

C ¼

cm;m cm;mþ1

cm;m. .

.

. ..

cm;mþ1

cm;m

2666664


cm;m ¼ 0; 0 6 m 6 S

cm;mþ1 ¼ l2; 0 6 m 6 S� 1;

D ¼

dm;m dm;mþ1

dmþ1;m dm;m. .

.

. .. . .

.dm;mþ1

dmþ1;m dm;m

2666664



dm;m ¼�ðk1 þ k2Þ; m ¼ 0

�ðk1 þ k2 þ l1Þ; 1 6 m 6 S� 1

�ðk2 þ l1Þ; m ¼ S

8><>:

dm;mþ1 ¼ k1; 0 6 m 6 S� 1

dmþ1;m ¼ l1; 0 6 m 6 S� 1;

E1 ¼

e1m;m e1m;mþ1

e1m;m. .

.

. ..

e1m;mþ1

e1m;m

e1mþ1;m e1m;m

26666666664

37777777775ðSþ1ÞðSþ1Þ

e1m;m ¼�ðk1 þ k2 þ l2Þ; 0 6 m 6 S� 2

�ðk2 þ l2Þ; m ¼ S� 1


8><>:

e1m;mþ1 ¼ k1; 0 6 m 6 S� 2

e1mþ1;m ¼ l1; m ¼ S� 1;

E2 ¼

e2m;m e2m;mþ1

e2m;m. .

.

. ..

e2m;mþ1

e2m;m

e2m;m

e2m;mþ1 e2m;m

26666666666664

37777777777775ðSþ1ÞðSþ1Þ

e2m;m ¼�ðk1 þ k2 þ l2Þ; 0 6 m 6 S� 3

�ðk2 þ l2Þ; S� 2 6 m 6 S� 1


8><>:

e2m;mþ1 ¼ k1; 0 6 m 6 S� 2

e2mþ1;m ¼ l1; m ¼ S� 1;

ES�1 ¼

em;m em;mþ1

em;m

. ..

em;m

emþ1;m em;m

266666664


em;m ¼�ðk1 þ k2 þ l2Þ; m ¼ 0

�ðk2 þ l2Þ; 1 6 m 6 S� 1


8><>:

em;mþ1 ¼ k1; m ¼ 0

emþ1;m ¼ l1; m ¼ S� 1

References

[1] C.A. Soman, D.P. van Donk, G. Gaalman, Combined make-to-order and make-to-stock in a food production system, Int. J. Prod. Econ. 90 (2004) 223–235.[2] A. Krishnamurthy, R. Suri, M. Vernon, Re-Examining the performance of MRP and kanban material control strategies for multi-product flexible

manufacturing systems, Int. J. Flex. Manuf. Sys. 16 (2004) 123–150.[3] I.J.B.F. Adan, J. ven der Wal, Combining make to order and make to stock, OR Spectrum 20 (1998) 73–81.


[4] V. Nguyen, A multiclass hybrid production center in heavy traffic, Oper. Res. 46 (1998) S13–S25.[5] A. Federguruen, Z. Katalan, Impact of adding a make-to-order item to a make-to-stock production system, Manage. Sci. 45 (1999) 980–994.[6] S. Carr, I. Duenyas, Optimal admission control and sequencing in a make-to- stock/make-to-order production system, Oper. Res. 48 (2000) 709–720.[7] A. Arreola-Risa, G.A. DeCroix, Make-to-order versus make-to-stock in a production-inventory system with general production times, IIE Trans. 30 (1998)

705–713.[8] S. Rajagopalan, Make to order or make to stock: model and application, Manage. Sci. 48 (2002) 241–256.

queueing analysis on a single-station make-to-stock/make-to-order inventory-production system

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