strategic wip inventory positioning for make-to-order production...

Research ArticleStrategic WIP Inventory Positioning for Make-to-OrderProduction with Stochastic Processing Times

Jingjing Jiang and Suk-Chul Rim

Department of Industrial Engineering, Ajou University, Suwon 16499, Republic of Korea

Correspondence should be addressed to Suk-Chul Rim; [email protected]

Received 15 June 2017; Revised 7 September 2017; Accepted 10 October 2017; Published 28 November 2017

Academic Editor: Emilio Jimenez Macıas

Copyright © 2017 Jingjing Jiang and Suk-Chul Rim. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

It is vital for make-to-order manufacturers to shorten the lead time to meet the customers’ requirements. Holding work-in-process(WIP) inventory at more stations can reduce the lead time, but it also brings about higher inventory holding cost. Therefore, it isimportant to seek out the optimal set of stations to holdWIP inventory to minimize the total inventory holding cost, while meetingthe required due date for the final product at the same time. Since the problem with deterministic processing times at the stationshas been addressed, as a natural extension, in this study, we address the problem with stochastic processing times, which is morerealistic in themanufacturing environment. Assuming that the processing times follow normal distributions, we propose a solutionprocedure using genetic algorithm.

1. Introduction

Nowadays, the manufacturing environment has shifted fromtraditional mass production to small order production,because the development of information technology hasresulted in an explosion of product types, shorter product lifecycles, a wider variety of customer requirements, and fasterchanges, which make demand forecasts more difficult.

In 1964, the material requirements planning (MRP) wasdeveloped by Joesph Orlicky, and in the following twodecades various investigations on MRP had been conducted.However, due to the lower demand forecast accuracy, theperformance of traditional MRP became barely satisfactory.Therefore, a new type of MRP called Demand Driven MRP(DDMRP) was introduced in 2011 by Ptak and Smith [1],which enables a manufacturer to respond more closely tothe actual market requirements and promotes better andquicker decision and actions in the planning and executionstage. In traditionalMRP inventory planning,most efforts arefocused on solving two questions: how to hold inventory andwhen to reorder goods. However, in DDMRP, the focus is onwhere to position the inventory, because “how” question ismeaningless until we can answer the “where” question appro-priately. DDMRP has its own logical and effective approach

to answer the “how to hold” question. Determining thebest inventory locations can solve the problems of satisfyinginventory performance and delivery performance.

One of the most important competitive factors in make-to-order manufacturing is to deliver the product to thecustomer within the due time that the customer requires.In order to shorten the delivery lead time, WIP inventoriesneed to be held at major processes. Meanwhile, the totalinventory holding cost should be minimized to decrease thefinancial burden in today’s cost-competitive manufacturingenvironment. It is noteworthy that, for some stations, holdinginventory does not contribute to reducing the lead time offinal product at all due to the BOM structure and processingtimes. Hence, determining the optimal set of stations to holdWIP inventory is a complicated and interesting problem.

This problem, namely, the strategic inventory positioning(SIP) problem, has been studied by many researchers. Why-bark and Yang [2] proposed a controlled simulation experi-ment to decide where to place inventory to achieve the bestservice level. Simpson Jr. [3] used the “all or nothing” policyto decidewhether to place the inventory or not in some pointsin serial line system. Graves and Willems [4–6] extendedSimpson Jr. [3]model to assembly, distribution, and spanning

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 8635979, 7 pageshttps://doi.org/10.1155/2017/8635979

https://doi.org/10.1155/2017/8635979

2 Mathematical Problems in Engineering

tree network structures. Lesnaia [7] considered the servicetime as a stochastic model instead of a deterministic modelassumedbyGraves andWillems [5].Magnanti et al. [8] solvedthe inventory positioning problem at production/assemblystages of components in an acyclic supply chain networkstructure which is not a spanning tree network structure.Many other researchers studied the inventory positioningproblem using different methods such as Kaminsky andKaya [9], Inderfurth [10], Alderson [11], and Georage etal. [12]. We also have studied SIP problem for simple andgeneral BOM using actively synchronized replenishment(ASR) lead time proposed in the DDMRP [13, 14], in whichwe present mathematical models and genetic algorithm tosolve SIP problems, assuming that the processing times areall deterministic. However, it will be more realistic in mostcases to consider the processing times to be stochastic due tovarious factors such as defectives, reprocessing, equipmentsetup, and/or calibration time variations. In this paper weaddress this stochastic SIP problem and present a solutionprocedure for simple BOM case.

This paper is organized as follows: Section 2 describesthe SIP problem and introduces the ASRLT, buffer profiles,and notations; Section 3 presents a genetic algorithm-basedsolution procedure and numerical examples; finally, theconclusion and potential future research issues are addressedin Section 4.

2. The Strategic InventoryPositioning Problem

In the authors’ earlier works [13, 14], the concept of a newlead time has been introduced, namely, actively synchronizedreplenishment lead time (ASRLT), which was proposed byPtak and Smith [1], in which ASRLT is defined as “thelongest unprotected or unbuffered sequence in the BOMfor a particular parent.” The strategic inventory positioning(SIP) problem is to determine the locations which holdWIP inventory in order to meet the lead time the customerrequires for the final product, while minimizing the totalinventory holding cost. The stochastic SIP problem is theSIP problem where processing times at the nodes follow acertain probability distribution. In this paper, we assume theprocessing times follow normal distributions.

2.1. Notations. For consistency, we will use in this paper thesame BOM and notations as used in the previous papers [13,14] as follows:

part𝑖,𝑗: 𝑗th part counted from the left in the 𝑖th levelof the BOM𝑝𝑖,𝑗: the number counted from the left in the BOM ofthe parent part of part𝑖,𝑗 (e.g., in Figure 1, 𝑝3,2 = 2, 𝑝3,4= 3)𝑟𝑖,𝑗: required quantity to make unit immediate parentof part𝑖,𝑗𝑎𝑖,𝑗: ASRLT of part𝑖,𝑗

101

201 202 203

302301 303 304P

401P 402 403P 404P

501P

2

5 7 6

4 4 3 3

3 4 3 7

11

Figure 1: An example of simple bill of materials.

pr𝑖,𝑗: processing time distribution of part𝑖,𝑗 that fol-lows a normal distribution with mean 𝜇𝑖,𝑗 and vari-ance 𝜎𝑖,𝑗2𝑛𝑖,𝑗: required quantity of part𝑖,𝑗 to make unit endproduct𝑠𝑖,𝑗: set of numbers counted from the left in the BOMof the subcomponents of part𝑖,𝑗 (e.g., in Figure 1, 𝑠2,2= {2, 3}; 𝑠3,2 = {3, 4})adu𝑖,𝑗: average daily usage of part𝑖,𝑗𝑐𝑖,𝑗: annual inventory cost of part𝑖,𝑗ai𝑖,𝑗: average inventory quantity of part𝑖,𝑗V𝑖,𝑗: unit price of part𝑖,𝑗ltp𝑖,𝑗: percentage usage of part𝑖,𝑗 over ASR lead timevp𝑖,𝑗: percentage of red zone base of part𝑖,𝑗 that redzone safety accounts forh: annual inventory holding cost rateu: average daily usage of the end productQ: lead time for the end product required by thecustomer (i.e., service time)𝑦𝑖,𝑗: yellow zone quantity of part𝑖,𝑗𝑔𝑖,𝑗: green zone quantity of part𝑖,𝑗rb𝑖,𝑗: red zone base quantity of part𝑖,𝑗rs𝑖,𝑗: red zone safety quantity of part𝑖,𝑗.

Decision variables:

𝑥𝑖,𝑗 ={{{

1 if part𝑖,𝑗 has inventory

0 if part𝑖,𝑗 has no inventory.(1)

In our earlier papers [13, 14], all the processing times areassumed to be deterministic, so theminimum total inventoryholding cost is also a deterministic value. In this paper,however, the processing times are assumed to follow normaldistributions. So we generate 𝑀 sets of random processingtimes to obtain 𝑀 different ASRLTs of the end product andcorresponding minimum total inventory holding costs. Then

Mathematical Problems in Engineering 3

the final minimum total inventory holding cost is the averageof 𝑀 different minimum total inventory holding costs. Wedefine that a solution X = {𝑥𝑖𝑗} is feasible if the probability 𝛼that ASR lead time of end product for 𝑋 is not greater than𝑄 is not smaller than 𝛼0, where 0 < 𝛼0 < 1, but close to 1; forexample, 𝛼0 = 0.95. Equation (2) is the total inventory holdingcost once the inventory location is determined as X = {𝑥𝑖𝑗}.Equation (3) is the annual inventory holding cost of part𝑖,𝑗.

TC = ∑∀𝑖

∑∀𝑗

𝑐𝑖,𝑗 ⋅ 𝑥𝑖,𝑗 (2)

𝑐𝑖,𝑗 = V𝑖,𝑗 ⋅ ai𝑖,𝑗 ⋅ ℎ. (3)

2.2. Inventory Control Scheme in DDMRP. In their book [1],Ptak and Smith proposed an inventory control scheme asfollows: there are five color zones that comprise the totalbuffer: green zone represents an inventory position thatrequires no actions; yellow zone represents an inventoryposition that requires replenishment; both red zone base andred zone safety represent an inventory position that requiresspecial attention. The average inventory quantity can becalculated as the total quantity of red zone plus half of greenzone, as (4). The magnitude of yellow zone is determined asthe average usagemultiplied by ASR lead time of part𝑖,𝑗 as (5).

ai𝑖,𝑗 = 0.5 ⋅ 𝑔𝑖,𝑗 + rb𝑖,𝑗 + rs𝑖,𝑗 (4)

𝑦𝑖,𝑗 = adu𝑖,𝑗 ⋅ 𝑎𝑖,𝑗. (5)

The magnitudes of red zone base and green zone aredetermined as the percentage usage of part𝑖,𝑗 over ASR leadtime multiplied by yellow zone quantity, as (6) and (7). Themagnitude of red zone safety is determined as percentageof red zone base of part𝑖,𝑗 that red zone safety accountsfor, multiplied by red zone base, as (8). Table 1 shows therecommended ranges of impact factor for the green and redzone bases, according to the relative length of lead time of thepart. Table 2 shows how red zone safety is sized according tothe demand variability. Note that minimum order quantity(MOQ) is not considered in this paper.

rb𝑖,𝑗 = ltp𝑖,𝑗 ⋅ 𝑦𝑖,𝑗 (6)

𝑔𝑖,𝑗 = ltp𝑖,𝑗 ⋅ 𝑦𝑖,𝑗 (7)

rs𝑖,𝑗 = vp𝑖,𝑗 ⋅ rb𝑖,𝑗. (8)

2.3. Mathematical Modeling. Using the above notations andinventory control scheme, we model the SIP problemmathe-matically as follows:

𝑛𝑖,𝑗 = 𝑟𝑖,𝑗 ⋅ 𝑛𝑖−1,𝑝𝑖,𝑗adu𝑖,𝑗 = 𝜇 ⋅ 𝑛𝑖,𝑗𝑎𝑖,𝑗

={{{

max𝑚∈𝑠𝑖,𝑗

[(1 − 𝑥𝑖+1,𝑚) 𝑎𝑖+1,𝑚 + 𝑡𝑖,𝑗] , if 𝑠𝑖,𝑗 = 0𝑎𝑖,𝑗 = 𝑡𝑖,𝑗, if 𝑠𝑖,𝑗 = 0 : end node.

(9)

Table 1: Recommended impact ranges for green and red zone base.

Long lead time 20–40% usage over LTMedium lead time 41–60% usage over LTShort lead time 61–100% usage over LT

Table 2: Recommended impact ranges for red zone safety.

High variability 60–100% of red zone baseMedium variability 41–60% of red zone baseLow variability 20–40% of red zone base

We want to find a set X = {𝑥𝑖𝑗} to minimize the averagetotal inventory holding cost, while the ASRLT of the endproduct is no longer than the customer’s requirement𝑄 withprobability 𝛼 ≥ 𝛼0, where 0 < 𝛼0 < 1, but close to 1; forexample, 𝛼0 = 0.95. From (2) to (9), the optimal solution tothe SIP problem can be obtained by solving the followingmathematical model:

Minimize average total cost:∑∑∀𝑖∑∀𝑗 V𝑖,𝑗 ⋅ ℎ ⋅ ai𝑖,𝑗 ⋅ 𝑥𝑖,𝑗

𝑀st. ai𝑖,𝑗 = (1.5 + vp𝑖,𝑗) ⋅ ltp𝑖,𝑗 ⋅ adu𝑖,𝑗 ⋅ 𝑎𝑖,𝑗

adu𝑖,𝑗 = 𝑢 ⋅ 𝑛𝑖,𝑗𝑛𝑖,𝑗 = 𝑟𝑖,𝑗 ⋅ 𝑛𝑖−1,𝑝𝑖,𝑗

𝑎𝑖,𝑗 ={{{

max𝑚∈𝑠𝑖,𝑗

[(1 − 𝑥𝑖+1,𝑚) 𝑎𝑖+1,𝑚 + 𝑡𝑖,𝑗] , if 𝑠𝑖,𝑗 = 0𝑎𝑖,𝑗 = 𝑡𝑖,𝑗, if 𝑠𝑖,𝑗 = 0 : end node

𝑎1,1 ≤ 𝑄𝛼 ≥ 𝛼0,

(10)

where𝑀 is the number of replications.The above model is not easy to solve because it is not a

LP problem. In the next section, we will present the geneticalgorithm-based solution procedure for the problem.

3. Solution Procedure and Numerical Example

In the deterministic SIP problem addressed in our earlierpapers [13, 14], the ASR lead time of the end product (W) canbe determined as a value, depending on the binary solutionX = {𝑥𝑖𝑗}, since all processing times are deterministic. So asolutionX = {𝑥𝑖𝑗} is feasible if𝑊 is not greater thanQ, the leadtime the customer requires. However, in the stochastic SIPproblem we address in this paper, since different processingtimes will be generated every time for each node,W will alsobe different even for a particular solution X = {𝑥𝑖𝑗}. So for thestochastic SIP we define that a solution X = {𝑥𝑖𝑗} is feasibleif 𝑊 is not greater than 𝑄 with probability 𝛼 ≥ 𝛼0, where0 < 𝛼0 < 1, but close to 1; for example, 𝛼0 = 0.95.

To implement this in the solution procedure, for aparticular solutionX = {𝑥𝑖𝑗} and parameters 𝜇𝑖,𝑗 and𝜎𝑖,𝑗2, rep-resenting the mean and variance of the normally distributedprocessing time of part𝑖,𝑗, respectively, we generate a set ofrandom processing times {𝑡𝑖,𝑗} for all parts and compute 𝑊by the genetic algorithm that we used in the earlier paper


Generate N validchromosomes

Evaluate all chromosomesChoose the best

Currentbest

Terminate?

Select N chromosomes(roulette wheel)

Crossover

Mutation Valid?

Put the chromosomeinto new population

Discard the chromosomeGenerate a valid chromosome

Yes

No

NoYes

For eachchromosome

To generate a valid chromosome(1) Generate random processing times for nodes(2) Calculate ASRLT of end product

(4) Repeat 100 times to compute

(5) If p ≥ 𝛼, the chromosome is a valid one.

(3) ASRLT ≤ Q?

p = avg.prob[ASRLT ≤ Q]

Set Pc, Pm, N

Figure 2: The flow chart of GA-based solution procedure.

for deterministic SIP problem. Repeat this for 100 times tocompute the probability (𝛼) that the ASRLT is not greaterthan 𝑄. If 𝛼 is greater than or equal to a given value 𝛼0, thenthe solution X = {𝑥𝑖𝑗} is regarded as a valid solution.Then thetotal inventory holding cost of X = {𝑥𝑖𝑗} is computed as theaverage of 100 inventory holding costs.

As shown in Figure 2, the solution procedure is that wegenerate𝑁 valid chromosomes, evaluate all chromosomes bycalculating the average total inventory holding cost, and keepthe best one. We select chromosomes by roulette wheel rule.Once two chromosomes are selected, they undergo crossoverandmutate.Then new population including𝑁 chromosomesis created. We test the validity of each chromosome, discardinvalid chromosomes, and generate new valid chromosomesto be put to the population so as to keep the number ofchromosomes in the populations unchanged to be 𝑁. Thenwe evaluate all chromosomes and choose the best one in thisnew population. If this best one is better than the current best,then replace the current best with the new best one (calledelitism rule). We set the termination condition as runningseveral generations, usually depending on the number ofparts in the BOM. The more the parts in the BOM are, themore the generation run is required. In order to evaluatethe performance of the proposed genetic algorithm-basedsolution procedure, two numerical examples are given asfollows.

3.1. Small BOM Case. Consider a small BOM comprising13 parts, as shown in Figure 1. The parameters are given inTable 3. We set the same variability percentage (vp) as 50%for all parts. The lead time percentage (ltp) is 30%, 50%, and80% for long, medium, and short lead time, respectively. 𝛼0 isset as 95%.We set the crossover rate𝑃𝑐 as 0.8 and themutationrate 𝑃𝑚 as 0.1. For small instances, we set the population size𝑁 as 80. The iteration will stop after 100 generations.

In order to evaluate the performance of the proposedGA-based procedure, the result is compared with that obtainedfrom enumeration. For each X = {𝑥𝑖𝑗}, we generate M = 100sets of processing times. Among the valid solutions out of213 possible solutions, we choose the twenty smallest averagetotal inventory holding costs using enumeration, as shown inTable 4.

Table 5 shows the optimal total inventory holding costof 10 replications of the proposed solution procedure. In theprevious problemwith deterministic processing time for eachnode, the optimal total inventory holding cost obtained usingGA can be matched with the ranking using enumeration.However, in the stochastic SIP problem, it is impossible tomatch the optimal total inventory holding cost from theproposed procedure with the ranking list obtained fromenumeration. Therefore, we run the proposed procedure for10 replications and list the result, as shown in Table 5, so thatwe can compare the result with the ranking list obtained fromenumeration. Note that the largest total cost (1,671.78) out ofthe 10 replications in Table 5 is approximately same as the 5thsmallest total cost from enumeration, which indicates that theperformance of the proposed procedure is satisfactory. Eachreplication takes about 30 seconds in the McIntosh PC.

3.2. Large BOM Case. In this example, we consider a BOMcomprising 50 parts, which means that there are 250 possiblesolutions. Table 6 lists the parameters in which vp, lvp, h, 𝑃𝑐,and 𝑃𝑚 assume the same values as those in small BOM case,but𝑄 is set as 16. For the problems of same size, computationtime depends on the population size (N) and the number ofiterations that we set as the termination condition. We setthe program to stop after 200 iterations. We can certainlyincrease the number of iterations, but due to the increasingcomputation time, we think that 200 iterations for eachreplicationwill be enough to show how the objective function


Table 3: Parameters for small BOM case.

Part 1 2 3 4 5 6 7 8 9 10 11 12 13𝜇 2 5 7 6 4 4 3 3 3 4 3 7 11𝜎 0.4 0.9 1.1 0.8 0.6 0.5 0.9 0.7 0.8 0.9 0.4 1.3 2.0p 0 1 1 1 1 2 2 3 1 1 2 2 2v 700 120 250 100 65 80 110 40 24 16 21 27 9Q 10h 0.25vp 0.5ltp Long lead time (26+ days) 30%

Medium lead time (11–25 days) 50%Short lead time (1–10 days) 80%

𝛼0 95%

Table 4: Twenty smallest average total costs obtained from enumer-ation.

Rank Total cost1 1492.952 1584.63 1607.174 1659.85 1687.716 1696.457 1702.68 1707.469 1722.1510 1763.811 1773.0712 1782.113 1785.314 1785.8715 1796.0916 1812.117 1816.0918 1828.6119 1831.3920 1838.2

Table 5: The results of 10 replications obtained from the proposedprocedure.

Number Total cost1 1671.782 1665.133 1529.414 1457.45 1611.376 1551.597 1489.038 1601.449 1493.8710 1597.08

40000

45000

50000

55000

60000

65000

70000

75000

80000

85000

0 20 40 60 80 100 120 140 160 180

Tota

l inv

ento

ry h

oldi

ng co

st

Iterations

Replication 1Replication 2Replication 3

Replication 4Replication 5

Figure 3: Decrease of objective function values for 5 replications forlarge BOM case.

values decrease as the iteration progresses.N is set as 300.Werun 5 replications, each of which takes about 13minutes in thesame PC.We chose the number of replications as 5 arbitrarily.We can certainly increase the number of replications, butdisplaying too many lines in a figure will make the figuremessy, without adding any implication.

Note that we cannot explicitly demonstrate the perfor-mance of the proposed solution procedure for a problemwithlarge BOM of 50 nodes since we cannot enumerate the 250possible solutions. Instead we show the decrease of objectivefunction values of five replications in Figure 3. Recall that thebest result obtained from GA when it terminates does notguarantee being optimal.

4. Conclusion

In this paper, we address the problem of determining thestations to hold work-in-process inventory in order to reduce


Table 6: Parameters for large BOM case.

Part 𝜇 𝜎 𝑟 𝑝 V1 2 0.2 1 0 12002 5 0.3 1 1 9003 7 2 1 1 8904 6 1.2 1 1 9105 4 0.8 2 1 8606 4 0.7 3 1 8807 3 0.4 2 1 8008 3 0.3 1 1 7809 3 0.3 2 2 79010 4 0.4 3 2 75011 3 0.4 2 3 69012 5 0.4 3 3 66013 5 0.4 4 4 60014 4 0.3 3 4 64015 2 0.2 1 4 70016 4 0.3 3 1 66017 6 0.9 2 1 58018 3 0.5 1 2 55019 3 0.4 1 3 57020 5 0.8 2 3 60021 7 1.1 3 4 54022 6 1.0 2 4 51023 3 0.5 2 4 59024 4 0.5 1 5 40025 3 0.5 1 6 49026 3 0.4 1 7 48027 5 0.7 2 7 52028 4 0.4 2 8 40029 3 0.5 1 9 48030 2 0.2 3 9 49031 5 0.6 1 1 46032 4 0.6 1 1 50033 3 0.3 3 3 47034 3 0.3 2 3 30035 5 0.6 2 5 38036 4 0.4 2 8 33037 3 0.4 3 8 34038 4 0.3 1 12 35039 6 0.6 1 12 36040 4 0.4 3 15 31041 4 0.5 2 2 33042 4 0.5 2 3 29043 3 0.3 2 3 27044 3 0.4 2 5 26045 2 0.2 1 7 27046 5 0.5 1 7 27047 3 0.4 3 9 10048 2 0.4 2 9 11049 3 0.6 4 5 5050 5 0.9 2 7 30

the production lead time of the end product to the onethe customer requires, while minimizing the total averageinventory cost inmake-to-ordermanufacturing. As an exten-sion of the earlier work of the authors, where processingtimes are assumed to be constant at each station, in thispaper we consider the stochastic version of the problemin which processing times follow certain probability dis-tribution, namely, stochastic strategic inventory positioning(SIP) problem. This will be more realistic than assumingdeterministic processing times. We formulate the stochasticSIP problemmathematically and present a genetic algorithm-based solution procedure. Numerical examples show thevalidity of the proposed solution procedure.

In most MTO manufacturing sites, it is not unusual tofind that almost all stations maintain WIP inventory, mainlydue to natural human behavior of protection. However,having WIP inventory at some stations does not contributeat all to reducing the total manufacturing lead time of endproduct because of the BOM structure and lead time of otherparts in the BOM. Tominimize theWIP inventory cost, suchstations should not have WIP inventory. The result of thispaper will help determine strategically where to have andwhere not to have WIP inventory. We use inventory controlscheme of DDMRP to compute the average inventory at astation. For further research, one may study the case wheresome parts are used more than once in the BOM.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This research was sponsored by Jeongseok Foundation. Theauthors would like to express their thanks to JeongseokFoundation for support during the research.

References

[1] C. Ptak and C. Smith, Orlicky’s Material Requirements Planning3/E, McGraw-Hill Education, New York, NY, USA, 2011.

[2] D. C. Whybark and S. Yang, “Positioning inventory in distri-bution systems,” International Journal of Production Economics,vol. 45, no. 1–3, pp. 271–278, 1996.

[3] K. F. Simpson Jr., “In-process inventories,”Operations Research,vol. 6, no. 6, pp. 863–873, 1958.

[4] S. C. Graves and S. P.Willems, “Strategic safety stock placementin supply chains,” in Proceedings of the MSOM Conference,Hanover, NH, USA, 1996.

[5] S. C. Graves and S. P. Willems, “Optimizing strategic safetystock placement in supply chains,” Manufacturing & ServiceOperations Management, vol. 2, no. 1, pp. 68–83, 2000.

[6] S. C. Graves and S. P. Willems, “Strategic inventory placementin supply chains: nonstationary demand,” Manufacturing &Service Operations Management, vol. 10, no. 2, pp. 278–287,2008.


[7] E. Lesnaia,Optimizing safety stock placement in general networksupply chains [Ph.D. thesis], MIT Operations Research Center,Cambridge, UK, 2004.

[8] T. L. Magnanti, Z. M. Shen, J. Shu, D. Simchi-Levi, and C.-P.Teo, “Inventory placement in acyclic supply chain networks,”Operations Research Letters, vol. 34, no. 2, pp. 228–238, 2006.

[9] P. Kaminsky and O. Kaya, “Inventory positioning, schedulingand lead-time quotation in supply chains,” International Journalof Production Economics, vol. 114, no. 1, pp. 276–293, 2008.

[10] K. Inderfurth, “Safety stock optimization in multi-stage inven-tory systems,” International Journal of Production Economics,vol. 24, no. 1-2, pp. 103–113, 1991.

[11] W. Alderson, “Marketing efficiency and the principle of post-ponement,” in Marketing Behavior and Executive Action, pp.423–427, 1957.

[12] I. Georage, G. Prastacos, and G. Skintzi, “Inventory position-ing in multiple product supply chains,” Annals of OperationsResearch, vol. 126, pp. 195–213, 2004.

[13] S.-C. Rim, J. J. Jiang, and C. J. Lee, “Strategic inventory position-ing for MTO manufacturing using ASR lead time,” in LogisticsOperations, Supply Chain Management and Sustainability, P.Golinska, Ed., EcoProduction, pp. 441–456, Springer, Basel,Switzerland, 2014.

[14] J. Jiang and S.-C. Rim, “Strategic Inventory Positioning in BOMwith Multiple Parents Using ASR Lead Time,” MathematicalProblems in Engineering, vol. 2016, Article ID 9328371, 2016.

Submit your manuscripts athttps://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

strategic wip inventory positioning for make-to-order production...

Documents