joint production and subcontracting planning of unreliable multi-facility multi-product production...
TRANSCRIPT
Joint production and subcontracting planning of unreliablemulti-facility multi-product production systems
M. Assid a, A. Gharbi a,n, K. Dhouib b
a Automated Production Engineering Department, École de Technologie Supérieure, Production System Design and Control Laboratory, University of Québec,1100 Notre Dame Street West, Montreal, QC, Canada H3C 1K3b Mechanical Engineering and Productique Department, École Nationale Supérieure des Ingénieurs de Tunis (ENSIT), LMSSDT Laboratory, University of Tunis,5Av. Taha Hussein, Tunis, Tunisia
a r t i c l e i n f o
Article history:Received 31 May 2013Accepted 21 July 2014This manuscript was processed by AssociateEditor KuhnAvailable online 2 August 2014
Keywords:Production/subcontracting controlPharmaceutical industryService levelSimulation modelingOptimizationResponse surface methodology
a b s t r a c t
This article addresses the problem of joint optimization of production and subcontracting of unreliableproduction systems. The production system considered presents a common problem in the pharma-ceutical industry. It is composed of multiple production facilities with different capacities, each of whichis capable of producing two different classes of medications (brand name and generic). The resort tosubcontracting is double: first, it involves the quantity of products received on a regular basis in order tocompensate for insufficient production capacity in existing facilities, second, when needed, urgentorders are also launched in order to reduce the risk of shortages caused by breakdowns of manufacturingfacilities. Failures, repairs and urgent delivery times may be represented by any probability distributions.
The objective is to propose a general control policy for the system under consideration, and to obtain,in the case of two facilities, optimal control parameters that minimize the total incurred cost for aspecific level of the customer service provided. Given the complexity of the problem considered, anexperimental optimization approach is chosen in order to determine the optimal control parameters.This approach includes experimental design, analysis of variance, response surface methodology andsimulation modeling. It allows the accurate representation of the dynamic and stochastic behaviors ofthe production system and the assessment of optimal control parameters. Other control parameterswhich represent the subcontracting are introduced and three joint production/subcontracting controlpolicies (general, urgent, regular) are compared to one another. The proposed joint production/regularsubcontracting control policy involves a cost decrease of up to 20%, as compared to results obtained byDror et al. [1], who used a simplified control policy in addition to a heuristic solution approach for a realcase study. This policy offers not only cost savings, but is also easier to manage, as compared to thatproposed by Dror et al. [1]. Numerical examples and a sensitivity analysis are also performed to illustratethe robustness of the proposed control policy and the solution approach.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Manufacturing systems require continuous control and mon-itoring [2]. Their role has become essential thanks to an economicenvironment which is getting more and more competitive. Inaddition, manufacturing systems are getting more complex due torandom fluctuations of production system components (demand,breakdowns, repairs, etc.). That is why the use of buffer stocksbetween workstations and at the end of the production cycle hasproven to be an effective tool for protecting against randomperturbations. These perturbations are often manifested through
the stoppage of production activities. Despite its usefulness inmaintaining customer satisfaction, having a high level of safetystocks has some disadvantages, such as increased operating andinventory costs. Low safety stock levels, however, increase the riskof shortage as well as customer dissatisfaction. The major dilemmathen resides in determining the optimal level of buffer stocks toadopt which allows a reduction of the total incurred cost andensures a high level of customer satisfaction.
The pharmaceutical field is characterized by increasingly fiercecompetition, a spectacular rise in the number of generic drugs andnew discoveries in biotechnology. In response to these develop-ments, pharmaceutical companies are continually seeking toimprove the planning and control of the supply chain [3] as wellas the efficiency of their production processes. They aim to thusminimize costs and successfully adapt their production activity tomarket needs. The U.S. Bureau of Labor Statistics (BLS) classifies
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Omega
http://dx.doi.org/10.1016/j.omega.2014.07.0070305-0483/& 2014 Elsevier Ltd. All rights reserved.
n Corresponding author. Tel.: þ1 5143968969; fax: þ1 5143968595.E-mail addresses: [email protected] (M. Assid),
[email protected] (A. Gharbi), [email protected] (K. Dhouib).
Omega 50 (2015) 54–69
pharmaceutical companies under three broad categories [4]: (1)producers of consumer products whose research and developmentactivities are primarily focused on the development of newformulas, (2) producers of generic drugs who specialize in pre-parations derived from branded drugs that are no longer protectedby patents, and finally, (3) designers of new molecules thatproduce two types of drugs (both brand name and generic), andwhich are usually engaged at all levels of the pharmaceutical chainof activities, namely, research and development, production andmarketing. This work concerns this last category. Note that genericdrugs are sold at lower prices, which explains the frequent use ofsubcontracting for such products, in order to reduce costs andincrease production capacity. That is not the case with brand namedrugs, for which patents have been issued. In this article, we areinterested in the case of those pharmaceutical companies that notonly produce both brand name and generic drugs, but must alsoresort to subcontracting in order to satisfy all customers andremain competitive. The goal is to find the production rate andthe storage capacities required for each product type, as well asthe subcontracting rate, which minimize the total incurred costand maintain a specific customer service level. More specifically, itis about a problem of optimal control of production and subcon-tracting. In fact and according to Kaplan and Laing [5], it is acommon problem in the pharmaceutical industry. Indeed, Booth[6] interprets the manufacturing priorities of pharmaceuticalcompanies as the balance between the supply and the demandas well as the increase of the use of subcontracting in order toreduce costs incurred.
In the literature, several research studies have been undertakento address the optimal control problem for different classes ofmanufacturing systems which are subject to random failures. Mostof them have focused on manufacturing systems with statesdescribed by a Markov process. Olsder and Suri [7] exploited theformalism of Richel [8] and developed the dynamic programmingequation of the optimal control policy for a manufacturing systemoperating in an uncertain environment according to the homo-geneous Markov process. They focus on the production planning ofa manufacturing system composed of a single machine thatproduces one part type and whose dynamics is described by thehomogeneous Markov Chain (constant transition rates). The workof Kimemia and Gershwin [9] and Akella and Kumar [10] haveshown that for such a system, the control policy known as theHedging Point Policy (HPP) is optimal. The HPP policy consists ofbuilding an optimal safety stock (threshold) during periods ofexcess machine capacity. As a result, future failures of the systemwill be prevented, leading to greater customer satisfaction. Severalauthors have extended the HPP policy in order to considerpractical aspects such as multi-state machines, random demand,preventive maintenance, quality problems, and simultaneousbreakdown and quality failures [11–16].
For systems with several product types, Caramanis and Shar-ifnia [17] used the results of Sharifnia [11] and proposed asuboptimal production control policy based on the decompositionmethod transforming the complex multi-product control problem(M1Pn) into several mono-product control problems (M1P1) thatcan be treated analytically. Sethi and Zhang [18] presented anexplicit formulation of the optimal control problem of a produc-tion system which consists of a single machine capable of produ-cing several part types with negligible setup times and costs. Thesame hypothesis was used by Gharbi and Kenné [19] when theystudied the production control problem of a manufacturing systemwith multiple machines and multiple product types. Bai andElhafsi [20] elaborated the optimality conditions described byHamilton–Jacobi Bellman (HJB) equations for a manufacturingsystem involving an unreliable machine. The production machineis capable of producing two part types, with non-negligible setup
times and costs. The authors then presented a suitable structure ofthe control policy, known as the Hedging Corridor Policy (HCP).Gharbi et al. [21] extended the results of Bai and Elhafsi [20] byproposing a near-optimal control policy called the ModifiedHedging Corridor Policy (MHCP).
Insufficient production capacity with respect to customerdemand forced some industrial companies from different fields(pharmaceutical, automobile, aeronautics, etc.) to seek otheralternatives, including subcontracting or the acquisition of newmanufacturing machinery, which are used to increase productioncapacity and to satisfy all customers in terms of quantity and time.Dellagi et al. [22] considered a production system composed of asingle machine producing one product type to satisfy a constantdemand. This system calls upon a second machine (the subcon-tractor) to ensure the satisfaction of the demand. In the samecontext, Ayed et al. [23] used subcontracting as an independentproduction system in order to meet a random demand. Gharbiet al. [24] studied a manufacturing cell capable of producing oneproduct type. Its total capacity changes depending on whether ornot a reserve machine (stand-by) is used; a machine with a higherproduction cost. The proposed control policy is called the StateDependent Hedging Point Policy (SDHPP), and is characterized bytwo thresholds.
Recently, Dror et al. [1] studied a common case in thepharmaceutical industry. Here, a complex production system(M2P2) consisting of two facilities prone to random breakdownsand repairs is examined. These facilities have different capacitiesand are able to produce two medication types. The brand namemedication must be produced internally, while the generic med-ication can be produced internally or supplied by subcontractors.The brand name and the generic medications are composed of thesame main chemical substances. However, the branded medica-tion contains special chemical additives that improve its perfor-mance characteristics. The subcontracting is intended tocompensate for the lack of production capacity with respect tocustomer demand, and to deal with failure occurrences. Thesubcontracting involves an amount of drugs received on a regularbasis, as well as other urgent orders which are initiated, whenneeded, in order to reduce the risk of shortages due to possiblerandom breakdowns. The urgent subcontracting depends on anon-negligible and random delivery time.
Based on the HPP policy, Dror et al. [1] proposed a controlpolicy that contributes to minimizing safety stocks and storagerequirements for the two medication classes in order to reach acustomer service level greater than or equal to 99.5%. The authorsused several restrictions and assumptions which affect the qualityof the results. They proposed a heuristic approach which ignoresthe dynamic aspects of the production system, the randomdelivery times and breakdowns, and above all, it also ignoresproduction, subcontracting, inventory carrying and shortage costs.However, the majority of the works in the scientific literaturecovering production systems indicate that it is important to takeinto consideration the incurred cost as the main decision criterionin tackling the concerns of the company's decision makers.
Our objective is to propose a better joint production/subcon-tracting control policy and an efficient solution approach to thiscomplex problem for a manufacturing system composed of multi-ple production facilities and producing two product types (MiP2,iA{1,…,m}). This will be achieved by simultaneously consideringcosts and customer satisfaction as two decision criteria in order todeal with the concerns of decision makers. We also explain therelationship between the customer service level, the total incurredcost and the control parameters of the production system. It isimportant to note that no other researchers, except Dror et al. [1],have addressed the control problem presented in this work. Theproposed experimental solution approach combines statistical
M. Assid et al. / Omega 50 (2015) 54–69 55
methods with simulation modeling in order to evaluate the bestcontrol parameters (i.e., control parameters that minimize thetotal incurred cost while respecting the customer satisfactionconstraint). It is based on experimental design, analysis of var-iance, response surface methodology, as well as simulation mod-eling. In fact, simulation is an effective tool which allows us to takeinto account the dynamic and stochastic nature of such a complexproduction system. The proposed approach will eliminate thesimplification and the unrealistic assumptions imposed by Droret al. [1], and will accurately reproduce the behavior of theproduction system [25]. The rest of this article is organized asfollows. In Section 2, the joint production/subcontracting controlproblem of the pharmaceutical production system (multiple facil-ities/two product types: MiP2, iA{1,…,m}) is formulated analyti-cally. Sections 3 and 4 respectively present the experimentalsolution approach adopted and the simulation model developed.Section 5 summarizes the solution approach proposed by Droret al. [1], system's data, and the main obtained results. In Section 6,we use the proposed solution approach to determine the optimalparameters of the control policy based on Dror's control policy.A comparative study of the results generated by the proposedapproach and that of Dror et al. [1] is also carried out. Section 7presents the comparison of three control policies which integratethe subcontracting parameters as decision variables. The best jointproduction/subcontracting control policy obtained is then studiedin depth in Section 8, and compared to the Dror's results.Sensitivity analyses are also carried out to confirm the robustnessof the joint control policy and the solution approach.
2. Notations and problem formulation
2.1. Notations
Let I¼ f1;2g and J¼ f1;2;…;mg.
Pi Type of product, iA I (P1: the brand name medication, P2:the generic medication)
xiðtÞ Inventory level (or backlog) of product Pi at time t; iA I;di Demand rate of the product Pi, iA Iuij(t) Production rate of the product Pi at the facility Mj at time
t, iA I; jA Jui(t) Production rate of the product Pi at time t, iA I
Uijmax Maximum production rate of the product Pi at the facility
Mj, iA I; jA JUR
st Regular subcontracting rate of product P2QU
stðtÞ Urgent subcontracting quantity of product P2 receivedat time t
UstðtÞ Subcontracting rate of product P2 at time tZi Storage space capacity of the product Pi iA IMTBFj Mean time between failures of the facility Mj, jA JMTTRj Mean time to repair of the facility Mj, jA Jcþi Product type i inventory cost per time unit per item, iA Ic�i Product type i backlog cost per time unit per item, iA Ice Storage space cost per time unit per itemcpij Production cost of the facility Mj per item of Pi, iA I; jA J
cRst Regular subcontracting cost per itemcust Urgent subcontracting cost per itemTD Random transportation delay of the urgent subcontracting
2.2. Problem formulation
The problem consists of multiple production facilities withdifferent capacities, and each of which is capable of producing twodistinct types of products: branded medication (P1) whose com-pany is the only producer, and generic medication (P2). Beforebeing delivered to customers, each type of product is stored in astorage space under ideal conditions in order to preserve itsquality [1]. Unlike other works as [26,27], which considered asystem with deteriorating item, both product types (P1 and P2) donot present any risk of degradation related to quality. Theproduction facilities are subject to random breakdowns andrepairs that can generate stock-outs. The company can providethe product P2 from external sources in order to increase theproduction capacity of facilities or perhaps to deal with theoccurrence of failures. The subcontracting consists of an amountof medications received on a regular basis ðUR
stÞ and of other urgentorders ðQU
stÞ launched to reduce the risk of shortages, whenneeded. It should be noted that urgent orders are characterizedby a transportation delay of TD time units, where TD is a randomvariable. This implies that urgent orders launched at time t arriveto the inventory of finished product P2 at time (tþTD). Fig. 1describes the structure of the production system.
Because of the random occurrence of breakdowns and repairactivities, the state of the system is modeled by two components,
Fig. 1. The considered pharmaceutical production system.
M. Assid et al. / Omega 50 (2015) 54–6956
the level of accumulated inventory of both products XðtÞ ¼ ðx1ðtÞ;x2ðtÞÞA R2 with a continuous nature in time and the stochasticprocess αjðtÞ A Bj ¼ f0;1g (1rjrm) that describes the operatingmode of the facility Mj. Such a facility is available when it is opera-tional ðαjðtÞ ¼ 1Þ and not available when it is under repair ðαjðtÞ ¼ 0Þ.The manufacturing system mode can be described by the randomvector αðtÞ ¼ ðα1ðtÞ;…; αmðtÞÞ0 with values in B¼B1xB2x…xBm.
The dynamic behavior of the system is described by the statevariables ðXðtÞ; αðtÞÞ; with XðtÞ ¼ ðx1ðtÞ; x2ðtÞÞ A R2 and αðtÞAB. Thefollowing differential equations represent the dynamics of finishedproduct stocks:
_x1ðtÞ ¼ ∑m
j ¼ 1u1jðtÞ�d1; x1ð0Þ ¼ x01
_x2ðtÞ ¼ ∑m
j ¼ 1u2jðtÞþUR
stþ QUstðtÞ�d2; x2ð0Þ ¼ x02
8>>>><>>>>:
ð1Þ
where x01 and x02 designate the initial level of in ventory of productsP1 and P2, respectively. The production rates at every momentmust satisfy the capacity constraint of the system given by:
0ruijðtÞrUijmaxnαjðtÞ; 8 iA I and jA J ð2Þ
Given the subcontracting of the product P2, the productioncapacity of the system increases in order to meet the customerdemand. The overall feasibility constraint is formulated as follows:
For the product P1
∑m
j ¼ 1U1j
maxnMTBFj
MTBFjþMTTRj
� �Zd1 ð3Þ
For the product P2
1� d1∑m
j ¼ 1U1jmax
0@
1An ∑
m
j ¼ 1U2j
maxnMTBFj
MTBFjþMTTRj
� �0@
1AþUR
stþQUstðtÞZd2 ð4Þ
Eq. (4) takes into account the prioritization of the product P1 whichcannot be provided from external sources (the subcontracting). Thisprioritization is represented by “d1=∑m
j ¼ 1U1jmax”. The latter corre-
sponds to the usage proportion of the production capacity of thefacilities which allow the satisfaction of the demand d1. Our decisionvariables are the production rates ðui1ðtÞ;…;uimðtÞÞ of facilitiesM1,M2,… Mm, the rate of regular ðUR
stÞ and the urgent subcontracting ordersðQU
stðtÞÞ. It should be noted that QUstðtÞ was used in order to restore the
level of safety stocks where applicable. When taking into accountEqs. (2)–(4), for each mode of αðtÞAB, i A f1;2g and j A f1;…;mg,the acceptable decisions are presented as follows:
where AR is the aggregation ratio converting product 1 into product2 from a capacity point of view. AR¼ ðU2j
max=U1jmaxÞ.
A facility failure may incur costs which are added to those forproduction, subcontracting, inventory and storage space. Suchcosts are associated with unmet customer demand, and includepenalties generated by insufficient stock, and even the loss ofcustomers. The problem is of finding an acceptable control policyin order to minimize the cost function J(.) by respecting the
customer satisfaction constraint.
Jð:Þ ¼ ceðZ1;Z2ÞþEZ 1
0e�ρtgð:Þ dt
� �ð6Þ
In addition to inventory costs ðcþi Þ, the model considers storagespace cost ðceÞ, depending on the storage capacity required tomaintain the two finished products. The cost penalizes the storagespace, and not the conservation actions targeting finished pro-ducts [28]. It represents the depreciation (investment/lifetime) ofthe equipment considered [29] (we assume a linear depreciation).It is in fact a step cost(1) in the form of an average cost per unit oftime. The discount rate is denoted by ρ and the instantaneous costfunction by g(.). This function is defined at the instant t as follows:
gðX;UÞ ¼ ∑2
i ¼ 1ðcþ
i xþi ðtÞþc�
i x�i ðtÞÞþ ∑
2
i ¼ 1∑m
j ¼ 1ðcpij uijðtÞÞþðcRst UR
stÞþðcust QUstðtÞÞ ð7Þ
where U denotes the vector of production rates, regular subcon-tracting rate and urgent subcontracting orders, X is the vector ofthe inventory levels (or of the shortage) of both products. Thus,
xþi ðtÞ ¼ maxð0; xiðtÞÞ
x�i ðtÞ ¼ maxð�xiðtÞ;0Þ
(
The production cost ðcp2j; j Af1;…;mgÞ and the subcontractingcost (cRst and cust) of product P2 must satisfy the following inequality:
0ocp2jocRstocust; 8 jA J
As mentioned above, the problem of production planning isconsidered to establish an acceptable control policy ðUnÞAΓ. Thiswill lead to the minimization of the updated total average cost (6),while respecting the customer satisfaction specification on the onehand, and considering constraints (1)–(5) on the other. This policyconstitutes a feedback that determines the amount of subcontract-ing and of production of each facility, depending on the state ofthe system. The value function of the optimization problem isdescribed by the following function:
vðXÞ ¼ infðuij ;UR
st ;QustÞAΓ
JðX; αðtÞ;uijðtÞ;URst;Q
UstðtÞÞ; iAf1;2g; jAf1;2;…;mg ð8Þ
The determination of the properties of the function (8) leads tothe optimality conditions described by the Hamilton–Jacobi Bell-man (HJB) equations. However, the analytical solution of theseequations is only possible in specific cases, namely, those involvingsimple production systems that are analyzed using the Markovchains [30]. In the following sections, we propose a control policywhich simultaneously combines the production and the subcon-tracting planning of the production system. We also present an
experimental solution approach based on experimental design,analysis of variance, response surface methodology, and simula-tion modeling in order to generate the optimal control parametersof the proposed policy.
Γ ¼
ðu11ðtÞ;…;u1mðtÞ;u21ðtÞ;…;u2mðtÞ;URst;Q
UstðtÞÞ : 0ruijðtÞrUij
maxnαjðtÞ;
d1nARþd2� ∑m
j ¼ 1U2j
max
!rUR
strd2; ∑m
j ¼ 1U1j
maxnMTBFj
MTBFjþMTTRj
� �Zd1;
1� d1∑m
j ¼ 1U1jmax
� �n ∑
m
j ¼ 1U2j
maxnMTBFj
MTBFjþMTTRj
� � !þUR
stþQUstðtÞZd2
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
ð5Þ
1 Fixed cost that increases to a new level in step with the significant changes inTank's capacity (Z1 and Z2).
M. Assid et al. / Omega 50 (2015) 54–69 57
2.3. General joint production/subcontracting control policy
The proposed general control policy, as in Wang and Chan whotreated the prioritization in the case of one machine and multipleproducts [31], takes into account the priority given to the innovativeproduct P1 whose company is the only producer. The manufacturing ofthis product depends only on its inventory level with respect tothreshold Z1 (see Eq. (9)). The threshold Zi, i A f1;2g defines thestorage space capacity of each type of product Pi. Using the term“∑m
j ¼ 1ðU2jmaxnαjðtÞÞ�d1nAR”, only the remaining production capacity
is allocated to the product P2 (see Eq. (10)). However, if this capacity isinsufficient to meet demand d2, the subcontracting is thus used toensure the satisfaction of the demand d2 and to face several randomphenomena such as failures and repairs time (see Eqs. (10) and (11)).Both regular and urgent subcontracting are considered in the pro-posed control policy (see Eq. (11)). The first is a fixed rate whose valuemust be optimized. While the second one is composed of orderswhich are activated in order to reduce the risk of shortages. We thusdefined the threshold SQ as the inventory level of product P2 at whichurgent orders are launched. We integrate urgent subcontracting,which depends on a random transportation delay (TD) into theproposed model based on Mourani et al. [32] whom considered thetransportation delay of the production.
Let UstðtÞ be the set of decision variables related to subcon-tracting and ðu1ðtÞ;u2ðtÞÞ the decision variables related to theproduction rate of part types P1 and P2, respectively. The structureof the proposed control policy, called the general joint production/subcontracting control policy, is expressed by Eqs. (9)–(11)
� For the product P1:
u1ðtÞ :
∑m
j ¼ 1ðU1j
maxnαjðtÞÞ if ðx1ðtÞoZ1Þ
min d1; ∑m
j ¼ 1ðU1j
maxnαjðtÞÞ !
if ðx1ðtÞ ¼ Z1Þ
0 otherwise
8>>>>>>><>>>>>>>:
ð9Þ
� For the product P2:
UstðtÞ :UR
stþQUstðtþTDÞ if x2ðtÞ ¼ SQ
URst otherwise
(
With,
URstþQU
stðtÞþ 1� d1
∑mj ¼ 1U
1jmax
0@
1An ∑
m
j ¼ 1U2j
maxnMTBFj
MTBFjþMTTRj
� � !24
35Zd2 ð11Þ
Recall that “∑mj ¼ 1ðU2j
maxnαjðtÞÞ�d1nAR” integrated in Eq. (10)represents the prioritization allocated to the product P1. The term“d2�UR
st” presented in Eq. (10) corresponds to the usage of theproduction capacity of the m facilities which allow to adjust
production of the product P2 to demand d2 and to avoid over-production. The urgent subcontracting orders considered inEq. (11) are activated when the inventory level of product P2decreases and reaches the threshold SQ ðx2ðtÞ ¼ SQ Þ, where SQ isthe activation threshold of rush orders and SQ rZ2.
To illustrate the effectiveness of the joint production/subcon-tracting control policy and the solution approach proposed in thisstudy, the results will be compared to those of Dror et al. [1] in thecase of two production facilities (see Sections 6.2, 7 and 8).
3. The experimental solution approach
The solution approach adopted in order to solve this problem isbased on that proposed by Gharbi et al. [24], and on a combination ofthe simulation modeling and statistical optimization methods. Theexperimental solution approach consists of the following main steps:
� Step 1: Mathematical definition of the problemAs presented in Section 2.2, the problem is formulated analy-tically in order to understand the dynamics of the productionsystem in terms of its states, as well as to define the expressionof the average total cost. It should be noted that customersatisfaction is considered as a constraint in the optimization ofthe control parameters that minimize the average total cost.
� Step2: Simulation modelThe simulation model uses the proposed general joint produc-tion/subcontracting control policy, as described in Section 2.3, asan input for conducting experiments and evaluating the produc-tion system performance. The model considers two responsevariables: the total cost and the level of customer satisfaction.The simulation model will be presented in Section 4.
� Step 3: Experimental design and response surface methodologyThe general simulation model was applied to the joint production/subcontracting control policy. The experimental design determinesthe main factors (Z1, Z2, U
Rst, Q
Ust(t), and SQ) and their interactions,
which have a significant effect on the output (i.e., the cost and thecustomer satisfaction). Subsequently, the response surface method-
ology will be used in order to define the relationship between eachof the response variables, the main factors and their significantinteractions. The resulting model is then optimized in order todetermine the best combination of parameters of the control policythat minimize the total cost while respecting the required custo-mer satisfaction level.
4. Simulation model
In this work, we use the “Flow Process” module of the ARENAsimulator and subroutines written in the VBA language in order tomodel the dynamics of the production system. The “Flow Process”
u2ðtÞ :
∑m
j ¼ 1ðU2j
maxnαjðtÞÞ if ðx1ðtÞ4Z1Þ and ðx2ðtÞoZ2Þ
max 0; ∑m
j ¼ 1ðU2j
maxnαjðtÞÞ�d1nAR
!if ðx1ðtÞ ¼ Z1Þ and ðx2ðtÞoZ2Þ
min d2�URst; ∑
m
j ¼ 1ðU2j
maxnαjðtÞÞ !
if ðx1ðtÞ4Z1Þ and ðx2ðtÞ ¼ Z2Þ
min d2�URst; max 0; ∑
m
j ¼ 1ðU2j
maxnαjðtÞÞ�d1nAR
! !if ðx1ðtÞ ¼ Z1Þ and ðx2ðtÞ ¼ Z2Þ
0 otherwise
8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:
ð10Þ
M. Assid et al. / Omega 50 (2015) 54–6958
module is based on a combined approach of continuous anddiscrete events of the ARENA simulator. Evidently, the choice ofthe “Flow Process” module is based not only on the large amountof finished products at stake, the continuity of the workflow andthe discrete nature of the dynamics of the manufacturing system,but also on its ability to dramatically shorten the execution timewhen compared to the purely discrete models [33]. Differentoffline separate replications of our simulation number are exe-cuted in order to determine the necessary time for the system toreach its steady state. The value found approximately correspondsto 300,000 days but we used 600,000 days which constitutes acushion. Note that the execution time in this case is equal to 12 s.
The simulation model diagram is presented in Fig. 2. It iscomposed of several networks interacting with each other andthat each one executes specific tasks (production, subcontracting,repair interventions, etc.). After the initialization ① of its variables(demand rates, subcontracting rates, production rates, simulationtime, etc.), the simulation model operates on the basis of a controlpolicy ② ((9)–(11)) and is executed through the ARENA simulationsoftware in order to reproduce the system dynamics and evaluateits performance. The block Sensors ③ continuously monitors theinventories of both finished products and sends signals after eachtriggering to the control policy block② each time a threshold 0, Z1,Z2 and SQ is reached. Then, the control policy block decides,through the opening and closing actions of the valves, what typeand how much of the finished products should be manufactured④ as well as the value of both regular and urgent subcontractingorders ⑤. Note that Z1 and Z2 represent the capacity of tanks andeach one is dedicated to one part type. The block ⑥ updates theinventory level of both products (xi, 8 iAf1;2g) in the tankswhenever a quantity of the finished products is manufactured orsubcontracted (xirZi, 8 iAf1;2g), or when a client request arrives.When a product type is not backlogged, its production increasesits inventory level; otherwise, it merely satisfies the overduedemand, which decreases the stock-out.
At the end of the simulation⑦ all the information needed in orderto calculate the average value of the total cost incurred (6), as well as
the percentage of orders fulfilled by the part types, from the stocks offinished goods without any delay. Customer satisfaction is linked tothe availability of finished products. Thus, for each part typePi; i¼ 1;2, the customer satisfaction is calculated as follows:
SðPiÞ ¼ 1�"�
∑iTNSi
�,TSim
#ð12Þ
where TNSi is the time during which the requests of the product Pi arenot satisfied on time, and Tsim represents the duration of the simula-tion. We note that the constraint of achieving a service level of 99.5% isequivalent to SðPiÞZ99:5% 8 iAf1;2g.
5. Control policy and heuristic approach proposed by Droret al. [1] (HEU)
Using two production facilities, Dror et al. [1] consider that themanufacturing system is complex and apply a heuristic method tosolve the problem based on simplifying assumptions. Their goalconsists in establishing a control policy which helps minimizesafety stocks and storage requirements. This should be done whilerespecting the client satisfaction constraint in order to ensure aspecific level of customer service greater than or equal to 99.5%.The customer satisfaction depends on the availability of finishedproducts. Thus, a customer is considered satisfied if the demand is
Fig. 2. Diagram of the simulation model.
Table 1Demand and production capacity parameters.
Demand (sta/day) Productioncapacity (sta/day)
Portion of the demandmet by subcontracting
URst(t) (st
a/day)d1 d2 Ui1
max Ui2max
2,710 2,540 3,200 1,5005,250 4,700 550 (5,250-4,700)
a st: standard tons.
M. Assid et al. / Omega 50 (2015) 54–69 59
met without delay; otherwise, the customer is not satisfied.Table 1 shows the production capacity of both facilities and thedaily customer demand amount for each part type. Note that sincethe products P1 and P2 have very similar chemical characteristics,the maximum production rates of the products P1 ðU1j
maxÞ and P2ðU2j
maxÞ at the facility Mj; jAf1;2g are identical ðU11max ¼U21
max ¼3200 st=day; U12
max ¼U22max ¼ 1500 st=dayÞ; so AR¼1.
It is worth noting that the control policy proposed by Dror et al.[1] represents a simplification of the general control policyexpressed in Section 2.3. Assuming that the cost of subcontractingis high, compared to that for production, these authors set theregular subcontracting rate at a value of 550 st/day which repre-sents the portion of the demand that cannot be produced in thefacilities due to a lack of capacity (Table 1). Because of break-downs, the demand cannot be satisfied at all. For that reason, Droret al. [1] took into consideration urgent subcontracting, which isactivated upon failure occurrence of one or more facilities. Urgentorders which concern only product P2 depend on the capacity ofthe broken down facility (M1 and/or M2) and the time of the repairactivity. They are intended to replace the production capacitywhich was lost during repair actions and to restore the inventorylevel of buffer stocks at the end of remedial actions. The entireproduction capacity is diverted to the restoration of the inventorylevel of product P1 since the company in question is the onlyproducer of product P1.
After taking into account the data of the problem presented inSection 2 and Table 1, the control policy of the production andsubcontracting was formulated by Eqs. (9), (13)–(15).
For the product P2:
u2ðtÞ ¼
∑2
j ¼ 1ðU2j
maxnαjðtÞÞ if ðx1ðtÞ4Z1Þ and ðx2ðtÞrZ2Þ
∑2
j ¼ 1ðU2j
maxnαjðtÞÞ�d1 if ðx1ðtÞ ¼ Z1Þ and ðx2ðtÞrZ2Þ
0 otherwise
8>>>>>><>>>>>>:
ð13Þ
URst ¼ ðd1þd2Þ�ðU21
maxþU22maxÞ ¼ 550 st=day ð14Þ
And
QUstðtÞ ¼
U21maxst=day if α1ðtÞ ¼ 0 and α2ðtÞ ¼ 1
U22max st=day if α1ðtÞ ¼ 1 and α2ðtÞ ¼ 0
U21maxþU22
max st=day if α1ðtÞ ¼ 0 and α2ðtÞ ¼ 00 st=day if α1ðtÞ ¼ 1 and α2ðtÞ ¼ 1
8>>>>><>>>>>:
ð15Þ
Given the complexity of this problem, Dror et al. [1] adoptedseveral approximations and hypotheses to reduce the difficulty ofmodeling:
� The mean time between failures (MTBF) and the mean time torepair (MTTR) facilities follow the exponential distribution(constant failure rate). The average time between two conse-cutive failures is equal to 6.4 months.
� No failure is allowed during the rebuilding of safety stocks.� The delivery time for urgent subcontracting orders is constant.� The setup time and cost are negligible.� The analysis is limited to the customer satisfaction criterion
and the incurred costs are not considered at all.
The solution obtained by Dror et al. [1] consists in constructing twostorage spaces with a capacity of 38,000 st and 34,000 st for thefinished products P1 and P2, respectively. However, this solution is theresult of a simplified approach, and is based on common sense, andrelies mainly on the 99th percentile of the repair-time distribution tocalculate the inventories of both finished products. These inventoriesare needed to ensure the immediate satisfaction of customer demand
if a facility fails, until the company can receive external supply.Furthermore, Dror et al. [1] roughly take account of possible failuresof both facilities at the same time. They calculated that the probabilitythat both facilities are down at the same time is approximately equalto 0.18% which is equivalent to one day in an 18 month interval.Therefore, an additional amount of finished product inventory equal toone day of operation is roughly estimated in order to compensate forthe situation where two facilities break down simultaneously. Basedon heuristic solution approach, the Dror's solution does not guaranteethe minimization of storage needs.
6. Experimental solution approach based on Dror's controlpolicy (EXP_S)
In this section, the experimental solution approach based onsimulation modeling, design of experiments, analysis of varianceand response surface methodologies proposed in Section 3, willfirst be applied to Dror's restrictive control policy presented inSection 5. Note that the number of the production facilitiesconsidered in the following is limited to two as adopted in [1].The main objective of this approach is to study the actual behaviorof the production system and to deduce the optimal parameters ofthe control policy defined in Section 5 (Z1 and Z2). These controlparameters minimize the total incurred cost (6), while respectingthe constraint of customer satisfaction. Thus, decision makers willbe able to take into account the cost factors and customersatisfaction simultaneously. The use of this approach has allowedthe relaxation of several assumptions made by Dror et al. [1]:
� According to the historical occurrences of failures, times betweenfailures of the two facilities follow the Weibull distribution whoseshape and scale parameters are equal to 1.53 and 6.72, respectively.The mean time to failure equals 6.4 months. The repair timedistribution also follows a Weibull distribution with a shape andscale parameters that are equal to 2.09 and 9.47, respectively. Themean time to repair equals 8.35 days.
� The two manufacturing facilities can fail simultaneously with-out any restriction.
� As in real situations, we consider that failures may occur beforethe end of the total restoration of buffer stocks.
� The delivery time for urgent subcontracting is a randomvariable. We represent it through a normal distribution withan average of 14 days, as in Dror et al. [1] and we consider astandard deviation of 3 days.
� We consider different costs associated with inventory holdingðcþ1 ; cþ
2 Þ, storage space ðceÞ, inventory backlogs ðc�1 ; c�2 Þ, pro-
duction ðcp1; cp2Þ, and regular and urgent subcontracting ðcRst; custÞ.
The experimental approach proposed in Section 3 is tailored toDror's restrictive control policy defined in Section 5 in order todeduce the optimal control parameters (Z1, Z2).
6.1. Validation of the simulation model
We consider the context of the pharmaceutical companystudied by Dror et al. [1] as a base case to validate the proposedsimulation model and to illustrate the concepts developed in thisarticle. In addition to the problem data defined in Section 5 [1], the
Table 2Values of system cost parameters.
Parameters ðcþ1 ; cþ
2 Þ ðc�1 ; c�
2 Þ ðcp1j; cp2jÞ ðcRst; custÞ
Values (1; 1) (80; 80) (30; 30) (60; 100)
M. Assid et al. / Omega 50 (2015) 54–6960
parameter values of the cost function (6) were chosen so thatcþi ≪ c�
i and cp2jocRstocust, 8 i; jAf1;2g2. Table 2 summarizes thevalues of these cost parameters. Recall that Dror et al. [1] ignoredin their work the production, the subcontracting, the inventoryand the backlog cost parameters. 8 j Af1;2g:
To validate the simulation model, a graphical representation ofthe variation in production/subcontracting rates and inventorylevels of finished products was generated (Fig. 3). The systemsimulation was performed for Z1¼12,000 st and Z2¼9000 st. FromFig. 3, the inventory level of both finished products increases totheir critical thresholds (Z1 and Z2) and stabilize at these values ①.This stability is possible if quantities of product P2 are received ona regular basis ðUR
stÞ from external sources ②, in order to address alack of capacity in relation to customer demand. During the failureof one or more Facilities (③: failure of M1), the stock leveldecreases until the end of the repair activities and/or the urgentsubcontracting reception of the quantity of product P2. In fact,when a facility is under repairs, urgent orders are automaticallylaunched in order to obtain additional quantities of product P2 ④.These quantities depend on the capacity of the failed facility (see(15)), and of course, the length of the repair activities. However,when facility M2 fails ⑤ the safety stock of product P2 is used inorder to satisfy the demand, which is why the correspondinginventory level decreases.
On the other hand, the inventory level of product P1 remainsstable. This is explained by the fact that the facility M1, which is
still in use, is able to satisfy the demand d1 along with a smallamount of product P2. Urgent orders are also activated in order torestore the inventory level of product P2 after a random deliverytime ⑥. Because the company is the only manufacturer of productP1, if the inventory level of the product is lower than the criticalthreshold (Z1), any facility in its operational state is used at itsmaximum capacity in order to produce this product ⑦. Suchbehaviors occur until the end of the simulation, and they showthat the simulation model adequately represents the control policydescribed in Section 5.
6.2. Experimental design and response surface methodology
This section presents the methodology adopted to optimize theparameters of the control policy. The objectives are: (1) to deter-mine the relationship between the output variables of the systemresponse (i.e., the cost and the customer satisfaction), the controlfactors (Z1 and Z2) as well as their interactions, which have asignificant effect on the output variables; (2) to calculate theoptimal values of the control policy parameters (control factors)which minimize the total cost considering the customer satisfac-tion constraint for each type of finished products.
Based on the work of Gharbi and Kenné [19], we assume thatthe value function (8) is convex, and we adopt a full factorialdesign with 2 factors at 3 levels each (32) [24]. The full factorial ofsuch a plan is often used for a model that assigns a small number
Fig. 3. Variation in inventories, subcontracting rates and production rates of both finished products (Z1¼12,000 st and Z2¼9000 st).
M. Assid et al. / Omega 50 (2015) 54–69 61
of factors. It gives more accurate results since each interaction isestimated separately. Five replicates were performed for eachcombination of factors, meaning therefore that 45 ð32 � 5Þ simula-tions were performed. The duration of the simulation is equal toTsim ¼ 600;000 days, which is long enough to reach the steadystate. The common random number technique is also used inorder to reduce the variability in the simulation results from oneexperiment to another [35]. From some preliminary simulationexperiments, the levels of the independent variables were estab-lished, such as in Table 3.
A multi-factorial analysis of variance (ANOVA) was performedusing the statistical software, Statgraphics. Fig. 4 shows the effectsof independent variables (Z1 and Z2), their interactions, and theirquadratic effect on the response variables (cost and customersatisfaction for each type of finished product).
Furthermore, the adjusted correlation coefficients ðR2Þ showthat more than 99% of the variability of the expected cost andcustomer satisfaction is explained by the response surface models(16), (17) and (18) [34]. An analysis of the residual normality and ofthe homogeneity of variance was also carried out to check theconformity of the model. The functions of the response surface ofthe dependent variables of the system are given by the followingequations:
CostEXP_S ¼ 141;837�1:07706 Z1�4:09158 Z2þ8:18192 10�5 Z21
þ9:22334n10�5 Z22 ð16Þ
SðP1ÞEXP_S ¼ 96:0052þ4:30682n10�4 Z1�1:19771n10�8 Z21 ð17Þ
SðP2ÞEXP_S ¼ 89:0623þ6:99364n10�4 Z2�1:13255n10�8 Z22 ð18Þ
These three functions are used to compute the near optimalcontrol policy that minimizes the cost (16) while respecting thecustomer satisfaction constraint given by Eqs. (17) and (18). Thenear optimal control policy is defined by Zn
1¼12,389 st andZn
2¼24,544 st, which correspond to an average total cost of270,244$/day and service levels equal to SðP1Þ ¼ SðP2Þ ¼ 99:5%.
In order to compare the above results generated by theproposed experimental approach (EXP_S) to those proposed byDror's heuristic approach (HEU) (Z1¼38,000 st and Z2¼34,000 st),10 replications were performed with the simulation model devel-oped. The proposal of Dror et al. [1] implies an average total cost of311,837 $/day and service levels equal to SðP1Þ ¼ 99:99% andSðP2Þ ¼ 99:83%. It is interesting to note that this proposal of Dror
et al. [1] exceeds the customer service target 99.5% without takinginto account any possible impact on the incurred costs. The resultsindicate that for a 99.5% customer satisfaction (the target level ofservice required), the experimental solution approach proposed inthis article allows a total cost reduction of 13.34% versus thesolution proposed by Dror et al. [1]. However, more analysis isneeded in order to assess the quality of the results. To cross-checkthe validity of the model represented by Eqs. (16)–(18), a Student'st-test is conducted [35]. The results confirm that the cost (270,244$/day) falls well within the 95% confidence interval [270,058;274,642] obtained using 100 replications of the simulation model.
7. Integration of subcontracting as decision variable
This section presents the general joint production/subcontract-ing control policy (EXP_G) presented in Section 2.3, and whichconsiders the security stocks for the two products (Z1, Z2), the ratesof regular and urgent generic medication subcontractingðUR
st; QUstðtÞÞ, as well as the threshold launch of urgent orders (SQ)
as decision variables. In fact, Dror et al. [1] consider that theregular subcontracting rate is set in advance to 550 st/day. How-ever, taking into account the steady state availabilities of produc-tion facilities, the minimum rate of subcontracting required tosatisfy the entire customer demand is 771 st/day, as calculated byEq. (19).
d1þd2� ∑2
j ¼ 1U2j
maxnMTBFj
MTBFjþMTTRj
� �¼ 771 st=day ð19Þ
Choosing a rate of a regular subcontracting which is equal to550 st/day effectively tends to increase the risk of stock-out. Incase of failure, the only source for restoring the safety stock levelby urgent subcontracting depends on relatively long and randomdelivery times. In addition, urgent orders are much more expen-sive than regular ones ðcRstocustÞ. On the other hand, the systemwhich deals with random perturbations of its production processis forced to protect itself by increasing the value of inventorycapacities (Z1 and Z2). This attitude generates high inventories andstorage space costs.
In order to take into account all aspects influencing theperformance of the production system, the general joint produc-tion/subcontracting control policy (EXP_G) considers the maindecision variables, Z1, Z2, U
Rst, Q
UstðtÞ and β. The new variable β is
considered in order to ensure that SQrZ2. So, SQ¼β * Z2 and0rβr1. Because of the high number of control parametersincluded in the model, we proceed in two steps. First, we use ascreening design to evaluate the importance of each of theindependent variables [34]. This step does not determine theoptimal control parameters, but measures the importance ofthese parameters and retains those which have a significant effecton the total incurred cost. Secondly, we determine the responsesurface functions of the dependent output variables of the system
Table 3Levels of the independent variables.
Factor Lower level Center Upper level Descriptions
Z1 2,000 10,000 18,000 Stock level of the product P1Z2 17,000 23,500 30,000 Stock level of the product P2
Fig. 4. Pareto Chart for the dependent variables of the system.
M. Assid et al. / Omega 50 (2015) 54–6962
based on the significant factors, after which we determine theoptimal solution. The adopted screening design is a full factorialdesign with 5 factors at 2 levels each (25) and with 3 center points.Three replications are conducted for each combination of factors,and so a total of 105 simulations are performed. An analysis ofvariance (ANOVA) is then performed based on the results of thesimulation using the statistical software, Statgraphics. Theadjusted correlation coefficient (R2) shows that the simulationmodel explains more than 93% of the variability of the expectedtotal cost [34]. The analysis of variance also shows that the twofactors (QU
st and β), their double interactions, and their quadraticeffects are not significant at a confidence level of 95% (see Fig. 5).This is normal since urgent subcontracting depends on a non-negligible random delivery time (see Section 2.2) and since theurgent subcontracting cost is relatively higher than the regular oneðcRstocustÞ. The response surface functions obtained using Stat-graphics are given as follows:
Cost EXP_G ¼ 639683:53�1:13 Z1�11:07 Z2�312:09 URst
þ6:55:10�2 UR2st �3:20:10�3 UR
st Z1þ5:50:10�3 URst Z2
þ1:04:10�4 Z21�2:33:10�3 Z1 Z2þ2:91:10�10 Z22 ð20Þ
SðP1ÞEXP_G ¼ 96:01þ6:09:10�5 URstþ4:07:10�4 Z1þ7:28:10�6 Z2
þ6:38:10�2 βþ8:87:10�5 QUst�1:75:10�8 UR2
st
�1:10:10�8 Z21�3:17:10�10 Z22�1:06:10�1β2
�4:67:10�8QU2st ð21Þ
SðP2ÞEXP_G ¼ 52:33þ1:68:10�2 URstþ7:65:10�4 Z1þ2:73:10�3 Z2
þ6:68 βþ9:45:10�3 QUst�1:15:10�6 UR2
st
�5:94:10�7 URst Z2�3:32:10�8 Z21�5:56:10�8 Z2
2
�11:43 β2�4:97:10�6QU2st ð22Þ
To further explore the contribution of subcontracting, two newjoint production/subcontracting control policies will be consid-ered, and thereby the subcontracting will be either urgent orregular. Therefore, in addition to Z1 and Z2, one or more decisionvariables which will represent the subcontracting activity(UR
st, QUst(t) and SQ) are also taken into account:
7.1. Joint production/urgent subcontracting control policy(EXP_Urg)
This policy uses only the urgent subcontracting to remedy thelack of production capacity of both facilities. When the level ofproduct inventory P2 decreases and then reaches the threshold SQ(x2¼SQ), a subcontracting order is issued in order to be delivered
after a random transportation delay (TD). The Eq. (11) becomesas follows:
UstðtÞ : QUstðtþTDÞ if x2 ¼ SQ
0 otherwise
(
With,
QUstðtÞZ0; QU
stðtÞþ 1� d1
∑mj ¼ 1U
1jmax
0@
1A
24
n ∑2
j ¼ 1U2j
maxnMTBFj
MTBFjþMTTRj
� � !#Zd2; 8 jA J
The order quantity ðQUstÞ is a decision variable to be optimized. It
is interesting to note that the non consideration of the regularsubcontracting implies that the inventory level (x2) cannot bestabilized at its threshold Zn
2 (see Fig. 3①) since the productioncapacity dedicated to produce P2 is less than the demand rate of P2.Therefore, the parameter Z2 is omitted from the optimizationmodel and the storage space cost is calculated according to thehigher value recorded by the x2.
In this section, we adopt the same approach considered inSection 6, except that the objective is to determine the optimalvalue of three control parameters of the EXP_Urg (Z1, β and QU
st(t))which minimize the incurred total cost (6) according to therequired level of service (12). Recall that β is considered in orderto ensure that SQrZ2, with 0rβr1 and SQ¼β� Z2. Consideringthe same base case numerical data (see Section 6), a full 33
experimental design is also used, and five replications performedfor each combination of factors. Therefore, a total of 135 ð33 � 5Þsimulations are performed. According to the resulting adjustedcorrelation coefficients (R2), over 95% of the system variability isexplained by the model (R2
Cost ¼ 95:38%, R2SðP1Þ ¼ 99:97% and
R2S ðP2Þ ¼ 99:56%). The response surface functions are given as
follows:
Cost EXP_Urg ¼ 463279�1:07 Z1�1:06 β�5:86 QUst
þ8:82:10�5 Z21�6:11:10�10Z1QUst
þ2:98:10�5β2þ3:26:10�6β QUst�4:94:10�5QU2
st ð23Þ
SðP1ÞEXP_Urg ¼ 96:22þ4:14:10�4 Z1�1:47:10�6 QUst�1:19:10�8 Z2
1
ð24Þ
SðP2ÞEXP_Urg ¼ 87:24þ3:56:10�4 βþ2:24:10�4 QUst
�4:24:10�9β2�1:40:10�9β QUst�1:43:10�9QU2
st ð25Þ
7.2. Joint production/regular subcontracting control policy(EXP_Reg)
This policy does not use the urgent subcontracting. It merelyconsiders the two manufacturing facilities and the regular sub-contracting in order to ensure the satisfaction of the demand. As aresult, the urgent subcontracting quantity to be ordered (QU
st(t))and the threshold for launching urgent orders ðSQ Þ are no longerconsidered in the joint production/regular subcontracting controlpolicy (EXP_Reg). Note that Eq. (19) becomes UR
stZ771 st=day andEq. (11) becomes as follows: UstðtÞ ¼ UR
st.In order to determine the optimal value of three control
parameters of the EXP_Reg (Z1, Z2 and URst), we adopt the same
approach and the same tools considered in Section 6.2. With morethan 96% of the system variability which is explained by the model(R2
Cost ¼ 96:44%, R2SðP1Þ ¼ 99:96% and R2
S ðP2Þ ¼ 99:61%). The response
Fig. 5. Pareto Chart for the total cost parameter of the EXP_G.
M. Assid et al. / Omega 50 (2015) 54–69 63
surface functions of the EXP_Reg are given as follows:
Cost EXP_Reg ¼ 474072�179:73 URst�1:13 Z1�10:79 Z2
þ3:69: 10�2 UR2st �3:20: 10�10 UR
st Z1
þ4:27: 10�3 URst Z2þ1:04:10�4 Z21
�2:33: 10�1 Z1 Z2þ1:40:10�4 Z22 ð26Þ
SðP1ÞEXP_Reg ¼ 96:12þ4:24n10�4 Z1�1:22:10�8 Z21 ð27Þ
SðP2ÞEXP_Reg ¼ 61:74þ2:29:10�2 URstþ2:03: 10�3 Z2
�3:45:10�6 UR2st �5:57: 10�7 UR
st Z2 �3:47:10�8 Z22ð28Þ
7.3. Comparison of the three control policies considering thesubcontracting as a decision variable
The response surface functions (20)–(28) of the three controlpolicies EXP_Urg, EXP_Reg and EXP_G, are used in order todetermine the optimal parameters which minimize the total cost(6) incurred while respecting the customer satisfaction constraint.Table 4 presents the optimization results, provided that thecustomer service level is greater than or equal to 99.5%.
To study the advantage in terms of costs of one policy overanother, a Student's t-test is conducted using the cost differencebetween the three considered control policies [35]. Table 5 pre-sents the 95% confidence intervals of the cost differencesCn
EXP_Urg�Cn
EXP_Reg, Cn
EXP_Urg � Cn
EXP_G and Cn
EXP_Reg�Cn
EXPG ; obtainedusing 20 replications of the simulation model. It shows that thelower and the upper bounds of the confidence intervals of the costdifferences Cn
EXP_Urg�Cn
EXP_Reg as well as Cn
EXP_Urg � Cn
EXP_G are
positive. These results can be interpreted as EXP_Reg and EXP_Gbeing better than EXP_Urg, in the sense that the latter generateshigher total cost than the other control policies. Table 5 also showsthat the confidence interval for Cn
EXP_Reg�Cn
EXP_G contains zero. Thismeans that there is no statistical evidence that one control policyis better than the other in terms of cost [35].
Further analyses were conducted in order to study the influ-ence of the urgent subcontracting cost ðcustÞ on the total cost of theconsidered joint control policies. In this sense, Fig. 6.a shows thatthe EXP_G becomes more advantageous in term of costs than theEXP_Reg when cust is less than 76 (both upper and lower bounds ofthe confidence intervals of the cost differences Cn
EXP_Reg � Cn
EXP_Gare positive). This value represents only 26.67% more than of thevalue of the regular subcontracting cost ðcRstÞ; this is unlikely, sincecust should be much more expensive. In the same context, the valueof the average total cost of EXP_Urg falls below that of EXP_Regonly when cust is less than 54 (see Fig. 6.b), which is less than thevalue of the regular subcontracting cost ðcRst ¼ 60Þ. These situationsare not possible in several industries such as the pharmaceuticalone. The contract agreement with regular subcontractor allowshim to deliver goods that cost less than those obtained withurgent orders from different subcontractors. Indeed, subcontrac-tors often aim to maximize their profit from occasional orders. Inaddition, regular subcontracting can be very useful to reduce theadverse effects of random transportation delays, to ensure anefficient control, an ease of management as well as an effectivetraceability of the subcontracted products and to demand thehighest standards of quality during contract negotiations thatshould cover all aspects related to the management of productsand services. For these reasons we chose the EXP_Reg as the bestjoint production / subcontracting control policy for the consideredmanufacturing system. This control policy will be studied in depthand compared to the solution recommended by Dror et al. [1] andto the control policy presented in Section 6 (EXP_S).
Table 4Optimization results for a service level higher than or equal to 99.5%.
Policy Zn
1 (st) Zn
2 (st) URnst (st/day) SQ (st) QUn
st (st) Cost ($/day)
EXP_Urg 12,598 – – 21,894 59,602 285,741EXP_Reg 12,365 10,922 1,924 – – 255,604EXP_G 12,463 20,425 1,604 5,914 978 253,449
Table 5Cost difference confidence interval (95%).
Policy Cn
EXP_Urg�Cn
EXP_Reg Cn
EXP_Urg�Cn
EXP_G Cn
EXP_Reg�Cn
EXP_G
Lower bound 29,772 29,580 �197Upper bound 29,991 29,982 99
Fig. 6. Comparisons between EXP_G and EXP_Reg (a) and between EXP_Urg and EXP_Reg (b) according to cust.
Table 6Solutions of the three models selected for a service level higher than or equalto 99.5%.
Solvingapproach
Policy Zn
1 (st) Zn
2 (st) URn
st (st/day) Cost($/day)
Heuristic (Drorpolicy)
HEU 38,000 34,000 550 311,837
Experimental(Dror policy)
EXP_S 12,389 24,544 550 270,244
Experimental(Proposedpolicy)
EXP_Reg 12,365 10,922 1,924 255,604
M. Assid et al. / Omega 50 (2015) 54–6964
8. Analysis of the joint production/regular subcontractingcontrol policy
Table 6 presents the solution proposed by Dror et al. [1] (HEU)as well as results of the optimization of both control policies:EXP_S and EXP_Reg, provided that the customer service level isgreater than or equal to 99.5%.
The results presented in Table 6 show that the joint production/regular subcontracting control policy (EXP_Reg) provides a sig-nificant economic gain when combined with the experimentalapproach. This gain reaches a value of 18% when compared to thesolution proposed by Dror et al. [1].
8.1. Effect of the customer service level on the control parameters
By using Eq. (26) with constraints (27) and (28), we havedetermined the behavior of the optimal parameters of the cont-rol policy as well as the associated cost, according to changesin customer satisfaction (Figs. 7–10). Fig. 7 shows variations inthe control parameter values based on the customer satisfac-tion level.
Fig. 7 shows that the parameter Zn
1 increases according tocustomer satisfaction in order to reduce stock-outs of the productP1. Unlike product P1, the required growth in customer servicelevel of P2 implies an increase of the value of the regular
subcontracting rate ðURnst Þ rather than that of Zn
2. Indeed, theincrease of URn
st in this case becomes more economical than thatof Zn
2, which generates very high costs especially those associatedwith storage. We should recall that the storage costs penalize thespace in the form of an average cost per unit of time for eachinterval depending on the value of Z2 (Section 2.2). The additionalquantities of the regular subcontracting are also used to reduce theimpact of the system's failure as well as to accelerate the restora-tion of the inventory of the product P2. This choice, which consistsin increasing URn
st implies a significant reduction of the value of Z2for the total cost incurred (Fig. 7). We note that the range in whichthe control parameters are constant corresponds to the situationswhere the values of Zn
1, Zn
2 and URnst generate a higher level of
service when compared to that required by the customer satisfac-tion constraint. Indeed, when the customer satisfaction constraintis deleted, the calculated values of Zn
1 and Zn
2, which minimize thetotal incurred cost, generate a service level of 98.07% and 98.99%for the two product types P1 and P2, respectively.
From Fig. 8, we note that the average total cost growsexponentially in terms of customer satisfaction without everexceeding that proposed by Dror et al. [1]. This growth isexplained by the increase of the control parameters Zn
1 and URnst ,
and consequently, by the growth of the inventory costs andstorage space for the product P1 and of the cost of regularsubcontracting for the product P2 (Fig. 7).
Fig. 8. Variation of the average optimal total cost in terms of customer satisfaction (EXP_Reg).
Fig. 7. Variation of control parameters of the EXP_Reg based on customer satisfaction.
M. Assid et al. / Omega 50 (2015) 54–69 65
To confirm the robustness of the solution generated by the model,we proceed with the same analysis, but adopting the followingconstraint: UR
str1745 (Fig. 7). This condition causes the system toincrease the value of Zn
2 in order to achieve a higher service levelðSðP1Þ&SðP2Þ498%). In fact, UR
st must remain less than or equal to1745 (optimal value obtained for SðP1Þ ¼ SðP2Þ ¼ 98%). This specificpolicy is denoted EXP_Reg2, and Fig. 9 summarizes the results.
Fig. 10 shows the difference between the total costs of the twopolicies EXP_Reg and EXP_Reg2, depending on the client satisfac-tion level. It demonstrates the advantage of increasing the value ofthe regular subcontracting rate ðUR
stÞ rather than that of Zn
2 in orderto achieve a higher service level. This difference in total cost isexpressed by: D:C:ðUR
stÞ ¼ C:T :EXP_Reg2 � C:T :EXP_Reg .
8.2. Effect of shortage cost ðc�2 Þ on control parameters
From Table 6, we note that the joint production/regular sub-contracting control policy (EXP_Reg) allowed the value of UR
st toincrease and the critical threshold Zn
2 of the product P2 to dropsignificantly. The threshold Z1 of the innovative product P1 hasdecreased slightly since the subcontracting concerns only thegeneric product P2. Given the value of the minimum subcontract-ing amount ðUR
st Z771 st=dayÞ, we note that the optimal value ofthe subcontracting rate is more than twice as high as its minimalrate, which affects the global utilization rate of the two facilities(Table 7). In fact, 25.78% of the facilities’ capacity is not in use, withthis value not including maintenance idle times. This unexpected
phenomenon is due to the reaction of the model, which mustsimultaneously satisfy the constraint of the feasibility of thesystem (5) on the one hand, and protect itself against thefluctuations of random events (failure occurrences, repair actiondurations, etc.), on the other. In addition, and due to high productshortage costs and the imposed customer service level (99.5%), themodel seeks to avoid stock-outs. Consequently, it minimizes theimpact of the stock-out on both the cost and the customersatisfaction by increasing the value of UR
st which reduces the useof the manufacturing facilities.
Figs. 11 and 12 represent the behavior of the system when varyingthe shortage cost of the product P2 and the customer satisfaction,respectively. They show that the growth of c�
2 for a customersatisfaction of 99.5%, does not affect the value of Zn
1 but increasesthe storage capacity of the product P2 (Z2 increases) so as to avoid theadditional shortage costs (Fig. 11). This behavior, observed by Gharbiet al. [21,24] and Lavoie et al. [28], reduces the risk of stock-outs andincreases the customer satisfaction level. As a result, the systemdecides to decrease the value of URn
st (Fig. 12) in order to reduce thecost of regular subcontracting while maintaining a minimum servicelevel of 99.5%. In addition, Fig. 12 shows that the decrease of URn
stincreases the percentage of capacity utilization of the two facilities.Indeed, URn
st is not only used to accelerate the restoration of theinventory level of the product P2, but also to satisfy an important partof customer demand d2. Likewise, when the value of URn
st decreases, thecapacity used to produce P2 increases in order to meet the overalldemand d2.
Fig. 10. Difference in the total cost between the policies EXP_Reg and EXP_Reg2, depending on customer satisfaction.
Fig. 9. Variation of the optimal parameters of the model EXP_Reg2 based on customer satisfaction when URstðtÞr1745.
M. Assid et al. / Omega 50 (2015) 54–6966
8.3. Sensitivity analysis
A sensitivity analysis based on a series of numerical exampleswas carried out in order to confirm the robustness of the solutionapproach and the effectiveness of the proposed control policy.Indeed, we studied the effect of different combinations of costparameters on the control parameters of the model and on thetotal cost incurred. Thus, the improvements are confirmed indifferent contexts (Table 7). These configurations are also evalu-ated with respect to the basic model in order to study the behaviorof the system.
8.3.1. Variation of cþ1 (cases 1 and 2)
The variation of cþ1 has no influence on the optimal value of the
control parameter Zn
1. In fact, the subcontracting takes only intoconsideration the product P2.
8.3.2. Variation of cþ2 (cases 3 and 4)
When cþ2 increases, the storage space capacity dedicated to the
product P2 ðZn
2Þ decreases in order to avoid additional inventorycosts. This reduction implies a greater risk of stock-out and a lowerlevel of customer satisfaction, hence the increase of the value ofURn
st . The growth of the total cost incurred was primarily due to thegrowth of the inventory costs as well as the storage and thesubcontracting costs of the product P2. The opposite occurs whencþ2 decreases.
8.3.3. Variation of ce (cases 5 and 6)This variation affects the control policy in the same way as the
variation of cþ2 .
8.3.4. Variation of c�1 (cases 7 and 8)
This variation does not affect the value of the control para-meters of the system. This is because the subcontracting only takesinto consideration the product P2.
Fig. 11. Variation of the optimal value of the thresholds according to c�2 for a 99.5%
service level.
Fig. 12. Variation of the regular subcontracting rate and the percentage of capacityutilization (C.U.) of facilities based on c�2 for a 99.5% service level.
Table 7Sensitivity analysis of model EXP_Reg for a 99.5% service level and improvement cost comparison.
Cases cþi ce c�
i cpi cRst cust Capacityutilizationof the 2facilities(%)
URn
st (st/day) Zn
1 (st) Zn
2 (st) Total cost ($/day) Cost improvement(%)
Remark
P1 P2 P1 P2 P1 P2 EXP_Reg(a)
EXP_S(b)
HEU (c) (c) - (b) (c) - (a)
Base 1 1 ceba 80 80 30 30 60 100 74.22 1,924 12,365 10,922 255,604.4 270,243.7 311,836.8 13.34 18.03 Basic case
1 0.8 1 ceba 80 80 30 30 60 100 74.22 1,924 12,365 10,923 253,178.4 267,793.5 304,288.7 11.99 16.80 Zn
12;Cn↓2 1.2 1 ceb
a 80 80 30 30 60 100 74.22 1,924 12,365 10,923 258,030.1 272,694.0 319,384.9 14.62 19.21 Zn
12;Cn↑3 1 0.8 ceb
a 80 80 30 30 60 100 75.77 1,855 12,365 12,369 253,338.3 265,596.4 305,634.1 13.10 17.11 Urnst ↓; Z
n
2↑;Cn↓
4 1 1.2 ceba 80 80 30 30 60 100 71.75 2,037 12,365 9,072 257,547.8 274,891.2 318,039.5 13.57 19.02 Urn
st ↑; Zn
2↓;Cn↑
5 1 1 0.8 ceb 80 80 30 30 60 100 75.70 1,858 12,365 12,295 250,532.6 262,827.0 303,236.8 13.33 17.38 Urnst ↓; Z
n
2↑;Cn↓
6 1 1 1.2 ceb 80 80 30 30 60 100 71.90 2,030 12,365 9,175 260,389.4 277,659.6 320,436.8 13.35 18.74 Urnst ↑; Z
n
2↓;Cn↑
7 1 1 ceba 60 80 30 30 60 100 74.22 1,924 12,365 10,918 255,278.5 269,912.9 311,836.5 13.44 18.14 Zn
12;Cn↓8 1 1 ceb
a 100 80 30 30 60 100 74.22 1,924 12,365 10,918 255,929.9 270,574.4 311,837.0 13.23 17.93 Zn
12;Cn↑9 1 1 ceb
a 80 60 30 30 60 100 73.34 2,010 12,365 9,927 255,397.1 270,063.8 311,664.5 13.35 18.05 Urnst ↑; Z
n
2↓;Cn↓
10 1 1 ceba 80 100 30 30 60 100 74.92 1,862 12,365 11,621 255,807.5 270,423.4 312,009.0 13.33 18.01 Urn
st ↓; Zn
2↑;Cn↑
11 1 1 ceba 80 80 25 30 60 100 74.22 1,924 12,365 10,918 242,073.7 256,709.6 298,305.4 13.94 18.85 Zn
12;Cn↓12 1 1 ceb
a 80 80 35 30 60 100 74.22 1,924 12,365 10,918 269,133.6 283,777.6 325,368.2 12.78 17.28 Zn
12;Cn↑13 1 1 ceb
a 80 80 30 25 60 100 75.82 1,853 12,365 12,421 252,310.9 260,826.8 302,418.7 13.75 16.57 Urnst ↓; Z
n
2↑;Cn↓
14 1 1 ceba 80 80 30 35 60 100 69.10 2,159 12,365 7,399 258,166.8 279,659.9 321,254.9 12.95 19.64 Urn
st ↑; Zn
2↓;Cn↑
15 1 1 ceba 80 80 30 30 55 100 69.02 2,161 12,365 7,370 245,461.0 267,493.9 309,086.8 13.46 20.59 Urn
st ↑; Zn
2↓;Cn↓
16 1 1 ceba 80 80 30 30 65 100 75.88 1,843 12,365 12,435 265,009.5 272,993.7 314,586.8 13.22 15.76 Urn
st ↓; Zn
2↑;Cn↑
Where EXP_Reg: joint production/regular subcontracting control policy, EXP_S: Experimental solution approach based on Dror's control policy, HEU: Heuristic solutionapproach proposed by Dror et al. [1].
a ceb: the storage space cost for the base case. It is represented by a step cost depending on intervals of Zi values. For example, if ZiA ½0;999� then ceb ¼ 1400, else ifZiA ½1000;1999� then ceb ¼ 1750, etc.
M. Assid et al. / Omega 50 (2015) 54–69 67
8.3.5. Variation of c�2 (cases 9 and 10)
The variation of c�2 has an inverse effect on the control
parameters, compared with the variation of the inventory costðcþ2 Þ. Indeed, when c�
2 increases, Zn
2 increases as well in order toavoid additional shortage costs. This increase implies a growth inthe value of the customer satisfaction level. Accordingly, thesystem decreases the value of URn
st in order to reduce the cost ofregular subcontracting while maintaining a minimum service levelof 99.5%. The increase in the total cost incurred is mainly due tothe shortage cost. The opposite occurs when c�
2 decreases.
8.3.6. Variation of cp1 (cases 11 and 12)This variation has no influence on the optimal value of the
control parameters ðZn
1; Zn
2 et URstÞ. This is explained by the priority
given to the product P1 which is manufactured solely by thecompany. In order to satisfy the customer demand d1, this type ofproduct is manufactured internally, whatever its production costs.
8.3.7. Variation of cp2 (cases 13 and 14)The variation in the production cost cp2 affects the rate of the
regular subcontracting ðURnst Þ and the threshold value ðZn
2Þ ofthe product (P2). This phenomenon is justified by the fact thatthe model EXP_Reg has allowed a percentage of the productioncapacity of the product P2 to be replaced by subcontracting. Thus,the increase in cp2 increases the value of URn
st in order to reduce themanufactured quantity of product; which accounts for thedecrease of Zn
2. The opposite occurs when reducing cp2.
8.3.8. Variation of cRst (cases 15 and 16)When cRst increases, U
Rnst decreases in order to avoid additional
regular subcontracting costs. This reduction of the URnst has the
effect of increasing the risk of stock-out. Therefore, the systemincreases the value of Zn
2 to reduce the shortage cost and to keep aservice level of 99.5%. The increase of the total cost is mainly dueto the growth in the cost of regular subcontracting. The oppositeoccurs when cRst decreases.
The data in Table 7 shows that the increase (decrease) ofregular subcontracting costs increases (reduces) the utilizationcapacity percentage of the two facilities. As already mentioned inSection 8.1, this is due to the fact that regular subcontracting isused to satisfy the customer demand d2 and to restore theinventory level of the product P2. It should be noted that onlythe production rate of the product P2 affects the variation in thepercentage of utilization capacity of the two facilities.
The results of the sensitivity analysis show that the approachadopted in this work is robust, and can significantly reduce thetotal cost of the studied production system. In fact, in comparisonwith the solution of Dror et al. [1], the joint production/regularsubcontracting control policy (EXP_Reg) developed in this article ismore advantageous in terms of the total incurred cost. Indeed, thegain achieved by combining the experimental solution approachwith EXP_Reg varies between 15.76% and 20.59% (Table 7 –
Improvement cost).
9. Conclusions
In this article, we studied a complex production and subcon-tracting planning optimization problem of a manufacturing sys-tem, which is a common problem in the pharmaceutical industry.The system considered consists of multiple facilities of differentcapacities, prone to random breakdowns and repairs. They pro-duce two separate part types: a brand name product protected bya patent and another generic product that may be obtained fromexternal sources to completely satisfy customer demand. A controlpolicy is proposed which simultaneously combines the production
and the subcontracting planning of the production system.Because of the complexity of the system considered and thelimitations of analytical solution approaches, we adopted anexperimental approach by integrating the simulation with statisticoptimization techniques. This solution approach consists inexperimentally determining the optimal value of the controlparameters that minimize the total cost incurred by taking intoaccount the customer satisfaction constraint. Thus simultaneouslyconsidering the cost and the customer satisfaction better approx-imates the concerns of the company's decision makers. The com-bination of the experimental approach with the joint production/regular subcontracting control policy achieved improvements interms of cost, up to 20% compared to an existing model in theliterature [1]. We have also demonstrated that for economicreasons, the system chooses to increase the value of the regularsubcontracting rate instead of internal production in order toachieve a greater service level. Other control policies using urgentsubcontracting parameters as decision variables are also inte-grated into the comparison. According to our policy, the pharma-ceutical company has no interest in considering urgentsubcontracting for the real case studied by Dror et al. [1]. Conse-quently, the joint production/regular subcontracting control policyproposed in this article outperforms Dror's proposal not only interm of total incurred cost, but also it is more easy to manage andvery useful to ensure an efficient control and an effective trace-ability of the subcontracted products. Sensitivity analyses wereperformed and some interesting behaviors were observed, con-firming the robustness of the solution approach used and theeffectiveness of the proposed control policy.
Further extensions of the proposed control policy could beconsidered. In fact, quality-review system can be adopted for thecase where the supplier manufactures items with defective ratio(see [36]). In addition, it should be noted that in this work, weignored the setup time and setup cost parameters because bothproducts have the same basic chemical characteristics since thebrand name and the generic medications are composed of thesame main chemical substances. However, in some situations,finished products are more distinct, and switching from oneproduct type to another requires significant setup time and cost(i.e., non-flexible machines). In this case, the control policy couldbe based on a single hedging point (SHP) structure (one hedgingpoint per product) as in [20] or on a multiple hedging point (MHP)structure as in [21] (two hedging points per product) and [37](three hedging points per product). The latter [37] shows that theMHP policy is more effective in terms of the costs than the SHPpolicy for the one-machine two-products case. However, using thecost and the customer satisfaction level as two key performanceindicators, the SHP may be more effective. Theses control policystructures will be considered in our future research work to checktheir effectiveness for the manufacturing system presented in thispaper (multiple-machine, two-products, subcontracting, etc.). Pre-ventive maintenance strategies could also be considered as in[13,38] in order to propose an effective joint subcontracting/production/maintenance control policy. The goal is to reduce thenumber of major failures and the risk of shortage.
References
[1] Dror M, Smith KR, Yano CA. Deux Chemicals Inc. Goes Just-in-Time. Interfaces2009;39(6):503–15.
[2] Whittle P. Optimal Control: Basics and Beyond. New York: John Wiley & Sons;1996 (474 p.).
[3] Meijboom B, Obel B. Tactical coordination in a multi-location and multi-stageoperations structure:Amodel and a pharmaceutical company case. OMEGA:The International Journal of Management Science 2007;35:258–73.
[4] eHow. Types of Pharmaceutical Companies. ⟨www.ehow.com/list_6873820_types-pharmaceutical-companies.html⟩; 1999 [accessed 02.06.12].
M. Assid et al. / Omega 50 (2015) 54–6968
[5] Kaplan WA, Laing R. Local production of pharmaceuticals: industrial policyand access to medicines. An overview of key concepts, issues and opportu-nities for future research. HNP Discussion Papers. World Bank, WashingtonDC; 2005.
[6] Booth R. The role of supply-chain re-engineering in the pharmaceuticalindustry. Logistics Information Management 1996;9(3):4–10.
[7] Olsder GJ, Suri R. Time-optimal control of parts-routing in a manufacturingsystem with failure-prone machines. In: Proceedings of the 19th conferenceon decision and control. Albuquerque, NM; 1980.
[8] Richel R. Dynamic programming and minimum principles for systems withjump Markov disturbances. SIAM Journal on Control 1975;13:338.
[9] Kimemia J, Gershwin SB. An algorithm for the computer control of a flexiblemanufacturing system. IIE Transactions 1983;15(4):353–62.
[10] Akella R, Kumar PR. Optimal Control of Production Rate in a Failure ProneManufacturing System. IEEE Transactions on Automatic Control 1986;31(2):116–26.
[11] Sharifnia A. Production control of a manufacturing system with multiplemachine states. IEEE Transactions on Automatic Control 1988;AC-33(7):600–26.
[12] Feng Y, Yan H. Optimal production control in a discrete manufacturing systemwith unreliable machines and random demands. IEEE Transactions on Auto-matic Control 2000;45(11):2280–96.
[13] Berthaut F, Gharbi A, Dhouib K. Joint modified block replacement andproduction/inventory control policy for a failure-prone manufacturing cell.OMEGA: The International Journal of Management Science 2011;39(6):642–54.
[14] Dhouib K, Gharbi A, Ben Aziza MN. Joint optimal production control/pre-ventive maintenance policy for imperfect process manufacturing cell. Inter-national Journal of Production Economics 2012;137(1):126–36.
[15] Chakraborty T, Giri BC, Chaudhuri KS. Production lot sizing with processdeterioration and machine breakdown under inspection schedule. OMEGA:The International Journal of Management Science 2009;37:257–71.
[16] Jeang A. Simultaneous determination of production lot size and processparameters under process deterioration and process breakdown. OMEGA:The International Journal of Management Science 2012;40:774–81.
[17] Caramanis M, Sharifnia A. Near optimal manufacturing flow controller design.The International Journal of Flexible Manufacturing Systems 1991;3:321–36.
[18] Sethi SP, Zhang Q. Average-cost optimal policies for an unreliable flexiblemultiproduct machine. The International Journal of Flexible ManufacturingSystems 1999;11:147–57.
[19] Gharbi A, Kenné JP. Optimal production control problem in stochasticmultiple-product multiple-machine manufacturing systems. IIE Transactions2003;31:941–52.
[20] Bai SX, Elhafsi M. Scheduling of an unreliable manufacturing systemwith non-resumable set-ups. Computers and Industrial Engineering 1997;32:909–25.
[21] Gharbi A, Kenné JP, Hajji A. Operational level-based policies in production ratecontrol of unreliable manufacturing systems with set-ups. InternationalJournal of Production Research 2006;44(3):545–67.
[22] Dellagi S, Rezg N, Xie X. Preventive maintenance of manufacturing systemsunder environmental constraints. International Journal of ProductionResearch 2007;45(5):1233–54.
[23] Ayed S, Dellagi S, Rezg N. Joint optimisation of maintenance and productionpolicies considering random demand and variable production rate. Interna-tional Journal of Production Research 2012;50(23):6870–85.
[24] Gharbi A, Hajji A, Dhouib K. Production rate control of an unreliablemanufacturing cell with adjustable capacity. International Journal of Produc-tion Research 2011;49(21):6539–57.
[25] Ponsignon T, Mönch L. Simulation-based performance assessment of masterplanning approaches in semiconductor manufacturing. OMEGA: The Interna-tional Journal of Management Science 2014;46:21–35.
[26] Dye CY. The effect of preservation technology investment on a non-instantaneous deteriorating inventory model. OMEGA: The InternationalJournal of Management Science 2013;41:872–80.
[27] Wee HM, Widyadana GA. A production model for deteriorating itemswith stochastic preventive maintenance time and rework process withFIFO rule. OMEGA: The International Journal of Management Science2013;41:941–54.
[28] Lavoie P, Gharbi A, Kenné JP. A comparative study of pull control mechanismsfor unreliable homogenous transfer lines. International Journal of ProductionEconomics 2010;124:241–51.
[29] De Garmo EP, Sullivan WG, Bontadelli JA. Engineering Economy. 8th ed.. NewYork: MacMillan; 1989.
[30] Kenné JP, Gharbi A. Production planning problem in manufacturing systemswith general failure and repair time distributions. Production Planning &Control 2000;11(6):581–8.
[31] Wang Z, Chan FTS. Independence of the hedging points in PHP control policy.Journal of the Operational Research Society 2013. http://dx.doi.org/10.1057/jors.2013.10.
[32] Mourani I, Hennequin S, Xie X. Simulation-based optimization of a single-stage failure-prone manufacturing system with transportation delay. Interna-tional Journal of Production Economics 2008;112(1):26–36.
[33] Bouslah B, Gharbi A, Pellerin R. Joint optimal lot sizing and production controlpolicy in an unreliable and imperfect manufacturing system. InternationalJournal of Production Economics 2013;144(1):143–56.
[34] Montgomery DC. Design and Analysis of Experiments. 7th ed.. New York: JohnWiley & Sons; 2008.
[35] Banks J, Carson JS, Nelson BL, Nicol DM. Discrete-Event System Simulation.Fourth Edition. Upper Saddle River, NJ: Prentice-Hall; 2005.
[36] Singer M, Donoso P, Traverso P. Quality strategies in supply chain alliances ofdisposable items. OMEGA: The International Journal of Management Science2003;31(6):499–509.
[37] Assid M, Gharbi A, Hajji A. Joint production and setup control policies: anextensive study addressing implementation issues via quantitative andqualitative criteria. The International Journal of Advanced ManufacturingTechnology 2014;72(5–8):809–26.
[38] Wee HM, Widyadana GA. A production model for deteriorating itemswith stochastic preventive maintenance time and rework process withFIFO rule. OMEGA: The International Journal of Management Science2013;41:941–54.
M. Assid et al. / Omega 50 (2015) 54–69 69