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Incorporating the cost of qualityin supply chain design

Amar RamudhinDepartment of Automated Manufacturing Engineering,

Ecole de Technologie Superieure, Montreal, Canada, and

Chaher Alzaman and Akif A. BulgakDepartment of Mechanical and Industrial Engineering, Concordia University,

Montreal, Canada

Abstract

Purpose This paper aims at exploring the challenges of introducing a model integrating the Cost ofQuality (COQ) into the modeling of a supply chain network.

Design/methodology/approach This paper introduces a comprehensive supply chain model thatminimizes a series of costs, in which COQ is integrated.

Findings The scenario of incorporating COQ in supply chain network design will ensure the lowestoverall cost, because it reduces the probability of defects and hence the probability of additional costwhich might be due to corrective action.

Practical implications With many industries today on the quest of improving their qualitysystems, finding ways to reduce nonconformities and failure of products is crucial. In industries suchas the aerospace industry, the variable production cost is high; hence producing extra parts tocompensate for defectives would be a costly option.

Originality/value While COQ is a very good indicator of how much poor quality is costing acompany, no work has been published in regard to integrating COQ into supply chain modeling.

KeywordsSupply chain management, Quality costs, Mathematical programming

Paper typeResearch paper

IntroductionA supply chain can be defined as an integrated process of various business entitiesinteracting with each other to source, process and distribute value added products orservices to customers. Those business entities can be generally categorized in fourcategories: suppliers, manufacturers, distributors and retailers (Beamon, 1998). Theinteractions between these entities insure the delivery of vital business processes.These vital processes can be described as follows (Min and Zhou, 2002):

. acquiring raw materials and parts;

. transforming raw materials and parts into finished products;

. adding value to these products;

. distributing and promoting these products to retailers and in-turn to customers; and

. facilitate information exchange among these entities.

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1355-2511.htm

This research was supported in part by the Natural Sciences and Engineering Research Councilof Canada (NSERC) as well as funds from the Faculty of Engineering and Computer Science,Concordia University, Montreal, Canada.

Incorporatingthe cost of

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Journal of Quality in MaintenanceEngineering

Vol. 14 No. 1, 2008pp. 71-86

q Emerald Group Publishing Limited1355-2511

DOI 10.1108/13552510810861950


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The supply chain not only includes the manufacturers and the suppliers, but also thetransporters, the warehouses, the retailers, and the customers themselves. The mainobjective of a supply chain is to maximize the overall value generated (Chopra andMeindl, 2001). From the customers point of view, a better definition of supply chain

can be thought of as a system consisting of all stages involved, directly or indirectly, infulfilling a customer request. The objective of a supply chain network is then tominimize the end customers total level of dissatisfaction, composed of price, quality,and delivery lead time (Cakravastia et al., 2002). So, a supply chain needs to beconfigured in such a manner as to minimize cost while still maintaining a good qualitylevel to satisfy the end user.

While supply chain network design problems have been addressed before by a fairnumber of researchers on the basis of operation costs, the idea of incorporating the costof quality into the network design is nonexistent in research. Knowing that mostsupply chain models employ some form of a cost variable, it would be advantageous ifone can find a cost indicator for quality which one could incorporate into the supplychain modeling. Cost of Quality (COQ) is such a cost indicator for quality and would benecessary to integrate into the supply chain models. Hence, this work aims atincorporating COQ into the modeling of supply chain.

The rest of the paper is organized as follows: the next section resents a literaturereview on supply chain network design. In the following section, the Cost of Qualityis defined and the relevant issues in this matter are discussed. In the next section, amodel that represents a single product, three-echelon system (i.e. suppliers, plants, andcustomers) aiming at the minimization the overall operational and quality costs isformulated. The model is nonlinear in nature, as it has nonlinearities in the objectivefunction and also in the constraints. The results are also shown and the significance ofincorporating COQ in supply chain network design is discussed. The final sectionpresents the conclusions and the future work.

Literature review in supply chain network designA supply chain network design model aims at determining the location of production,stocking, and sourcing facilities, and paths, which the product(s) take. Such models areof large scale and require strong computational power. The earliest work in this areawas by Geoffrion and Graves and can be traced back to 1974. They introduced amulti-commodity logistics network design model for optimizing the flow of finishedproducts from plants to distribution centers. Later, Breitman and Lucas (1987)presented comprehensive models of a production-distribution system. The system theyprovided was named PLANETS. PLANETS is a framework that decides whatproducts to produce, where and how to produce these products, and which market totarget with these products. PLANETS was an ambitious system which models almost

the whole scope of the supply chain. Some parts of their project were implementedsuccessfully at General Motors. Cammet al.(1997) presented a model analyzing Procter& Gambles (P&Gs) supply chain. Their work sought improving the efficiency of allwork processes and eliminating non-value added costs at P&G. The developedmethodology involved mathematical modeling. More specifically, they developed amodel, which lumps integer programming, network optimization, and geographicalinformation system (GIS) together. The modeling strategy was to decompose theoverall supply-chain problem into two easily solvable sub-problems: a

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distribution-location problem and a product-sourcing problem. In the first problem,distribution location attributes are determined and then inputted into the productionsourcing problem for results.

As a general look at papers published with regards to supply chain modeling,

Beamon presented his 1998 review on the literature of supply chain modeling. Heclassified supply chain models into four categories:

(1) analytical deterministic models;

(2) analytical stochastic models;

(3) economic models; and

(4) simulation models.

For the analytical models, one can note the works of Cohen and Lee (1989), Arntzenet al. (1995), Voudouris (1996), and Camm et al.(1997). For the stochastic models, theworks of Cohen and Lee (1988), Svoronos and Zipkin (1991), Lee and Billington (1993),and Pyke and Cohen (1993) can be noted. The works of Christy and Grout (1994), forthe economic models, and of Towill (1991), for the simulation models, can be regardedas further examples.

Pivotal to this study, Arntzenet al.(1995) provided a deterministic model for supplychain network design. The model went as far as considering duty and, morespecifically, options for avoiding drawback duty charges. In their paper, the objectivefunction minimized a combination of cost and time elements. The function minimizedsuch cost elements as purchasing, manufacturing, pipeline inventory, transportationbetween different plants or sites, and duty costs. The time elements in the objectivefunctions were manufacturing lead and transit times. Implementation of this modelsresulted in magnificent savings at Digital Equipment Corporation as the paperclaimed. Additionally, Vidal and Goetschalck (2000) developed a mixed integer

programming (MIP) model that models a global logistic system (GLS). The modelattempts to include supplier reliability in the design. Supplier reliability, was modeledusing historic data, and estimates the probability that a supplier will send shipmentson time. They also studied the effect of exchange rates, changes in demand, andinternational transportation lead times, on the model.

In recent studies, Jayaraman and Ross (2005) presented PLOT, which is aproduction, logistics, outbound, and transportation design system. The overall systemproduces near optimal distribution system utilizing simulated annealing. In their work,the objective function minimizes fixed costs to open warehouses and cross-docks, coststo transport products from warehouses to cross-docks and costs to supply productsfrom cross-docks to satisfy the demand of customers. Meloet al. (2005) work focused onthe strategic design of supply chain networks. They proposed a mathematical

framework that captures dynamic planning horizon, generic supply chain networkstructure, external supply of materials, inventory opportunities for goods, distributioncommodities, facility configuration, availability of capital for investments, and storagelimitations. Santoso et al. (2005) presented a stochastic programming model andsolution algorithm for solving supply chain network design problems. Themethodology, used in their work, is the sample average approximation (SSA)scheme with an accelerated bender decomposition algorithm. Their objective functionminimized total investment and operational costs.


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COQOur literature survey suggests that no work to this date had sought the integration ofCOQ into the supply chain network design. A quality cost is defined as the expenditureincurred by the producer, by the user and by the community, associated with the

quality of a product or a service (British Standards Institution, 1991). A quality-relatedcost is defined as the expenditure incurred in defect prevention and appraisal activitiesplus the losses due to internal and external failure (British Standards Institution, 1991).Moreover, quality-related costs are defined as those costs incurred in ensuring andassuring satisfactory quality as well as the losses incurred when satisfactory quality isnot achieved (British Standards Institution, 1995).

Quality costs are categorized into prevention, appraisal, and failure costs.Prevention and appraisal costs are cost incurred to insure conformance (i.e.conformance costs). Failure costs are costs incurred due to nonconformance (i.e.nonconformance costs). According to Campanella (1991), quality costs are categorizedand defined, as follows:

(1) Prevention costs.The costs of all activities specifically designed to prevent poorquality in products or services.

(2) Appraisal costs. The costs associated with measuring, evaluating or auditingproducts or services to assure conformance to quality standards andperformance requirement.

(3) Failure costs. The cost resulting from products or services not conforming torequirement or customer/user needs. Failure costs are divided into internal andexternal failure cost categories:. Internal failure costs. Failure costs occurring prior to delivery or shipment of

the product, or the furnishing of a service, to the customer.. External failure costs. Failure costs occurring after delivery or shipment of

the product, and during or after furnishing of a service, to the customer.(4) Total quality costs. The sum of the above costs. It represent the difference

between the actual cost of a product or service and what the reduced cost wouldbe if there were no possibility of substandard service, failure of products, ordefects in their manufacture.

Methods of collecting COQ are well documented in literature. Each component of COQcan be divided further into primary costs of activities that happen as a result ofprevention, appraisal, and failure. The list below shows some examples of quality costsin each COQ category (Roden and Dale, 2001).

(1) Prevention costs:.

quality engineering and training;. design and development of equipment;. quality planning by other functions;. maintenance and calibration of production and inspection equipment;. supplier assurance; and. administration, audit and improvement (ISO 9002 registration, customer

audits).

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(2) Appraisal costs:. inspection of components coming from supplier (inbound inspection);. in-process inspection;.

after process inspection and testing (outbound inspection);. evaluating of field stock;. production control; and. quality control.

(3) Internal failure costs:. cost of spoilage;. cost of rework; and. cost of scrap.

(4) External failure costs:.

cost of troubleshooting the problem;. product liability cost;. repair on return material cost;. warranty cost; and. reputation loss cost.

Juran (1979) presented a graphical demonstration of how quality costs affect the overallquality conformance of a given system (Figure 1). One can observe that as the qualitylevel rises, failure costs decline and appraisal plus prevention costs increase. We can alsoinfer that a relationship exists between the quality of conformance percentage, (1 2 y),and prevention and appraisal costs on one count and with failure costs on another.

Model and findingsMethodologyThe model represents a single product, three-echelon system (i.e. suppliers, plants, andcustomer groups) that aims at minimizing the overall operational and quality costs.

Figure 1.Juran model of quality


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Here, the perspective will be that of the supply chain as a whole, meaning suppliers aresubsidiaries of the supply chain and are an integral part of it; so the production costand other operational costs at the supplier would also be cost attributes for the supplychain and would be accounted for in the model. The objective function of the model will

minimize a series of costs: total cost of production at the supplier, total cost oftransportation from suppliers to plants, cost of quality at the supplier, total productioncost at the plant, total cost of transportation from plants to customers. The inputparameters, decision variables, and constraint parameters are explained as follows:

Decision variables:

Si;j;b Number of good component b shipped from supplier i to plant j;i[ I;j [ J; b [ B.

Si;b Number of component bmanufactured at supplier i;i[ I; b [ B.

S Number of component b manufactured at supplier i for plant j;i[ I;j [ J; b [ B.

Xj;k Number of product produced at plant j to satisfy customer k demand;j [ J; k [ K.

zj Binary variable; 1 if plantj is open and running; 0 if plantj is shut;j [ J:.

yi;b Percentage of defectives at supplier ifor component b; i[ I; b [ B.

Parameters (input data):

PcCi;j;b Direct cost of component b procured from supplier i to plant j;i[ I;j [ J; b [ B.

Pr Cj;k Cost of producing one product at plant j for customer k;j [ J; k [ K.

TrCi;

j;

b Cost of transporting component b from supplier i to plant j;i[ I; j [ J; b [ B.

fcoqyi;b Total cost of quality (including prevention and appraisal costs) forsupplieriper good componentbas a function ofyi, the level of proportionof non-quality components. This is equivalent to the total cost of qualityin Figure 1; i[ I; b [ B.

TCj;k Transportation cost of transporting a product from plantjto customerk;j [ J; k [ K..

Constraints input data:

Dk Number of products demanded by customer k; k [ K.

SCi;b Capacity of Supplier ifor component b; i[ I; b [ B.

CAPj Allowable production capacity at plant j; j [ J.

Nb Number of components b required to make a product; b [ B.

ADi;b Maximum acceptable proportion of defective of component b at supplieri; i[ I; b [ B.

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.The sets to which these parameters belong are the following:

B: Set of component types required for one product.

I: Set of suppliers.

J: Set of plants.

K: Set of customers.

The general non-linear mixed integer-programming model is as follows:

Mini[I

X

j[J

X

b[B

XPcCi;j;bS

i[I

X

j[J

X

b[B

XTrCi;j;b Si;j;b

i[I

X

b[B

Xfcoqyi;b1 2yi;bSi;b

j[J

X

k[K

XPr Cj;kXj;k

j[J

XFCjzj

j[J

X

k[K

XTrCj;kXj;k

1

Subject to:

j[J

XXj;k Dk ;K 2

k[K

XXj;k # CAPj*zj ;j 3

k[K

XNbXj;k

i[I

XSi;j;b ;j;;b 4

Si;j;b # S0

1 2yi;b ;i;;j;;b 5

j[J

XS

0

# SCi;b ;i;;b 6

Si;b j[J

XSi;j;b ;i;;b 7

yi;b # ADi;b ;i;;b 8

zj1 0; 1f g; Si;j;b $ 0; S0

$ 0;Xj;k $ 0;yi;b $ 0:

The objective function minimizes the following costs respectively:

(1) direct cost of procuring components from suppliers;

(2) transportation cost from suppliers to plants;


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Moreover, different data sets were used to solve the model. Those data setsrepresented many different cost scenarios and practical situations. In this paper weused one of the data sets, as an example, in order to be consistent, especially whencomparing the three different cases we have contested in the model. Other data sets

will yield solutions with several variations from the illustrative example given.Tables I-IV contain the data sets for our problem as well as optimal values of the

decision variables in our model. The value ofN, which is the number of componentsrequired to make a product, is set to a value of 3. And the optimal value of the objectivefunction is obtained as $641,730.179. In the tables, it is important to note thatSG i;j

Si;j, XT j Xj (the sum of all products made at plant j), and

X j; k

Xj;k.Figure 2 shows the logistic routes the model chooses once it reaches optimality. The

model optimizes the network in accordance to operational costs and quality costs.The solutions presented are for a network of six-suppliers, three-plants, andtwo-customers and were fetched by Lingo in less than a second timing (Figure 2). The

quality functions are tabulated in Table III, among other parameters and weresimulated to resemble polynomial functions of second order, inspired by Jurans

Customers (k)Plants (j) 1 2

TC( j,k)1 1.0 1.12 1.0 1.03 1.1 1.0

X(j,k)1 1,250 250

2 0 1,0503 0 0

Table I.

Input and output modelparameters

Plants (j)Suppliers (i) 1 2 3

SG(i,j), decision variable1 2,400 0 02 0 2,244 03 0 729 04 2,100 177 05 0 0 0

6 0 0 0

TrC(i,j), input parameter1 1.00 1.20 1.302 1.20 1.10 1.253 1.20 1.00 1.254 1.10 1.15 1.055 1.20 1.10 1.006 1.20 1.10 1.00

Table II.Input and output model

parameters


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graphical presentation (Figure 1), COQ as a function ofyi. One can observe that themost prevalent (i.e. constitutes high COQ) quality cost function is that of supplier 1,produced at zero percent of defectives (Table II). This validates the model when thecost of failure is high, consequently the model produced at zero defectives.

Special caseLingo was able to solve the small model of six suppliers, three plants, and twocustomers. However, when bigger models were contested (i.e. models with moresuppliers, plants, and customers) Lingo failed to bring about an optimal solution. Withnonlinearities present both in the objective function and in the constraints, the problemis challenging to solve. Also the nonlinear constraints are binding and hence are moredifficult to relax. Many solution methodologies in literature propose the linearization of

Figure 2.Optimal logistic routes

Model input parametersDecisionvariables Input parameter

Plants F PrC Capacity XT Z Customers Demand

1 11,000 100 1,500 1,500 1 1 1,2502 10,000 102 1,500 1,050 1 2 1,3003 9,000 104 1,500 0 0

Table IV.Input and output modelparameters

Model input parameters Decision variablesSuppliers PcC SC fcoq (quality function) S y

1 35.5000 2400.0000 119y2 2 39y 7 2,400.0000 0.00002 35.7000 2400.0000 118y2 2 60y 12 2,400.0000 0.06513 35.0000 2400.0000 60y2 2 60y 17 830.0259 0.12184 35.2000 2400.0000 45y2 2 44.6y 14.5 2,400.0000 0.05125 34.0000 2400.0000 49y2 2 63y 23.8 0.0000 NA6 34.0000 2400.0000 68y2 2 76.5y 25 0.0000 NA

Table III.Input and output modelparameters

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the constraint successively. In doing so, an optimal solution can be fetched. For therelatively small model presented in this paper, the linear approximation is viable andcan produce good results. As the model gets bigger, the linearization of the constraintswill be more difficult and more calculation extensive. For this, exact solution might not

be economical and hence the simplification of the model would be needed.The model poses several challenges to solve due to:

(1) the total cost of quality function fcoqyiin the objective function is non-linear;

(2) both Si and yi are variables creating another non-linearity in the objectivefunction;

(3) constraint (5) involves a multiplication of two unknown quantitiesSi*yiand ishence non-linear; and

(4) variableszj are binary in nature.

If they values for all suppliers were fixed, then issues 2 and 3 above disappeared and

the problem becomes a special case one. yi is now an input to the model and henceconstraint 8 would be eliminated. There are two motives for this simplification; asolution methodology motive and a practical application-oriented motive. The firstmotive is based on the fact that most approaches, in solving a nonlinear model withnonlinearity both in the objective function and in the constraints, rely on some form ofsuccessive linearization. As the model size of the model increases, optimality becomesdifficult to achieve in such cases. Whenyis fixed, the model becomes linear; hence easyto solve. The other motive is an application-oriented motive. Management of the supplychain network can wish to have all entities in the supply chain working at the samepercent of defectives and might focus on the overall cost of achieving such a scenario.

In Figure 3, one can observe that at high defect ratios (i.e. high ys) the overallobjective value increases. An increase in defectives will have a consequence of a

corresponding increase in production cost, as we need to compensate for defectiveproducts by producing more. Alternatively, a decrease in y will cause an increase inquality costs and hence an increase the overall objective value. The graphdemonstrates an initial decrease of the objective value for low values of y and

Figure 3.Graph of the values of the

objective function vspercent of defectives


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sequential increase later for the objective values for higher y values, which concurswith the previous two statements.

Referring back to the original problem, for smaller problems Lingo was capable ofsolving the problem. But for larger problems, there are possible directions for solving

the model. The method of Lagrangian relaxation can possibly be beneficial in solvingthe model. Also, as mentioned in this section, when y is fixed the model becomes linearand easy to solve. Hence, other methods can be constructed to exploit this specialcharacteristic in the model.

Impact of COQThis section illustrates the impact of incorporating COQ into the supply chainnetwork. In order to address this, the model presented previously (i.e. the COQintegrated into the supply chain network model) was modified by removing theCOQ terms from the model (i.e. the basic supply chain network model). Lookingback at the integrated model, if we were to remove the cost of quality term, theP

i[IP

b[Bfcoqyi;b1 2yi;bSi;b term, from the objective function then the objectivefunction will be free of quality related terms. In addition, the constraint 5, Si;j;b #

S01 2yi;bwould also need to be eliminated. In doing so, the integrated model will be

transformed into the basic supply chain network model. The basic model was solvedwith the same data sets used above. Tables V-VIII present the results of the basic model.

As both models were solved using the same data sets, the results can be comparedas follows: In the integrated model, the value of the objective function was obtained tobe $641,730. In the latter case (i.e. COQ not incorporated), the value of the function was$552,295; indicating a difference of approximately 16 percent. This difference is due tothe contribution of COQ into the overall operation. This cost would have been hidden ifone were not to include COQ in the model.

Moreover, when COQ was not incorporated into the model (Figure 4), it can be seenthat suppliers 1 and 2 were no longer furnishing any material to any of the plantsaccording to the optimal solution from the basic model. Suppliers 1 and 2 with abetter COQ structure no longer dispensed any material to any plant. Alternatively, theoptimal network solution from the basic model chooses the suppliers 5 and 6 despitethe fact that the failure costs at these two suppliers were high. Hence, solutions fromthe basic model, if implemented, will be prone to potential future problems at the plantsas well as throughout the network. Nonconformance costs will eventually outweigh the

Customers (k)Plants (j) 1 2

TC( j,k)1 1.0 1.12 1.0 1.03 1.1 1.0

X(j,k)1 1,250 2502 0 1,0503 0 0

Table V.Input and output modelparameters

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cost savings in the latter network. When one also considers issues of delays, inventory,and reputation, the benefits of incorporating COQ will outweigh the latter case.

With COQ incorporated into supply chain network design, suppliers behavior inregard to quality issues will be scrutinized. Suppliers whom are incurring high cost ofquality will be less competitive or attractive even if they were to produce at lower costs.As shown, if we were to compare the two different scenarios, one with COQ notincluded in the design of the supply chain network and one with COQ included. In thefirst, the final optimal network will choose key suppliers who have low costs. Noinformation is inferred in regard to the quality nonconformance cost. In this scenario asupplier that is running at a high quality nonconformance cost is treated similarly to

Plants (j)Suppliers (i) 1 2 3

TrC(i,j)

1 1.00 1.20 1.302 1.20 1.10 1.253 1.20 1.00 1.254 1.10 1.15 1.055 1.20 1.10 1.006 1.20 1.10 1.00

SG(i,j)1 0 0 02 0 0 03 0 2,400 04 450 0 05 2,400 0 06 1,650 750 0

Table VI.Input and output model

parameters

Model input parameters Decision variablesSuppliers PcC SC fcoq (quality function) S y

1 35.50 2400 119y2 2 39y 7 0 NA2 35.70 2400 118y2 2 60y 12 0 NA3 35.00 2400 60y2 2 60y 17 2,400 NA4 35.20 2400 45y2 2 44.6y 14.5 450 NA5 34.00 2400 49y2 2 63y 23.8 2,400 NA6 34.00 2400 68y2 2 76.5y 25 2,400 NA

Table VII.Input and output model

parameters

Model input parametersDecisionvariables Input parameter

Plants F PrC Capacity XT Z Customers Demand

1 11,000 100 1,500 1,500 1 1 1,2502 10,000 102 1,500 1,050 1 2 1,3003 9,000 104 1,500 0 0

Table VIII.Input and output model

parameters


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the one that is operating at a lower quality nonconformance cost, given they both have

the same production cost. Even if that supplier has a lower production cost, if chosen,additional costs would result due to product failures; this in turns would disburse thesavings realized initially. Hence, choices made solely on production cost could sacrificequality and incite additional quality nonconformance costs. The scenario ofincorporating COQ in supply chain network design will ensure the lowest overallcost, because it reduces the probability of defective and hence the probability ofadditional cost which might be due to corrective action.

Conclusions and future workSupply chain network design (SCND) is an important problem and attracts theattention of many researchers. However, no research has been done so far to integratethe vital concept of Cost of Quality into the network designs. This study attempts to

incorporate COQ in SCND. We have modeled COQ into SCND using a nonlinearmathematical programming model. The non-linearity in the objective function,represented by the convex quality functions, and the nonlinearity present presented inthe constraints have not deterred us from achieving insightful results. The qualityfunctions that were used are valuable as they can represent mathematically the qualitysystem of a given supplier. While, COQ costing differ among different companies, onecan most certainly analyze the behavior of COQ with respect to y, and infer amathematical function to represent it. Hence, using functions, we can bypass thecomplications of accounting systems and infer costs, which correspond to percent ofdefectives, which can be related to production costs. Our model had not only sought theoptimal quality level but had realized a solution which takes into account day to daytradeoffs in supply chain operations. By minimizing the overall cost of the supplychain we have fetched a reasonable y that is low enough to deter extra cost ofreproduction and high enough as not to drive COQ up.

Just like COQ was modeled at the suppliers, it can be also modeled at the plant.Although in this study COQ was ignored at the plant, further research could modelCOQ at both supplier and plants simultaneously. Also, further research could address amulti-product sourcing and distribution network. This will increase the number ofdecision variables and the number of constraints in the model and will add morecomplexity to the bill of material parameters, producing more interrelations between

Figure 4.Optimal network for thecase when COQ is notincorporated into themodel

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parameters and decision variables. The complexity of such a problem would perhapsbe implausible to solve by Lingo and would eventually require the formulation of asolution methodology that might take advantage of any special characteristics of ourobjective function and constraints. As alternative means of solving more complex and

larger models, we will investigate using some stochastic search procedures such astabu search or genetic algorithms.

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Further reading

Geoffrion, A. and Graves, G. (1974), Multi-commodity distribution system design by Bendersdecomposition, Management Science, Vol. 29 No. 5, pp. 822-44.

Newhart, D.D., Stott, K.L. and Vasko, F.J. (1993), Consolidating product sizes to minimizeinventory levels for a multi-stage production and distribution systems, Journal of theOperational Research Society, Vol. 44 No. 7, pp. 637-44.

Corresponding authorAmar Ramudhin can be contacted at: [email protected]

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