Profit oriented supply chain network optimization

CEJOR (2013) 21:455–478DOI 10.1007/s10100-012-0240-0

ORIGINAL PAPER

Profit oriented supply chain network optimization

Turan Paksoy · Eren Özceylan ·Gerhard-Wilhelm Weber

Published online: 26 February 2012© Springer-Verlag 2012

Abstract This paper proposes a novel mixed integer linear programming model tosolve a supply chain network design problem. The proposed model deals with majorissues for supply chains; product quality and cost. These issues are usually solved sepa-rately, but in this paper, we investigate effects of product quality on supply chain designand transportation flow. A trade-off between raw material quality, its purchasing andreprocessing costs was considered. Assuming decision maker (DM) wishes to workwith a supplier which serves a low quality raw material; this raw material should be inneed of reprocessing. To avoid the reprocessing costs, a supplier which serves a highquality raw material should be chosen but at this time the DM has to face a high purchas-ing cost. A supply chain network which consists of multiple suppliers, manufacturers,distribution centers and retailers is tried to be designed to accomplish aforementionedabove trade-offs. The paper examines and discusses the relationship between productquality and supply chain design and offers several managerial insights.

Keywords Mixed integer linear programming · Profit maximization · Quality ·Trade-off · Supply chain optimization

1 Introduction

Supply chain management is a kind of process to increase the efficiency of a supplychain network (SCN) via conducting and integrating money, information and goods

T. Paksoy · E. Özceylan (B)Department of Industrial Engineering, Selçuk University, Campus, 42031 Konya, Turkeye-mail: [email protected]

T. Paksoye-mail: [email protected]

G.-W. WeberInstitute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkeye-mail: [email protected]

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456 T. Paksoy et al.

flows from suppliers to end-customers. The questions and problems connected with theoptimizing a SCN are linked to procurement, production, distribution, transportationand suppliers’ selection etc. In today’s global market, if enterprises wish to succeed ina business environment, they have to manage and design their entire SCN. It is certainthat traditional and usual SCN methods are not sufficient to satisfy conscious cus-tomers anymore. To overcome the aforementioned problem, enterprises must handleintegrated manufacturing and distribution processes together and present alternativeoptions to customers via strong relationships with the suppliers.

The issue of how to select the best suppliers has been one of the important researchquestions in the fields of operations management and supply chain management fordecades. Supplier selection criteria suggested in the literature encompass much onlyfactors related to operational performance, such as quality, cost and delivery besidesthose associated with organizational infrastructure, such as financial and organizationalstability, technology and manufacturing capability (Shin et al. 2009). It is known thatsupplied materials’ quality levels are directly affected the agreements with suppliersand also indirectly effected the material flow through SCN. Bad quality materialsshould lead to the production processes which are not defect free, and result in itemsthat require reworking. Rework occurs when a product or a service does not meet theinternal or external quality requirements, and could be defined as doing something atleast one extra time due to non-conformance to these requirements. This additionaleffort adds cost and not value to the product in manufacturers (El Saadany and Jaber2008). At this point, DM faces with a purchasing decision under purchasing costs,quality levels and probability of reprocessing costs which are occurred because of badquality level.

In this paper, we emphasize and point suppliers’ efficiencies on a SCN consideringtheir agreement contracts and the product quality options they present. Quality is animportant factor in the value-adding process involved in the production and deliveryof products along the supply chain. The production of defect-free components andparts that meet the requirements of customers along the supply chain is critical for thequality of the final products. Sustaining quality efforts throughout the chain also hassignificant implications for reducing costs (Bulgak et al. 2008). We consider that rawmaterials which are provided from suppliers to manufacturers have different qualityoptions. These options could be of high, low or bad quality. While a DM wants to dealwith a supplier which provides and presents a low quality raw material, the DM has tostand remanufacturing process for this raw material. To avoid the reprocessing costs,the DM can choose a supplier which serves a high quality raw material by facing a highpurchasing cost. Since these disruptions can cause materials not to reach their qualitytarget for manufacturers we focus on the use of option contracts to hedge against theloss. The DM has to work manufacturers and suppliers according to money whichthey budget for considering suppliers’ engagement fees. The question then becomesfor which quality level should these option contracts be placed and with which sup-pliers? By implication, a tradeoff is occurred that involves losing one quality levelin return for gaining purchasing low prices. It implies the decision to be made withfull comprehension of pros and cons of a particular choice. This is a difficult questionin part because of the need to trade-off between the purchasing different quality rawmaterials, purchasing costs and reprocessing costs while aiming maximizing profit.

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Profit oriented supply chain network optimization 457

This paper is organized as follows. Section 2 provides a literature review of thesupply chain network design problems and related issues. The proposed mixed inte-ger programming model that maximizes total profit via minimizes the total cost ofoverall SCN is presented in Sect. 3. For illustrative purposes, a numerical exampleis given, the results and the scenarios that are obtained under various conditions arediscussed, and a sensitivity analysis is examined in Sect. 4. Finally, Sect. 5 concludesthe paper and proposes future research directions.

2 Literature review

This brief literature review will cover some of the relevant previous work in supplychain modeling and quality issues in supply chains, respectively. Supply chain manage-ment has received considerable attention from academicians, researchers and operatorsduring the last several decades. Designing and optimizing of SCNs is one of the mostpopular problems in this research field. For that reason, there are a lot of mathematicaland heuristics models have been proposed. Most of these models are related to thetransportation/distribution networks with following entities (Paksoy and Chang 2010):

• Location of facilities (plants, distribution centers, etc.) to be opened,• Design of the network configuration,• Satisfy customer’s demand with minimization of the total cost including purchas-

ing cost, transportation cost, fixed operating cost, etc. (Cohen and Lee 1989; Pykeand Cohen 1993; Syarif et al. 2002; Paksoy et al. 2007 and Altiparmak et al. 2009).

In their pioneering paper, Cohen and Lee (1989) developed a deterministic, mixedinteger, non-linear mathematical programming model. The objective function of themodel maximizes the total after-tax profit under resource, production and demandconstraints. Output of the model determines global resource deployment policy forthe manufacturing plants, distribution centers and customer regions. Pyke and Cohen(1993) present a stochastic mathematical programming model for a tree-stage sup-ply chain consisting of one manufacturing plant, one warehouse and one retailer.The model minimizes the total cost under a service level constraint and deter-mines the economic reorder interval, replenishment batch sizes and the order-up-tolevels. The model, however, does not consider multiple products and has only one man-ufacturing plant, one warehouse and one retailer, so failed to represent the complicatedsupply chain networks of the real world.

Additionally to the stochastic model of Pyke and Cohen (1993); Petrovic et al.(1999) consider a serial supply chain including inventories and production facilitiesbetween them in fuzzy environment. For each inventory, proposed fuzzy model deter-mines the order quantities giving an acceptable service level of the supply chain atreasonable total cost. They use simulation techniques to examine the dynamics andperformance of all the parts of the supply chain. As a result, they claim that uncertaincustomer demand and uncertain supply delivery along the supply chain have greatimpact on the behavior of the supply chain. Although, the model deals multi-stagesupply chain, it is confined to single-period, single-product problems. To examine thesensitivity of supply chain performance to three inventory planning parameters: (i) theforecast error, (ii) the mode of communication between echelons, and (iii) the planning

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frequency, Ganeshan et al. (2001) develop a simulation model based on a case study.They show that all the three parameters have a significant effect on performance.Increasing forecasting errors and the re-planning frequency (for example, planningevery quarter instead of every month) decreases service, return on investment, andincreases cycle time.

Syarif et al. (2002) defined this kind of problem as NP-hard because of the combi-nation of multiple choice Knapsack problems with the capacitated location-allocationproblem simultaneously. Thus, some heuristics and meta-heuristics models are devel-oped and proposed by Jayaraman and Pirkul (2001); Syarif et al. (2002); Jayaramanand Ross (2003); Gen and Syarif (2005); Altiparmak et al. (2009) and Jolai et al.(2011). Syarif et al. (2002) propose a minimum spanning tree based genetic algo-rithm approach for a multi-stage supply chain problem formulated by 0–1 mixedinteger linear programming model. The efficacy and the efficiency of the method areanalyzed by comparing its numerical experimental results with those of traditionalmatrix-based genetic algorithm and a professional software package. Although exper-imental results show that the proposed algorithm yields better heuristic solutions thantraditional genetic algorithm, the comparison with the performances of other meta-heuristic techniques (tabu search, simulated annealing etc.) are remained unanswered.Jayaraman and Ross (2003) consider a mixed integer linear programming model todetermine locations (for distribution centers and cross-docks) and distribution strate-gies in supply chain management. They propose a simulated annealing approach forthe model which is characterized by multiple product families, a central manufacturingplant site, multiple distribution center and cross-docking sites, retail outlets, and eval-uate the computational performance of the proposed approach under a variety ofproblem scenarios.

Supplier policy in a supply chain design problem is studied by Paksoy et al. (2007).They develop a multi-echelon mixed integer linear programming model. The modellets DM reflect his supply policy by determining VIP suppliers and also includes thechoice of the facilities [plants or distribution centers (DCs)] to be opened and designof network configuration with minimum total cost. The total cost involves supply cost(combination of purchasing cost and shipping cost from suppliers); transportationcosts between plants and DCs; distribution cost between DCs and customer zones;fixed operating costs; and opportunity cost not have the material at the right time.Altiparmak et al. (2009) introduce a solution procedure based on steady-state geneticalgorithms for the design of a single-source, multi-product, multi-stage supply chainnetwork. Their model determines the choice of facilities (plants and distribution cen-ters) to be opened and the distribution network design to satisfy the demand withminimum cost.

In supply chain optimization models, costs, customer service and inventories areconsidered mostly as quantitative purposes. According to Mula et al. (2010), regardingcosts, the minimization of costs (Ozdamar and Yazgaç 1997; Azaron et al. 2008), themaximization of revenues and the maximization of benefit (Cohen and Lee 1989;Tsai et al. 2008) are contemplated, while the maximization of the service level (Chenand Lee 2004; Torabi and Hassini 2008), the minimization of backorders (Tuzkayaand Onut 2009; Paksoy and Chang 2010), the flexibility in volume or delivery dates(Sabri and Beamon 2000) or the cost and quantity discount (Tsao and Lu 2012) are

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considered for customer services. The maximization of safety inventories is also takeninto account. Also multi-objective papers are studied by Pokharel (2008); Tsai et al.(2008); Xu et al. (2008) and Farahani and Elahipanah (2008). Pokharel (2008) proposea deterministic model which involves two objectives, one for minimizing the costs andthe other for maximizing the reliability of supply from an echelon to the followingechelon. Farahani and Elahipanah (2008) develop a mixed integer linear program-ming model with two objective functions: minimizing costs, and minimizing the sumof backorders and surpluses of products in all periods. They also consider deliverylead times and capacity constraints in a multi-period, multi-product and multi-channelnetwork.

This section has presented a review of mathematical programming models onlyfor SCN problems until now. Besides modeling supply chain problems, quality is asignificant concern in the supply chain because of the historical problems that havecome from supplier selection. The issue of supplier selection based on the quality ofraw material is an important topic and has gained more focus in designing supplychains. Raw material quality is usually an objective that can be achieved by sourcingmaterial from better quality suppliers. In most cases, those higher quality suppliersare more costly due to their tighter quality specifications. By analyzing only thosetwo separate measures of profit and quality, it can be seen that they are contradictory.With the exception of the aforementioned papers, Choi and Hartley (1996) focus sup-plier’s selection according to companies’ position. For that reason, they comparedsupplier-selection practices based on a survey of companies at different levels in theauto industry. Their findings rebut the common thinking that indirect suppliers whoare more involved in commodity purchasing emphasize initial price and de-emphasizerelational considerations. Also they learned that quality and delivery in the context ofsupplier selection are part of the same underlying construct, consistency, which wasthe most important factor for supply chain design. Persson and Olhager (2002) presenta supply chain simulation study for a real case, concerned with the manufacturing ofmobile communication systems. The purpose of the study is twofold. One objectiveis to evaluate alternative supply chain designs with respect to quality, lead-times andcosts as the key performance parameters. A second objective is to increase the under-standing of the interrelationships among these and other parameters, relevant for thedesign of the supply chain structure. In the model three experiments with three differ-ent quality levels were executed. The quality levels ranges from a low level of qualityup to a high level. The results are discussed and a model capturing the relationshipsamong cost, quality, and lead-times, based on the simulation study, is presented.

Singer et al. (2003) study a single-product distribution channel where a suppliermanufactures items of a given type, some of which are defective, that are sold bya retailer who only detects a subset of the defective items, passing the rest along tocustomers. They conjecture the structure of the demand and cost functions, assum-ing customers to have a decreasing marginal aversion to bad quality while both thesupplier and the retailer make marginally increasing efforts to avoid bad quality.Ho Ha and Krishnan (2008) outlined a hybrid method, incorporating multiple tech-niques into an evaluation process, in order to select competitive suppliers in a sup-ply chain and it enables a purchaser to do single sourcing and multiple sourcingby calculating a combined supplier score, which accounts for both qualitative and

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quantitative factors that impact on supply chain performance. Xu and Nozick (2009)formulated a two-stage stochastic program and a solution procedure to optimizesupplier selection to hedge against disruptions. Their model allows for the effectivequantitative exploration of the trade-off between cost and risks to support improveddecision-making in global supply chain design. Shin et al. (2009) provided buyingfirms with a useful sourcing policy decision tool to help them determine an optimumset of suppliers when a number of sourcing alternatives exist. They proposed a probabi-listic cost model in which suppliers’ quality performance is measured by inconformityof the end product measurements and delivery performance is estimated based on thesuppliers’ expected delivery earliness and tardiness. Luo et al. (2009) developed amodel that helps overcome the information-processing difficulties inherent in screen-ing a large number of potential suppliers in the early stages of the selection process.Based on radial basis function artificial neural network, the model enables potentialsuppliers to be assessed against multiple criteria using both quantitative and quali-tative measures. Kheljani et al. (2009) considered the issue of coordination betweenone buyer and multiple potential suppliers in the supplier selection process. In theobjective function of the model, the total cost of the supply chain is minimized ratherthan only the buyer’s cost. Franca et al. (2010) demonstrate a method for evaluatingthe systemic supply chain risk of poor quality. They introduce a multi-objective sto-chastic model that uses Six Sigma measures to evaluate financial risk. Results frommodeling suggest quality, profit, and customer satisfaction can be evaluated. Paksoyet al. (2011) applies fuzzy sets to integrate the supply chain network of an edible veg-etable oils manufacturer. The model has been formulated as a multi objective linearprogramming model where data are modeled by triangular fuzzy numbers. Sadjadyand Davoudpur (2012) deal with a two-echelon supply chain network design problemin deterministic, single-period, multi-commodity contexts. They have formulated theproblem as a mixed integer programming model and intends to minimize total costsof the network including transportation, lead-times, and inventory holding costs forproducts, as well as opening and operating costs for facilities.

The presented review shows that, the literature has mainly considered quality fromthe perspective of the supply chain and acknowledged the lack of a consistent visionpertaining to quality throughout the supply chain. Therefore, the proposed model isfilling this gap via considering different quality levels of materials, reprocessing costsand their effects on transportation flow. The proposed mixed integer programmingmodel is presented in following section.

3 Proposed model

3.1 The trade-off in supply chain

The emergence of management science in the early part of the last century was func-tionally structured and resulted in local optimization centered on cost. In this waythe level of inspection and the size of batches were determined via cost models thatprovided a means of optimizing the conflicting requirements (Fig. 1).

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Fig. 1 The traditional conformance quality cost model (Stratton and Warburton 2006)

Fig. 2 Traditional trade-off models in the form a conflict resolution diagram (Stratton and Warburton 2006)

Supply chain management is a growing issue that many enterprises are beingincreasingly interested in minimizing, postponing, maximizing or otherwise man-aging in the networks. However while there are such a lot of important points andaims in supply chains, controlling and making promising decisions become a veryarduous condition. That’s why the DMs always face to face the trade-offs to answerand satisfy customer’s demands. Some of the aforementioned trade-offs which aremostly encountered are the cost versus response, the cost versus quality, the qualityversus the purchasing cost, the re-processing cost versus quality level etc.

In an attempt to illustrate how the trade-off concept may be used to focus improve-ment, the batch size model has been presented (Fig. 2) in such a way that the policiesand associated thinking underpinning the model may be challenged systematically.This is in the form of an Evaporating Cloud Diagram otherwise known as a con-flict resolution diagram (CRD) (Stratton and Warburton 2006). At Fig. 2, CR dia-gram shows the purchasing cost versus quality level trade-off in our model. Figure 2represents the logic of the traditional batch optimization diagram, where require-ments B (minimizing purchasing costs) and C (maximizing total quality) are necessary

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Table 1 Indicesi ∈ I Set of potential suppliers

j ∈ J Set of manufacturers

k ∈ K Set of distribution centers

l ∈ L Set of retailers

q ∈ Q Set of quality options

(but not sufficient) to achieve the objective A. Similarly the prerequisites D and D′ arenecessary (but not sufficient) to achieve the requirements at B and C, respectively. Thepre-requisite requirements are normally shown as opposites to make the contradictionas explicit as possible, so enabling the logic underpinning it to be challenged (Strattonand Warburton 2006).

3.2 Network structure and basic features

Here, the constituted model represents four echelons, multi suppliers, multi-manu-facturers, multi-DCs, multi-retailers and multi-quality raw material options problem.Manufacturers would be working with the logistics service provider, which executetransportation functions, and deals with delivering on time supplied from supplierswith an engagement contract. Products which are produced with multi quality optionalraw materials supplied from suppliers are moved to DCs and retailers from manu-facturers, respectively. At that point, a DM faces a trade-off between low-mid-highquality raw materials, their purchasing costs and reprocessing costs according to qual-ity options in manufacturers. Such a supply chain network problem can include alltransportation costs between all echelons, raw material costs, reprocessing costs andtotal supplier engagement fees which could not be greater than total money DM budgetsfor. In this paper, we try to optimize and design the supply chain considering thesecost factors together. Before giving the model formulation, the following conditionsshould be satisfied and a few basic assumptions should be made:

• The supplier selection is binary,• The demand of each retailers must be satisfied,• The flow is only allowed to be transferred between two consecutive echelons,• The capacities of facilities are limited,• The reprocessing rates of each quality level and the estimated selling amounts of

each retailer are given.

Because of the supplier selection’s binary situation, the developed model can bedefined a mixed integer and also linear programming model. The proposed mixed-integer linear programming model is formulated by including aforementioned indices,variables, parameters, objective function and constraints as follows (Tables 1, 2, 3):

Objective function (maximize)

∑

l

Sl Prl

−∑

i

∑

j

∑

q

Xi jq .Ci jq −∑

j

∑

k

Y jk .C jk −∑

k

∑

l

Zkl .Ckl

−∑

i

∑

q

Piq .Ciq −∑

j

∑

q

Rp jq .C jq −∑

i

δi .�i (1)

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Table 2 Variables

Xi jq Quantity of qth quality raw material shipped from supplier i to manufacturer j

Y jk Quantity of product shipped from manufacturer j to distribution center k

Zkl Quantity of quality product distributed from distribution center k to retailer l

Sl Estimated quantity product sold from retailer l

Piq Purchased quantity of qth quality raw material from supplier i

Rp jq Quantity of qth quality product reprocessed in manufacturer j

�i If an agreement is signed mutually, 1; otherwise, 0

Table 3 Parameters

δi Agreement contract fee of supplier i

Caiq Capacity of supplier i to supply qth quality raw material for manufacturers

Ca j Capacity of j th manufacturer

Cak Distribution capacity of DC k

El max, El min Estimated maximum and minimum demand at retailer l

B Maximum allowable money budget for dealing with suppliers

Prl Unit selling price at retailer l

Ci jq Unit transportation cost of qth quality raw material between supplier i andmanufacturer j

C jk Unit transportation cost between manufacturer j and DC k

Ckl Unit transportation cost between DC k and retailer l

Ciq Purchased price of qth quality raw material from supplier i

C jq Reprocessing cost of qth quality product at manufacturer j

λq Percent value of total qth quality raw material which needs reprocessing

Constraints

∑

j

Xi jq ≤ Caiq .�i ∀i,q (2)

∑

k

Y jk ≤ Ca j ∀ j (3)

∑

l

Zkl ≤ Cak ∀k (4)

El min ≤ Sl ≤ El max ∀l (5)

λq .∑

i

Xi jq = Rp jq ∀ jq (6)

∑

j

Xi jq = Piq ∀i,q (7)

∑

i

∑

q

Xi jq =∑

k

Y jk ∀ j (8)

∑

j

Y jk =∑

l

Zkl ∀k (9)

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∑

k

Zkl = Sl ∀l (10)

∑

i

δi .�i ≤ B (11)

�i ∈ {0, 1} ∀i (12)

Xi jq , Y jk, Zkl , Piq , Rp jq ≥ 0 ∀i, j,k,l,q (13)

The objective function aims at maximizing the profit of distribution network.The objective function Constraint (1) includes total cost of sold amount from retailersminus transportation cost between all echelons, purchased amount of raw materials,reprocessed costs in manufacturers and total contract fees. Constraint (2) describes theraw material supply restriction if there is an agreement between supplier and manufac-turer. Constraint (3) is the manufacturer production capacity restriction. Constraint (4)is the distribution capacity constraint for DCs. Constraint (5) ensures that the totalproducts transported from different DCs to a retailer must be greater than minimumestimated sales amount and smaller than maximum estimated sales amount of thatretailer. Constraint (6) ensures that a percent value of total raw material transportedsuppliers to a manufacturer must be reprocessed at that manufacturer for each qualityoptions. Constraint (7) shows that the total raw material transported from a supplierto manufacturers equals the total purchased amount of raw material from that sup-plier for each quality options. Constraint (8) sets a balance between the raw materialstransported to a manufacturer from suppliers and the products transported from thatmanufacturer to DCs. Constraint (9) sets a balance between products transported to aDC from manufacturers and the products transported from that DC to retailers. Con-straint (10) sets a balance between products transported to a retailer from DCs and theproducts sold from that retailer. Constraint (11) limits the total contract fees accordingto maximum allowable money for agreements. Finally, Constraints (12) and (13) referto non-negativity.

4 A numerical example

In this section we present a numerical example to illustrate the proposed model men-tioned in previous section. The application of the model is performed for a logical datawhich was inspired from related cases in the real world. The considered supply chainnetwork includes five suppliers which are located different places, three manufactur-ers, three distribution centers and four retailers for selling (Fig. 3). The network isstructured to supply raw materials and transport products from suppliers to end-usersis constituted from multi echelon, multi quality raw material, supplier contract feesand capacitated elements of network.

As mentioned in Sect. 3, the proposed model includes different sub-problems inan integrated structure. When this model is considered, to maximize total profit, DMhas to minimize total transportation costs between all echelons, reprocessed costs inmanufacturers, purchasing rates of raw material and contract fees. While DM is looking

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Fig. 3 The multi-echelon supply chain network in the example

Fig. 4 Material flows in the example network

for the answers of these problems, he/she has to consider quality options raw materialswhich are presented from suppliers (Fig. 4).

For example, a DM can purchase low quality raw material with a low price, buthe/she faces a reprocessing cost because of low quality of raw material. A percentvalue of the total raw material may be broken or may not be fit while they are manu-facturing at machines according to their quality. The DM has to reprocess these rawmaterials which are broken or not fixed to provide demands of retailers. This percentvalue is directly proportional with quality of raw materials. Also there is a trade-offbetween quality options (high-mid-low) and its purchasing costs while supplying rawmaterial (Fig. 5). So if the DM desires to maximize total profit, he/she has to considerreprocessing costs or high purchasing rates. Also he/she has to deal with supplierswith a contract fee under allowable money for agreements. The trade-off shown inFig. 5 constitutes equity like volatility smile function (Derman 2006) in Black andScholes (1973) model analogously. At the money point in volatility smile functionequals the optimum and balanced point of our model factors (purchasing costs, qual-ity level and also reprocessing costs). As in volatility function, the DM tries to findthe equity between cost and quality.

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Fig. 5 The trade-off between quality levels and purchasing, reprocessing costs

Table 4 Unit transportation costs between suppliers and manufacturers (TL)

Suppliers Manufacturer 1 Manufacturer 2 Manufacturer 3

High Mid Low High Mid Low High Mid Low

1 0.9 0.9 0.9 0.7 0.7 0.7 0.8 0.8 0.8

2 1.1 1.1 1.1 1 1 1 1.1 1.1 1.1

3 0.8 0.8 0.8 0.9 0.9 0.9 1 1 1

4 0.7 0.7 0.7 0.75 0.75 0.75 0.8 0.8 0.8

5 1.1 1.1 1.1 0.9 0.9 0.9 1 1 1

Table 5 Unit transportation costs between manufacturers-DCs (TL) and capacities of manufacturers

Manufacturers DC 1 DC 2 DC 3 Capacity

1 1.2 1.3 1.1 9,400

2 1.4 1.3 1.5 8,100

3 1.1 1.5 1.4 7,800

In the numerical example, maximum allowable money budget (B) for dealing withsuppliers is given 2000 TL and percent values (λ1,2,3) of total qth quality raw materialwhich needs reprocessing are determining 5, 10 and 30%, respectively. The informa-tion on three kinds of quality options including transportation costs, capacities, sellingprices, demand etc. is given Tables 4, 5 and 6. The purchasing values, raw materialcapacities and contract fees of suppliers are also given in Tables 7 and 8. Mentionedpresented parameters are shown below.

4.1 Solution

According to data obtained LINDO 11.0 package program, results are given in Table 9.Table 10 gives the sub-objective values of the example. When the proposed modelis solved for the given example, total cost is found out to be 102,553.20 TL (total

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Table 6 Unit transportation costs between DCs-retailers, selling prices and demand of retailers, capacitiesof DCs (TL)

Retailers DC 1 DC 2 DC 3 Selling Demandprice

1 1.6 1.3 1.2 15 4,200 <· · · .< 5,000

2 1.3 1.6 1.5 14 4,000 <· · · .< 4,500

3 1.4 1.2 1.1 16 4,200 <· · · .< 4,700

4 1.5 1.4 1.7 17 4,400 <· · · .< 4,900

Capacity 8,000 6,500 7,700 – –

Table 7 Purchasing values of a unit raw material from suppliers and reprocessing costs of a unit productat manufacturers for each quality (TL)

Quality options Suppliers Manufacturers

1 2 3 4 5 1 2 3

High 2 2.2 2.1 1.9 2.3 1 0.9 0.8

Mid 1.8 2 1.9 1.8 2 1.2 1.2 1.1

Low 1.6 1.8 1.6 1.6 1.8 1.4 1.3 1.25

Table 8 Raw material capacities and contract fees of suppliers (unit)

Quality options Suppliers

1 2 3 4 5

High 1,600 1,900 1,800 1,550 1,850

Mid 1,600 1,900 1,800 1,550 1,850

Low 1,600 1,900 1,800 1,550 1,850

Contract fees (TL) 500 550 480 450 530

sales-total profit). Total profit which includes total sales minus total transportation,purchasing, reprocessing costs and contract fees is 65% of the total sales of retailers.Total purchasing cost has the highest percentage in total profit as 11.88%.

Considering allowable money constraint (2000 TL) for supplier contracts, DMagreed and signed a contract with first, third, fourth and fifth suppliers about 1960TL. 7,700, 3,600 and 7,800 mixed-quality raw materials are transported to manu-facturers, respectively. As high quality 5,500, as mid quality 6,800 and as low qual-ity 6,800 unit raw materials are purchased from suppliers except third one (Fig. 6).At first manufacturer, 128 units of high quality, 335 units of mid quality and 540 unitsof low quality raw material are needed to reprocess and extra costs are shouldered.Totally at all manufacturers, 275 units of high, 680 units of mid and 2,040 units of lowquality raw material were reprocessed because of quality source production errors.7,700, 3,600 and 7,800 units are transported to DCs and produced in manufacturers,respectively. Finally, 193,946.80 TL profit is provided by selling 5,000, 4,500, 4,700and 4,900 units from retailers, respectively.

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Table 9 The results obtained by LINDO package program

Variable Value Variable Value Variable Value Variable Value

X1,2,1 1,600 Y1,3 7,700 P3,3 1,800 R3,3 1,500

X1,3,2 1,600 Y2,2 3,600 P4,1 1,550 C1 1

X1,3,3 1,600 Y3,1 7,800 P4,2 1,550 C3 1

X3,1,1 1,800 Z1,2 4,500 P4,3 1,550 C4 1

X3,1,2 1,800 Z1,4 3,300 P5,1 550 C5 1

X3,1,3 1,800 Z2,3 2,000 P5,2 1,850 S1 5,000

X4,1,1 750 Z2,4 1,600 P5,3 1,850 S2 4,500

X4,1,2 1,550 Z3,1 5,000 R1,1 1,275 S3 4,700

X4,3,1 800 Z3,3 2,700 R1,2 335 S4 4,900

X4,3,3 1,550 P1,1 1,600 R1,3 540 Objective 193,946.80TL

X5,2,1 550 P1,2 1,600 R2,1 1,075

X5,2,2 1,450 P1,3 1,600 R2,2 145

X5,3,2 400 P3,1 1,800 R3,1 40

X5,3,3 1,850 P3,2 1,800 R3,2 200

Table 10 Sub-objective values of the example

Sub-objectives Definition Value (TL) % of OBJ1

OBJ1 Total sales of retailers 296,500.00 100

OBJ2 Total transportation costs between suppliers and manufacturers 15,540.00 5.24

OBJ3 Total transportation costs between manufacturers and DCs 21,730.00 7.33

OBJ4 Total transportation costs between DCs and retailers 24,410.00 8.23

OBJ5 Total purchasing costs 35,230.00 11.88

OBJ6 Total reprocessing costs 3,683.25 1.24

OBJ7 Total contract fees 1,960.00 0.66

Total profit 193,946.80 65.22

4.2 Scenario analyses for managerial insights

When the results are examined, it is seen that each retailer’s sales affects supply chainnetwork and transportation flow directly. By changing the sales limitations of retailers,different scenarios can be applied to the model to examine the relations and trade-offsamong model variables. Also some scenario analyses are considered in this section tosee the effects of reprocessing rates and capacity of facilities on product distributionamounts from suppliers to retailers. The solution of the model without any scenario(current solution) and the results of the developed 4 main scenarios are compared inthis section. The main scenarios are listed below (Table 11):

The first scenario (A) is created to consider the effect of decreasing andincreasing sales of the retailers on the performance measures (agreed suppliers and

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Fig. 6 The optimal distribution network

Table 11 Main scenarios

Scenarios Definition

A Decreasing and increasing 10 times the retailers’ sales from −25 to +25%

B Decreasing and increasing 10 times reprocessing rates from −100 to +100%

C Decreasing and increasing 9 times the suppliers’ capacity from −15 to +30%

D Decreasing and increasing 8 times the capacity of manufacturers’ and DCssimultaneously from −20 to +20%

Table 12 Scenario A with sub-scenarios on decreased and increased sales

Changing (%) Suppliers OBJ1 OBJ2 OBJ3 OBJ4 OBJ5 OBJ6 OBJ7 Total profit

−25 1-3-4 222,375 11,220 15,757.5 18,277.5 25,832.5 2,713.5 1,430 147,144.00

−20 3-4-5 237,200 13,034 16,820 19,502 28,844 2,857.5 1,460 154,682.50

−15 2-3-5 252,025 15,651.5 18,005.5 20,750 31,810.5 3,040.8 1,560 161,206.70

−10 2-3-5 259,200 16,025 18,545 21,313 32,765 3,059.5 1,560 165,932.50

−5 1-3-4-5 281,675 14,680.5 20,468.5 23,210 33,236 3,502.5 1,960 184,617.40

Current 1-3-4-5 296,500 15,540 21,730 24,410 35,230 3,683.25 1,960 193,946.80

+5 1-3-4-5 311,325 16,399.5 22,971.5 25,630 37,426.5 3,726.22 1,960 203,211.30

+10 2-3-4-5 326,150 18,789 24,213 26,850 40,313 3,893.7 2,010 210,081.30

+15 1-2-3-5 333,765 19,385 24,785 27,452 41,405 3,928.5 2,060 214,749.50

+20 1-2-3-5 333,900 19,385 24,785 27,465 41,405 3,928.5 2,060 214,871.50

+25 1-2-3-5 334,350 19,385 24,785 27,439 41,405 3,928.5 2,060 215,347.50

sub-objective values). Agreement budget is also decreased and increased with the salesto find feasible solutions Scenario A, which is applied to reflect the real life problems,is given in Table 12 with its sub-scenarios.

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Fig. 7 Variation of suppliers according to scenario A

Table 12 gives the results of scenario A. The results presented in Table 12 have thefollowing implications: As expected, while the sales are increasing, total profit andtotal sales are also increasing. According to Table 12, when the sales are decreasedwith 25%, this reduction has a reflection on total profit with a 24.13% decline. Justthe opposite of this situation, an increase on sales with 25% created an increase ontotal profit with 11.03%. The same situation is also valid and observed for agreedsuppliers. We can explain this reason that increasing the sales systematically increasesthe number of suppliers which are agreed with because of meeting the demand. Sothe model selected to change current suppliers or agree with new suppliers to satisfyeach retailer under growing sales. Figure 7 shows the suppliers variation according tochange of sales. When the sales are decreased 25%, the model pulls the suppliers from4 to 3 via terminating fifth supplier contract. While the sales are increasing, the modelis changing the number of suppliers to increase the raw material capacity. The modelnever permits 5 suppliers because of budget constraint although increased sales. In allsub-scenarios, third and fifth (except only one) suppliers are constant. DM works withsecond and first suppliers at least, respectively.

In scenario B, we try to find the relation between reprocessing rates and total profitvia changing reprocessing rates. Table 13 gives the reprocessed quantities and totalprofit according to different reprocessing rates.

The increase on reprocessing rates results in a linear decrease in total profit. In allsub-scenarios, there is no third quality product reprocessing in second manufacturer.Maximum reprocessed quantity (2,960 units) is occurred in third manufacturer. If DMmakes an improvement on reprocessing rates from 20 to 100%, total profit is increas-ing 0.38, 76, 1.14, 1.52 and 1.9%, respectively. Figure 8 gives the total reprocessedquantities in each manufacturer. According to this figure, maximum reprocessing isactualized in third manufacturer, first and second manufacturers are coming frombehind. Too much transportation of third quality raw materials to third manufactureris the main reason of this situation.

Scenario C is applied to see the effects of suppliers’ capacity changing on eachsub-objective. Table 14 gives capacity changing rates and results of the scenario.

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Table 13 Scenario B with sub-scenarios on decreased and increased reprocessing rates

Changing (%) Reprocessed quantity Total profit (TL)

λq λ1 λ2 λ3 R11 R12 R13 R21 R22 R23 R31 R32 R33

−100 − − − − − – − − – − − – 197,630.00

−80 1 2 6 27.5 67 108 21.5 25 – 6 44 300 196,893.30

−60 2 4 12 67 134 216 43 26 – − 112 600 196,156.70

−40 3 6 18 100.5 201 324 64.5 39 – − 168 900 195,420.00

−20 4 8 24 134 204 432 86 116 – − 224 1,200 194,683.40

Current 5 10 30 127.5 335 540 107.5 145 – 40 200 1,500 193,946.80

+20 6 12 36 201 306 648 129 174 – − 336 1,800 193,210.10

+40 7 14 42 234.5 385 756 150.5 175 – − 392 2,100 192,473.50

+60 8 16 48 204 536 864 276 24 – 64 528 1,776 191,794.00

+80 9 18 54 243 603 972 310.5 − – 58.5 621 1,998 191,145.80

+100 10 20 60 255 670 1,080 345 30 – 80 660 2,220 190,497.50

0

500

1000

1500

2000

2500

3000

3500

-100% -80% -60% -40% -20% Current 20% 40% 60% 80% 100%

Changing on reprocessing rates

Rep

roce

ssed

qu

anti

ty

Manufacturer 1 Manufacturer 2 Manufacturer 3

Fig. 8 Reprocessed quantities in each manufacturer

Table 14 Scenario C with sub-scenarios on supplier’s capacity changing

Changing (%) Suppliers OBJ1 OBJ2 OBJ3 OBJ4 OBJ5 OBJ6 OBJ7 Total Profit

−15 1-2-3-4 272,156 14,918.7 19,605.8 22,369.19 32,582.1 3,213.59 1,980 177,486.60

−10 1-2-3-4 287,925 15,751.5 20,943.5 23,673.5 34,501.5 3,387 1,980 187,688.00

−5 1-3-4-5 296,500 15,661.8 21,710 24,430 35,666.2 3,537.7 1,960 193,534.20

Current 1-3-4-5 296,500 15,540 21,730 24,410 35,230 3,683.25 1,960 193,946.80

+5 1-3-4-5 296,500 15,418.8 21,730 24,410 34,983.7 3,700 1,960 194,297.10

+10 1-3-4-5 296,500 15,297 21,710 24,430 35,012.5 3,529.72 1,960 194,560.80

+15 1-3-4-5 296,500 15,175.8 21,710 24,430 35,020.6 3,387.18 1,960 194,816.40

+20 1-3-4-5 296,500 15,054 21,730 24,410 34,882 3,438.3 1,960 195,025.70

+25 1-3-4-5 296,500 14,932.8 21,710 24,430 34,744.1 3,489.2 1,960 195,233.90

+30 1-3-4 296,500 14,831.5 21,710 24,430 34,585 3,554.6 1,430 195,958.80

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Table 15 Scenario D with sub-scenarios on manufacturers’ and DCs’ capacity changing

Changing (%) Suppliers OBJ1 OBJ2 OBJ3 OBJ4 OBJ5 OBJ6 OBJ7 Total profit

−20 1-3-4-5 276,860 14,178 20,592 22,736 32,543 3,369.9 1,960 181,481.00

−15 1-3-4-5 293,280 15,216 21,879 24,149.5 34,701 3,684.6 1,960 191,689.90

−10 1-3-4-5 296,500 15,462 22,022 24,427 35,230 3,691 1,960 193,708.00

−5 1-3-4-5 296,500 15,501 21,885 24,409.5 35,230 3,687.15 1,960 193,827.30

Current 1-3-4-5 296,500 15,540 21,730 24,410 35,230 3,683.25 1,960 193,946.80

+5 1-3-4-5 296,500 15,579 21,575 24,410.5 35,230 3,679.35 1,960 194,066.20

+10 1-3-4-5 296,500 15,618 21,420 24,411 35,230 3,675.9 1,960 194,185.00

+15 1-3-4-5 296,500 15,657 21,265 24,411.5 35,230 3,688.35 1,960 194,288.20

+20 1-3-4-5 296,500 15,696 21,110 24,412 35,230 3,701.8 1,960 194,390.20

Decreasing of capacities is ended with 15% rate because of the infeasible solution.When the capacities of suppliers have a 10% reduction, total sales (OBJ1) is alsodecreasing to 287,925.00 TL with a 2.9% reduction. Every increment on capacityafter this point, total sales is stabled with 296,500.00 TL because of the maximumestimated sales constraint. Another inference from this scenario should be about num-ber of suppliers. The model behaves to minimize total contract fees while the capacitiesare increasing. DM agreed first, second, third and fourth suppliers when the supplier’scapacity has 10% reduction but after a 5% increment, DM selects fifth supplier, whichcontract fee is cheaper than second supplier, instead of second supplier. When capac-ities have a 30% increment, DM finds 3 suppliers (first, third and fourth) sufficient.

Scenario C also shows that increasing suppliers’ capacity does not affect trans-portation, reprocessing and purchasing costs directly, although it affects a little totalsales, total profit and contract fees. While OBJ2, OBJ3, OBJ4, OBJ5 and OBJ6 areconstant, OBJ1 and total profit have 8.94 and 10.41% increment, and OBJ7 has a27.77% decrease via changing capacity from −15 to +30%. Except suppliers’ capac-ity, capacities of manufacturers and DCs are also affect the transportation flow andobjective values. Table 15 shows these effects while the capacities of manufacturersand DCs are changing 8 times from −20 to +20% in step of 5%.

According to scenario D, objectives are not affected significantly by manufacturerand DC capacity increases. While the other parameters are constant, a 20% increaseon current situation leads to only 0.22% increment on total profit. Maximum increas-ing is occurred with 5.6% increment on total profit, when the capacities are increasedfrom −20 to −15%. Defined sales between minimum and maximum rates cause thissituation. The number of agreed suppliers is not affected by manufacturers’ and DCs’capacity increases (Table 15-suppliers column). The above-mentioned scenario results(except scenario B) on total profits can be followed clearly in Fig. 9. Accordingto Fig. 9, the most influential factor which affects total profit is sales of retailers(Scenario A). Scenario A gives both minimum and maximum profit values 154,682.50and 215,347.50 TL. Changing the capacity of manufacturers and DCs in other wordsScenario D is the most ineffective element on total profit. After −5% changing rate,Scenario C and D are watching closely.

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160000

170000

180000

190000

200000

210000

220000

-15% -10% -5% Current 5% 10% 15% 20%

Changing on rates

To

tal p

rofi

t (T

L)

Scenario A Scenario C Scenario D

Fig. 9 The changes in total profit values according to the scenarios

Further numerical experiments have been carried out in order to test the behaviorof the proposed model under large scale problems. Therefore we designed new testproblems with different sizes to test the proposed model applicability for real worldproblems and calculation time required for academicians. It is known that there are afew supply chain design models deal with real case studies in literature (Mula et al.2010). Also the linear programming or integer programming problems are difficult tosolve in reasonable time for these kinds of problems. In those situations, there are somemethods (such as decomposition techniques) that translate the problem into a new onethat is easier to solve. However, the linear programming problem solvers have levelmodeling capabilities to solve large and difficult problems in short calculation times(Tuzkaya and Onut 2009). In this study, the proposed model is solved by using B&Btechnique as a LINDO 11.0 solver, which is a powerful solver of complex and largeproblems and is a successful tool in supply chain optimization problems (Syarif et al.2002; Liang 2011). Solving the original problem including 499 constraints and 106variables related with 5 suppliers, 3 manufacturers, 3 DCs and 4 retailers requires only0.01 s. with a Pentium IV 1.66 GHz and 2 GB ram notebook. However, it is importantto realize the validity of the proposed model on solving larger problems in terms ofthe time requirements. A set of 7 different problems with the same input data struc-ture are solved to achieve a reasonable level of confidence about the performance andvalidation of the solution procedure. Parameter intervals used in producing randomproblems for experimental analysis can be seen in Table 16.

Different 7 test problems with dimensions are listed in Table 17. Number of sup-pliers and retailers are increased 7 times in steps of 100%. Number manufacturersand DCs are set between 3 and 7. Budget for contract fees is also increased 7 timesaccording to number of suppliers to keep the solution under feasible.

The obtained results and calculation times of these models are given in Table 18.As expected, each objective including total profit is also increased linearly while thesize of problem is enlarging (Fig. 10).

According to Table 18, enlarging the problem 8 times leads to an increase of 755%on total profit. In Table 17, contract fee column represents and restricts the maximumnumbers of potential suppliers that can be agreed. For example, in Table 18, for thelargest problem size there are 31 agreed suppliers of 40, and for the problem TP4there are 19 agree suppliers of 25. Figure 11 shows the variation of agreed suppliers

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Table 16 Parameter intervals used in generating random problems

Parameter Interval

Unit selling price 14–19 TL

Unit transportation costs between suppliers and manufacturers 0.7–1.1 TL

Unit transportation costs between manufacturers and DCs 1.1–1.5 TL

Unit transportation costs between DCs and retailers 1.1–1.7 TL

Unit purchasing costs 1.6–2.3 TL

Unit reprocessing costs 0.8–1.4 TL

Contract fees 450–550 TL

Supplier capacity 1,550–1,900 units

Manufacturer capacity 1.2 times of total supplier capacity

DCs capacity 1.2 times of total manufacturer capacity

Estimated sales of retailers 4,000–5,000 units

Reprocessing rates 5, 10, 30%

Table 17 Test problem sizes

Test problem Suppliers Manufacturers DCs Retailers Contract fee (TL) Variables Constraints

Current problem 5 3 3 4 2,000 106 499

TP 1 10 3 4 8 4,000 224 809

TP 2 15 4 4 12 6,000 328 1,378

TP 3 20 5 5 16 8,000 516 2,214

TP 4 25 5 6 20 10,000 660 3,546

TP 5 30 6 6 24 12,000 882 4,794

TP 6 35 6 7 28 14,000 1,057 5,726

TP 7 40 7 7 32 16,000 1,333 8,599

Table 18 Obtained results of different test problems

Test problems

Current TP1 TP2 TP3 TP4 TP5 TP6 TP7

Number of agreedsuppliers

4 8 11 16 19 23 27 31

OBJ1 296,500 602,200 884,400 1,252,900 1,561,400 1,880,600 2,125,700 2,481,000

OBJ2 15,540 32,720 48,450 68,000 84,820 100,020 113,420 134,420

OBJ3 21,730 45,040 63,140 91,060 113,420 137,920 154,120 179,200

OBJ4 24,410 45,370 63,680 91,340 116,640 141,780 160,920 189,940

OBJ5 35,230 70,555 98,930 141,110 175,075 210,595 236,390 276,310

OBJ6 3,683.25 7,063 9,910.5 14,467 16,763 19,782 23,144 27,118

OBJ7 1,960 3,990 5,540 7,980 9,600 11,560 13,590 15,550

Total profit 193,946.8 397,462 594,749.5 838,943 1,045,082 1,258,943 1,424,116 1,658,462

CPU time (s) 0.1 0.3 0.7 1 1.2 1.3 1.5 1.6

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1900121900241900361900481900601900721900841900961900

108190012019001321900144190015619001681900180190019219002041900216190022819002401900

OBJ1 OBJ2 OBJ3 OBJ4 OBJ5 OBJ6 OBJ7 Total Profit

Objectives

Ob

ject

ive

valu

es (

TL

)Current TP1 TP2 TP3 TP4 TP5 TP6 TP7

Fig. 10 The response of the objective values to changes in test problems

048

1216202428323640

Current TP1 TP2 TP3 TP4 TP5 TP6 TP7

Test problems

Val

ues

CPU time Number of Agreed Suppliers Number of Potential Suppliers

Fig. 11 Changes in agreed suppliers and CPU time according to each test problem

and CPU times according to each test problem. The obtained results also show thatthere are negligible increases in the calculation time. Increasing the number of vari-ables 12 times and the constraint equations 17 times increases the calculation timeonly 1.5 s. As a result, this kind of model can be quickly solved by powerful optimizeralgorithms as LINGO and GAMS. However, some linear programming problems thatinclude integer values and that have very limited solution space can cause inappro-priate calculation times. In that situation, decomposition or similar methods can beapplied to the problem (Tuzkaya and Onut 2009).

5 Conclusion

In this paper, we studied on product quality analysis within the context of a supplychain network and showed the general relation between quality and transportation flowon supply chain planning. After reviewing introduction and the most recent literature,

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mathematical formulation of developed model is presented to identify the characteris-tics of supply chain network optimization. The proposed model is illustrated throughan example by using a set of data reflecting a real life business situation. Differentlarge-scale problems are solved within an experimental design scheme in order to val-idate the performance and sensitivity of the proposed model. Different scenarios havebeen established in order to compare the possible performance of different scenariosand to allow the plan maker to take a better decision as well. Therefore, a unique toolis provided so as to understand how the system behaves under different situations. Ourmathematical model is developed base on a trade-off which is quality factors versus itsreprocessing and purchasing costs. We try to describe general trade-offs of the supplychains via giving a traditional conformance quality cost model and conflict resolutiondiagram.

To make the problem more realistic, we select the suppliers according to their con-tract fee with a money constraint when an agreement will actualize between potentialsuppliers and manufacturers. Also we consider multi-quality options for raw materialto pay attention to the quality factor. Quality is an important issue on production costsfor manufacturers. DMs always prefer high quality raw material but they also have toshoulder its high purchasing costs. At the same time, if DM wishes buying low qualityraw material to avoid high raw material costs, he/she could meet production errorswhich are sourced by quality factors such as breaking, cracking and misfit. So whilethe DM is avoiding raw material costs, he/she has to pay reprocessing costs accordingto the quality level of raw materials. Proposed model helps and solves the trade-offbetween cost and quality factors mentioned above.

Although, in most of them the structure of the supply chain network is consider-ably simplified with cost minimizing and static context, in this paper, we handled andplanned a network via maximizing the total profit while minimizing total transporta-tion costs, production costs, raw material costs and supplier contract fees. Followingare the significant findings which are obtained from proposed study:

• Increasing sales of all retailers together will result in transporting many more prod-ucts from suppliers to successor facilities. Thus, while the sales are increasing, totalprofit and total sales are also increasing.

• The increase on reprocessing rates results in a decrease in total profit linearly. Thus,DMs should reduce the impact of trade-offs between quality and cost factors orimprove the purchasing processes not be affected by low quality raw materials.

• While the suppliers’ capacity increases are not affected the transportation costbetween suppliers and manufacturers, it causes an increase about 10% on trans-portation costs between manufacturers, DCs and retailers.

• Increasing manufacturers’ and DCs’ capacity does not make any sense on agree-ments with suppliers.

• Emerging new decision variables for enlarging the network increases the modelcomplexity. However, this kind of model can be quickly solved by powerful opti-mizer algorithms.

• The most effective way is changing the demand (sales of retailers) to see signifi-cant changes in whole supply chain network. Then, suppliers’, manufacturers’ andDCs’ capacity changes affect the transportation flow, respectively.

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For further researches, to get better solutions with trade-offs, more mathemat-ical statements [inverse problems (Aster et al. 2004); regressions models (Weberet al. 2009) etc.] can be added the model formulation. For the future studies therobust optimization techniques (El Ghaoui 2003; Werner 2007) can be used toincrease the models’ robustness. This type of large sized models is NP-hard. Dueto exponentially increasing time with increasing problem sizes, a heuristic proce-dure to solve the mixed-integer programming model in a reasonable time needs to bedeveloped.

Acknowledgments The authors express their gratitude to the anonymous reviewers for their valu-able comments on the paper. In carrying out this research, the first and second authors were supportedby the Selcuk University Scientific Research Project Fund (BAP). These funds are hereby gratefullyacknowledged.

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Profit oriented supply chain network optimization