Modeling Flexible Supplier Selection Framework

ORIGINAL ARTICLE

Modeling Flexible Supplier Selection Framework

Nilesh R. Ware • Surya Prakash Singh •

D. K. Banwet

Received: 7 March 2014 / Accepted: 11 July 2014 / Published online: 13 August 2014

� Global Institute of Flexible Systems Management 2014

Abstract Due to highly competitive market environment,

the supply chain network of business organization are not

only efficient but also flexible. The flexible supply chain of

business organization is greatly influenced by the suppli-

ers. Hence, the selection of suppliers has to be flexible

which not only take into account the quantitative factors

but also take into account qualitative factors. In this paper,

a novel attempt is conceptualized to model a flexible sup-

plier selection (FSS) problem by integrating the qualitative

and quantitative models for supplier selection problem.

A MINLP, a quantitative model, is considered in the pro-

posed FSS framework where factors such as lead time,

quality, supplier’s capacity and transportation cost are

considered. Similarly, in qualitative model factors such as

loyalty, technology adaptability, CSR and environmental

factors are considered. AHP and IRP, qualitative models,

have been applied and compared for supplier’s ranking. An

integrated ranking of AHP and IRP is used for modeling

FSS problem. Finally, Integration of quantitative and

qualitative model provides the set of deviation which

measure the level of flexibility from pure solution of sup-

plier selection problem by quantitative and qualitative

model. The methodology of FSS problem is presented and

demonstrated through an illustrative example of multi-

product, multi-source and multi-period case.

Keywords AHP � Flexible supplier selection �Interpretive ranking process �MINLP � Qualitative model �Quantitative model

Introduction

Supplier Selection problem deals with the selection of right

supplier with right products at right time to minimize the

total procurement cost while satisfying quantitative and

qualitative parameters. Every business organization largely

depends on the suppliers and therefore suppliers plays a

vital role to make an organization cost efficient and prof-

itable. This is done by right orders at the right place and at

the right time in the right quantity with quality. This not

only ensure the smooth production but also minimizes

unnecessary delays in production and delivery of finished

goods. In addition, suppliers should be loyal and ready to

adopt new technology to manufacture the products to the

procuring organization so that the total cost can be further

optimized. This can be achieved through a flexible supplier

selection process. In past, several work on supplier selec-

tion has been reported. However, those work either focused

on developing ranking of the suppliers (qualitative models)

or on developing mathematical models (quantitative mod-

els) to optimize a given objective function. Detailed review

on the research of supplier selection can be found from

Weber et al. (1991), Boer et al. (2001), Wadhwa and

Ravindran (2007), Bruno et al. (2012) and Ware et al.

(2012).

Ample work on developing qualitative models have

been done and available in literature. The main focus on

qualitative model is to prioritize the suppliers based on

certain criteria. Some of the criteria used in supplier

selection literature are loyalty, environment, availability,

CSR, environment, risk factor, suppliers profile, flexibility,

new technology awareness etc.

Some of the multi-criteria decision making tools widely

applied in ranking are AHP, ANP, TOPSIS, Fuzzy-AHP,

Fuzzy-TOPSIS, Fuzzy-DEMATEL, weighted method etc.

N. R. Ware (&) � S. P. Singh � D. K. Banwet

Department of Management Studies, Indian Institute of

Technology Delhi, New Delhi, India

e-mail: [email protected]

123

Global Journal of Flexible Systems Management (September 2014) 15(3):261–274

DOI 10.1007/s40171-014-0070-0

can be referred in the literature (Saaty 1980; Soukup 1987;

Bayazit 2006; Chia-Wei and Allen 2009; Demirtas and

Ustun 2009; Gencer and Gurpinar 2007; Sanjay Kumar

et al. 2009; Shahanaghi and Yazdian 2009; Wang et al.

2009; Chan and Chan 2004; Mithat et al. 2011). Interpre-

tive ranking process (IRP) includes the significance of both

intuitive and rational choice process to minimize cognitive

overload. IRP is more focused on interpretive logic given

by the expert’s judgment rather the weightage of one ele-

ment over other with intensity, thereby IRP making the

interpretive logic of the decision opaque the implementer

(Sushil 2001, 2005, 2009a). Sushil (2009b) presents basic

fundamental research on IRP which uses Interpretive

Matrix as a basic tool and pair comparison of interpreta-

tions in the matrix. This results into interpretive logic-

knowledge base and a dominance matrix. This process is

also illustrated by SAP-LAP interactions (Sushil 2009a).

All the dominance relationships and interpretations can be

diagrammatically represented in the form of an ‘Interpre-

tive Ranking Haleem et al. (2012) analyzed a critical

success factors of world class manufacturing system where

ISM and IRP model have been applied for a solution

methodology.

Quantitative models in supplier selection mostly

involves linear programming (LP), mixed-integer non-lin-

ear program (MINLP), mixed-integer linear program

(MILP), goal programming, fuzzy goal programming, data

envelopment analysis (DEA), etc. In some of the work,

researches have also applied meta-heuristic such as genetic

algorithm (GA) (Ding et al. 2005; Che 2010; Yeh and

Chuang 2011). Development of quantitative models in

supplier selection can be seen from literature (Dempsey

1978; Shipley et al. 1991; Weber and John 1993;

Ghodsypour and O’Brine 1998, Guneri et al. 2009, 1998;

Ghodsypour and O’Brien 2001; Ozgen et al. 2006; Thomas

and Srinivas 2008; Sanayei et al. 2008; Demirtas and Ustun

2008; Razmi and Rafiei 2010; Amin et al. 2011; Ware et al.

2014a). In the dynamic market scenario Ware et al. (2014b)

analyze the effect of demand variation on final supplier

assignments considered for multi-product, multi-source

and multi-period case.

In this paper, a framework of flexible supplier selec-

tion is presented where the quantitative model for sup-

plier selection is integrated with qualitative model. As a

result of integration, the solution obtained get deviated

from the pure solution of qualitative and quantitative

model. The deviation is measured for all possible cases

of integration which provide a deviation matrix. The

resulted deviation matrix capture the flexibility aspect of

the supplier selection. This will assist a decision maker

to finally decide the selection of suppliers which mini-

mize deviation from qualitative model and quantitative

model.

Modelling Supplier Selection Problem

Hence, we reviewed literature on qualitative and quanti-

tative model for supplier selection problem. ‘‘Qualitative

model (AHP and IRP)’’ section discuss AHP and IRP a

multi-criteria decision making process to solve qualitative

model. Whereas ‘‘Quantitative model (MINLP)’’ section

presents a mixed integer non-linear programming (MINLP)

as a quantitative model for solving supplier selection

problem.

Qualitative Model (AHP and IRP)

To handle qualitative factors, this paper consider two

techniques such as Analytical Hierarchy Process (AHP)

and Interpretive Ranking Process (IRP). AHP consider

weight vector for each criteria, sub-criteria for alternative

whereas, IRP focused on interpretive logic given through

decision judgment by experts. Both these methods gives

the final suppliers ranking which may or may not be vary.

By analyzing these ranking manager can interpret the most-

preferred, not-preferred and absolute/preferred suppliers

while implementing flexible supplier selection model.

Analytical Hierarchy Process (AHP)

One of the most used qualitative technique for supplier

selection is Analytical hierarchy process (AHP) (Saaty

1990). AHP is used in multi-criteria decision making

problem. It is developed by Saaty in 1970 which provided a

framework to handle with multiple-criteria, situations

involving intuitive, rational, qualitative (Bhutta and Huq

2002). Generally, the AHP has three levels: the goal, the

criteria and the alternatives, there could be sub-criteria

also. Selection of best supplier is the ultimate goal; the

criteria can be cost of ownership, product quality, after

sales services, etc. and the alternatives are the sets of

available suppliers. The AHP is used as a framework to

formulize the evaluation of tradeoffs between the con-

flicting selections criteria associated with the various sup-

pliers. It involves many intangible factors, but still requires

a logical and rational control of decisions (Nydik and Hill

1992). Akarte et al. (2001) developed a web-based AHP

system to evaluate the casting suppliers with respect to

eighteen criteria. Chan and Niraj (2007) developed an

interactive selection model with fuzzy extended AHP

considering risk factors to facilitate decision makers in

selecting global suppliers.

Saaty (1980) developed the following steps for applying

the AHP:

1. Define the problem and determine its goal, keep it as a

top level.

262 Global Journal of Flexible Systems Management (September 2014) 15(3):261–274

123

2. Structure the hierarchy from the top then the interme-

diate levels then to the lowest level which usually

contains the list of alternatives.

3. Construct a set of pair-wise comparison matrices (size

n 9 n) for each levels. The pair-wise comparisons are

done in terms of which element dominates the other.

This matrix is given by decision maker.

4. There are n (n - 1) judgments required to develop the

set of matrices, Reciprocals are automatically assigned

in each pair-wise comparison.

5. Hierarchical synthesis is now used to weight the

eigenvectors by the weights of the criteria and the sum

is taken over all weighted eigenvector entries corre-

sponding to those in the next lower level of the

hierarchy.

6. Having made all the pair-wise comparisons, the

consistency is determined by using the eigenvalue,

kmax, to calculate the consistency index, CI as follows:

CI = (kmax - n)/(n - 1), where n is the matrix size.

Judgment consistency can be checked by taking the

consistency ratio (CR) of CI with the appropriate

value. The CR is acceptable, if it does not exceed 0.10.

If it is more, the judgment matrix is inconsistent. To

obtain a consistent matrix, judgments should be

reviewed and improved.

7. Steps 3–6 are performed for all levels in the hierarchy.

Apart from many available criteria in the literature, this

paper considered following 8 criteria for supplier selection,

as shown in the following Table 1:

Interpretive Ranking Process (IRP)

To overcome the limitations of intuitive process and

rational choice process, IRP (Sushil 2009b) uses the

strengths of both the processes of decision making and

complementing the limitations of each one by the other.

Sushil (2001) has recommended the use of SAP–LAP

methodology for critical examination of case studies. A

generic interpretive framework for analyzing managerial

context with example of SAP-LAP linkages has been given

to illustrate the IRP process (Sushil 2009b). All the

required steps in hierarchy form for IRP is shown in Fig. 1.

The steps of the Interpretive Ranking Process (IRP) are

as follow:

1. Identify two sets of variables—one to be ranked with

reference to the other, e.g. Alternatives and Criteria,

Actions and Performance, Actors and Processes, & so

on.

2. Clarification of contextual relationship between the

two sets of variables.

3. Develop a cross-interaction matrix between the two

sets of variables.

4. Convert the binary matrix into an interpretive matrix

(Sushil 2005) by interpreting the interactions, i.e. ‘1’

entries in various cells.

5. Convert the interpretive matrix into an interpretive

logic of pair-wise comparisons and dominating

interactions matrix by interpreting the dominance of

one interaction over the other.

6. Develop ranking and interpret the ranks in terms of

dominance of number of interactions.

7. Validation of ranks derived.

Fig. 1 Interpretive ranking process (Sushil 2009b)

Table 1 Qualitative factors influencing on supplier selection used for

AHP

C1 Current technology awareness

C2 Corporate social responsibility

C3 Environmental management system

C4 Flexibility and responsiveness

C5 Loyalty and attitude

C6 Regulatory measures (Government & Legal Regulations)

C7 Financial status

C8 Firm market image

Global Journal of Flexible Systems Management (September 2014) 15(3):261–274 263

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8. Displaying ranking diagrammatically in the form of

an ‘Interpretive Ranking Model’.

9. Decision about ranks with interpretation and recom-

mendation for action.

10. Knowledge management for further use.

Quantitative Model (MINLP)

A MINLP to model quantitatively supplier selection is as

follow (Ware et al. 2014a). The development of the MINLP

for multi-period, multi-source and multi-products integrat-

ing quality and lead time is based on the following notations:

T Set of time period; 1, 2, …, t

S Set of supplier; 1, 2, …, s

P Set of product type; 1, 2, …,p

Xtsp Product quantity j supplied by supplier s in time

period t

UCtsp Unit cost of product j for supplier s in time

period t

TCts Total transportation cost for product (irrespective

of product type) by supplier s in time period t

Yts Supplier s assignment in time period t

SCtsp Capacity of supplier s for product p in time

period t

Dtp Demand for product p in time period t

UPCtsp Unit penalty cost to supplier s for product p in

time period t

UDCtsp Unit delay cost to supplier s for product p in time

period t

DLTtsp Delay lead time of supplier s for product p in

time period t

Qtsp Quality level in time period t of supplier s of part

p

Objective of the problem is to minimize overall cost for

entire period.

Fig. 2 Flowchart of flexible

supplier selection framework


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Objective 1: Unit cost of product;

Z1 ¼XT

t¼1

XS

s¼1

XP

p¼1XtspUCsp

þXT

t¼1

XS

s¼1TCtsYts ð1Þ

Transportation cost incurred only when the particular

supplier is assigned for any product for any time period.

Objective 2: Unit delay cost;

Z2 ¼XT

t¼1

XS

s¼1

XP

p¼1UDCtspDLTtspXtspYts ð2Þ

Objective 3: Unit penalty cost:

Z3 ¼XT

t¼1

XS

s¼1

XP

p¼1ð1� QtspÞUPCtspXtspYts ð3Þ

Hence, Total and final objective minimize function is

given by Eq. (4)

Z ¼ Z1 þ Z2 þ Z3 ð4Þ

Here, supplier capacity for a product for that time period

is taken into account,

Xtsp� SCtsp 8 t 2 T ; 8 s 2 S; 8 p 2 P ð5Þ

We consider here that all suppliers fulfill the order for

that product in respective time periodXS

s¼1Xtsp�Dtp 8 t 2 T ; 8 p 2 P ð6aÞ

XS

s¼1XtspYts ¼ Dtp 8 t 2 T ; 8 p 2 P ð6bÞ

Supplier assignment constant is a binary constant

Ys ¼1 for Xtsp [ 0

0

�8 s 2 S ð7Þ

We can set a quality standard and do not allowed the

product quality below a specified by the manufacturer

Qtsp�Qo 8 t 2 T ; 8 s 2 S; 8 p 2 P: ð8Þ

Finally, non-negativity constant given to the final

product p by the supplier s in time period t

Xtsp� 0 8 t 2 T ; 8 s 2 S; 8 p 2 P: ð9Þ

For this model, limited supplier capacity is to be considered,

all supplier combine fulfill the demand of product during that

period. Deterministic data set is considered here for this model.

Manufacturer can set the basic level of quality, below which

product will not be accepted. This MINLP formulation solved

by LINGO and results analysis have done. Implication of

results leads to the allocation of available suppliers for a

product and during given time period.

Modeling Flexible Supplier Selection Framework

In this section, a flexible supplier selection framework is pre-

sented which is an integration of the qualitative and quantita-

tive model. Quantitative model is primarily a mathematical

model of either LP or MINLP. While, qualitative model

basically prioritize the set of supplier or rank the suppliers

based on qualitative criteria using any MCDM approaches.

Table 2 Final priority weights for given suppliers using AHP

C1 C2 C3 C4 C5 C6 C7 C8 Final weights Supplier rank

0.074 0.097 0.184 0.12 0.122 0.129 0.14 0.136

S1 0.180 0.072 0.180 0.329 0.100 0.169 0.089 0.133 0.158 V

S2 0.065 0.113 0.074 0.105 0.249 0.062 0.134 0.120 0.115 VI

S3 0.178 0.234 0.122 0.192 0.201 0.278 0.165 0.243 0.198 I

S4 0.045 0.123 0.311 0.118 0.118 0.202 0.319 0.131 0.190 II

S5 0.251 0.206 0.135 0.176 0.205 0.077 0.116 0.180 0.160 IV

S6 0.281 0.252 0.178 0.081 0.128 0.212 0.176 0.193 0.181 III

Table 3 Dominating interaction matrix

Dominating ? S1 S2 S3 S4 S5 S6

Being dominated ;

S1 – C3, C4, C6, C8 C1,C3, C4 C1, C4, C5, C6 C7 C4

S2 C2, C5, C7 – C7 C8 C5 –

S3 C2, C5, C6 C1, C2, C4, C6, C8 – C1, C2, C7, C8 C2, C6, C8 C4, C5

S4 C3, C7, C8 C3, C7 – – C2, C3, C7 C3, C6, C7

S5 C5 C1, C2 C7 C1, C5, C8 – C1, C2, C3, C4, C5, C8

S6 C1, C2, C3, C5, C6, C7, C8 C1, C3, C6, C7 C2, C8 C1, C2 C6 –


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Quantitative model provides a quantified solution hav-

ing one objective function value under given sets of con-

straints. Similarly, qualitative model provides a ranking of

suppliers from one to last or from top to bottom signifying

the top prioritized to least prioritized supplier. But the pure

solution of qualitative and quantitative model contradicts

each other. Contradiction occurs when least preferred

suppliers from qualitative model get selected in quantita-

tive model due to the presence of certain constraints.

Contradiction can also be appeared when a supplier pro-

viding parts at economical rate is being ranked at last in the

qualitative model. This contradiction is very obvious due to

the presence of different attributes and their parametric

values. During ranking process using any MCDM quanti-

fication is done based on either Saaty’s scale or any other

likert scale whereas in developing quantitative (MINLP)

model past data or real time data are considered where

scaling is not done at all. This results in different supplier

selection in quantitative and qualitative models. To avoid

this contradiction, a flexible supplier selection framework

is proposed which minimizes this difference through a

deviational analysis. The deviation analysis is carried out

from the deviation matrix being generated from all possible

cases of integration of qualitative and quantitative models.

The proposed flexible supplier selection framework is

shown in Fig. 2. The heuristic approach to generate the

deviation matrix as a result of integration of quantitative

and qualitative models are discussed below. Following are

the steps of the heuristic approach.

Step 1 Prepare the list of highly preferred supplier from

qualitative model.

Step 2 Identify the set of least prioritized supplier from

qualitative model.

Step 3 Prepare the possible replacement of least preferred

supplier by highly preferred supplier.

Step 4 Reformulate the quantitative model by consider-

ing each possible replacement of least preferred

supplier.

Step 5 Solve the reformulated quantitative model at step

4.

Step 6 Generate deviation matrix from the original

solution of quantitative model and the solution

obtained from the Step 5 for each reformulated

quantitative model.

Step 7 Prepare set of flexible supplier and select the most

preferred.

An Illustrative Example

To demonstrate the proposed flexible supplier selection

(FSS) framework an example consisting of four suppliers

having eight criteria is considered here. Two qualitative

models namely AHP and IRP applied to rank the suppliers

as shown in ‘‘Analytical hierarchy process’’ and ‘‘Inter-

pretive ranking process’’ sections respectively. The

Table 4 Dominance matrix—ranking of alternatives with respect to criteria

S1 S2 S3 S4 S5 S6 Number dominating (D) Net dominance (D-B) Rank dominating

S1 – 4 3 4 1 1 13 -4 IV

S2 3 – 1 1 1 0 6 -11 V

S3 3 5 – 4 3 2 17 10 I

S4 3 2 0 – 3 3 11 -3 III

S5 1 2 1 3 – 6 13 4 II

S6 7 4 2 2 1 – 16 4 II

No. being dominated (B) 17 17 7 14 9 12 76 (Total interactions)

Fig. 3 Interpretive Ranking Models (a) and (b)


123

combined ranking of suppliers obtained from AHP and IRP

is considered finally. MINLP is used as a quantitative

model to solve the supplier selection problem as shown in

‘‘Mixed integer non-linear programme’’ section. Finally,

the proposed flexible supplier selection framework is

applied as shown in ‘‘Flexible supplier selection’’ section.

Analytical Hierarchy Process

Pair-wise comparison matrices generated in AHP is pro-

vided in Appendix (From Table 10 to Table 18). Final

weights calculated by AHP is shown in Table 2. Ranking

shows supplier S3 gets highest weight and considered as the

best supplier.

Thus, the final ranking of the supplier would be S3, S4,

S6, S5, S1, and S2.

Interpretive Ranking Process (IRP)

IRP consider here same criteria and alternatives as con-

sidered in AHP. Interpretive logic taken from the expert

opinion. Dominating interactions matrix generated as

shown in the Table 3 through interpretive logic for IRP

(Table 19).

For the final ranking of alternatives with respect to

criteria, dominance matrix drawn as shown in Table 4. The

ranking provided be the IRP is S3, S5, S6, S4, S2 and S1 or

S3, S6, S5, S4, S2 and S1.

Interpretive ranking models for the final interaction from

IRP is shown in Fig. 3. Upward and downward interaction

is shown in Fig. 3a, b. Table 5 show the rank comparison

for AHP and IRP. Here, IRP gives same/equal rank for the

suppliers S5 and S6 as second position therefore, two sep-

arate ranking of the suppliers is considered here as IRP

(I) and IRP (II). The ranking by IRP (I) is S3, S5, S6, S4, S2

and S1 and by IRP (II) is S3, S6, S5, S4, S2 and S1,

Based on the rankings provided by AHP, IRP (I) and

IRP (II), a combined ranking is considered which is shown

in Table 5. From Table 5, it is evident that the ranking of

all suppliers except supplier S4, S5, and S6 all are same.

This combined ranking provides the list of most-preferred

(S3) and least-preferred (S1 and S2) suppliers which will be

used in the proposed flexible supplier selection framework

(FSS).

Mixed Integer Non-Linear Programme (MINLP)

MINLP (Ware et al. 2014b) is applied as a quantitative

model to solve supplier selection problem. ‘‘Quantitative

model (MINLP)’’ section provides the details of MINLP.

To apply MINLP, this paper consider three different pro-

ducts for three time period for six suppliers. Data generated

for this model is random and taken from Ware et al.

(2014b).

Xtsp is the variable which indicates product quantity to a

particular supplier for a specific time period. Another

variable ‘‘Y’’ indicates that the final assignment of suppliers

for a particular product and given time period which is a

binary variable. Transportation cost incurred only when

supplier gets assigned. Table 6 shows the final results

obtained using MINLP model solved by LINGO. The

optimal objective function value is 173,580 for the given

problem.

Flexible Supplier Selection

Here, we apply the proposed FSS framework which is

discussed in ‘‘Modeling Flexible Supplier Selection

Framework’’ section. All steps for FSS framework is

described below.

Table 5 Final ranking comparison

Ranking AHP IRP (I) IRP (II) AHP \ IRP

suppliers preference

I S3 S3 S3 Most-preferred

II S4 S6 S5 Preferred

III S6 S5 S6

IV S5 S4 S4

V S1 S1 S1 Least-preferred

VI S2 S2 S2

Table 6 Final supplier assignment by MINLP model

Supplier P1 P2 P3 Y

T1 S1 0 0 500 1

S2 100 550 0 1

S3 0 250 200 1

S4 0 0 0 0

S5 400 0 0 1

S6 0 0 0 0

T2 S1 0 500 500 1

S2 400 600 100 1

S3 0 300 800 1

S4 450 100 0 1

S5 350 100 0 1

S6 0 0 100 1

T3 S1 0 350 300 1

S2 300 200 250 1

S3 0 250 200 1

S4 0 200 0 1

S5 300 0 0 1

S6 200 0 250 1


123

Step 1: Prepare list of highly preferred supplier from

qualitative model.

From both qualitative methods (AHP and IRP) S3 gets

ranking I. So, as per Table 5 supplier S3 considered as

most-preferred supplier

Most� preferred supplier : S3f g:

Step 2: Identify the set of least prioritized supplier from

qualitative model.

From both qualitative methods (AHP and IRP) S2 and S1

gets least ranking. So, as per Table 5 supplier S2 and S1

considered as least-preferred supplier

Least� preferred suppliers : S2 and S1f g:

Supplier S4, S5 and S6 are preferred and manager can

keeps their assignment for that time period and for a

particular product

Preferred suppliers : S4; S5 and S6f g:

Step 3: Prepare the possible replacement of least

preferred supplier by highly preferred supplier.

Here, all possible replacement strategy would be ana-

lyzed one by one. As it can be seen from step 2, S1 and S2

are least-preferred supplier. Therefore, in equal time period

the assignment of these suppliers has to be minimized.

Following are the possible replacement strategies.

Replacement Strategies 1 Replacing S2 in all time period

Set constraint) Y12 ¼ 0; Y22 ¼ 0; Y32 ¼ 0:

Replacement Strategies 2 Replacing S1 in all time period

Set constraint) Y11 ¼ 0; Y21 ¼ 0; Y31 ¼ 0:

Replacement Strategies 3 Replacing S1 and S2 in all

time period

Set constraint) Y12 ¼ 0; Y22 ¼ 0; Y32 ¼ 0; Y11 ¼ 0;

Y21 ¼ 0; Y31 ¼ 0:

Step 4: Reformulate the quantitative model by considering

each possible replacement of least-preferred supplier.

Here, based on replacement strategies implemented here

at step 3 following modes are reformulated.

Reformulated Model 1

This model does not considered S2 as it is least-preferred.

Z ¼ Z1 þ Z2 þ Z3 ð10ÞXtsp� SCtsp 8 t 2 T ; 8 s 2 S; 8 p 2 P ð11Þ

XS

s¼1Xtsp�Dtp 8 t 2 T ; 8 p 2 P ð12Þ

XS

s¼1XtspYts ¼ Dtp 8 t 2 T ; 8 p 2 P ð13Þ

Ys ¼1 for Xtsp [ 0

0

�8 s 2 S ð14Þ

Qtsp�Qo 8 t 2 T ; 8 s 2 S; 8 p 2 P ð15Þ

Xtsp� 0 8 t 2 T ; 8 s 2 S; 8 p 2 P ð16Þ

(17)

Since here, t = 3 the reformulated model 1 is solved

(tC1 ? tC2 ? tC3) times separately considering Y12 = 0,

Y22 = 0, Y32 = 0, Y12 = Y22 = 0, Y12 = Y32 = 0,

Y22 = Y32 = 0 and Y12 = Y22 = Y32 = 0.


This model does not considered S1 as it is least-preferred.

Z ¼ Z1 þ Z2 þ Z3 ð18ÞXtsp� SCtsp 8 t 2 T ; 8 s 2 S; 8 p 2 P ð19ÞXS


XS


Ys ¼1 for Xtsp [ 0

0

�8 s 2 S ð22Þ

Qtsp�Qo 8 t 2 T ; 8 s 2 S; 8 p 2 P ð23Þ

Xtsp� 0 8 t 2 T ; 8 s 2 S; 8 p 2 P ð24Þ

(25)



Y21 = 0, Y31 = 0, Y11 = Y21 = 0, Y11 = Y31 = 0,

Y21 = Y31 = 0 and Y11 = Y21 = Y31 = 0.


This model does not considered S2 and S1 as it is least-

preferred.

Z ¼ Z1 þ Z2 þ Z3 ð26ÞXtsp� SCtsp 8 t 2 T ; 8 s 2 S; 8 p 2 P ð27Þ


123

XS


XS


Ys ¼1 for Xtsp [ 0

0

�8 s 2 S ð30Þ

Qtsp�Qo 8 t 2 T ; 8 s 2 S; 8 p 2 P ð31Þ

Xtsp� 0 8 t 2 T ; 8 s 2 S; 8 p 2 P ð32Þ

(33)

(34)



Y22 = 0, Y32 = 0, Y12 = Y22 = 0, Y12 = Y32 = 0,

Y22 = Y32 = 0, Y12 = Y22 = Y32 = 0; and Y11 = 0,

Y21 = 0, Y31 = 0, Y11 = Y21 = 0, Y11 = Y31 = 0,

Y21 = Y31 = 0 and Y11 = Y21 = Y31 = 0.

Step 5: Solve the reformulated quantitative model at

step 4.

For all possible replacement strategies (in the illustrative

example it will be 21), the reformulated quantitative model

shown in step 4 is solved. The objective function value is

shown in Table 7 for all possible combinations.

Shaded regions in Table 7 indicate infeasibility due to

other factors such as limitation on supplier’s capacity etc.

Step 6: Generate deviation matrix from the original

solution (Z = 173,580) and the solution obtained from the

step 5 for each reformulated quantitative model. This is

shown in Table 8 given below:

Table 8 shows various deviations for different combi-

nation of replacement strategy. The Table 8 is seen as

flexibility in terms of deviations and decision maker based

on organization’s policy can decide the final supplier

selection having minimum number of least-preferred sup-

plier and minimum deviation.

Step 7: Prepare set of flexible supplier and select the

most preferred.

Decision maker such as company’s procurement man-

ager can make inference and take decision to choose the set

of suppliers (by eliminating not-preferred suppliers) using

Tables 7 and 8.

For an example, a situation where manager is flexible up to

10 % deviation from the original objective function without

considering the least-preferred suppliers. This can be

observed from Table 8 if suppliers S1 and S2 is not considered

for the time period T1 and T2, the goal of 10 % deviation can

be easily achieved. Thus, incorporating this strategy decision

maker can get final supplier selection as shown in Table 9.

Table 7 Objective function values for respective possible replacement models

Strategy T1 T2 T3 T1, T2 T1, T3 T2, T3 T1,T2,T3

S1 Reformulated

Strategy 1

Y11=0 Y21=0 Y31=0 Y11=Y21=0 Y11=Y31=0 Y21=Y31=0 Y11=Y21=Y31=0

Objective: Z 175800 -- 177105 181865 180191 183170 185390

S2 Reformulated

Strategy 2

Y12=0 Y22=0 Y32=0 Y12=Y22=0 Y12=Y32=0 Y22=Y32=0 Y12=Y22=Y32=0

Objective: Z 175278 178867 -- 181118 177855 -- --

S1, S2

Reformulated

Strategy 3

Y13=0 Y23=0 Y33=0 Y13=Y23=0 Y13=Y33=0 Y23=Y33=0 Y13=Y23=Y33=0

Objective: Z 178455 185977 179324 190852 184199 191721 196596

Optimum objective function value obtained from MINLP is Z = 173,580;

Table 8 Deviational matrix for objective function

T1 T2 T3 T1,T2 T1,T3 T2,T3 T1,T2,T3

S2 0.98 3.05 -- 4.34 2.46 -- -- S1 1.28 -- 2.03 4.77 3.81 5.52 6.80

S2, S1 2.81 7.14 3.31 9.95 6.12 10.45 13.26

Shaded regions indicate infeasibility


123

The deviational matrix can be seen as a sensitivity analysis

where decision maker can vary the flexibility from the lowest

deviation i.e. 0.98 % to the highest deviation i.e. 13.26 % as

shown in the Table 8. Similarly, the variation in the objective

function value due to the flexibility in the deviation can be

also determined from the Table 7. In this situation when

decision maker accept the flexibility of 10 % deviation then,

the objective function value would be 190,852 as compare to

the original objective function value of 173,580.

Conclusion and Future Scope

In this paper, novel attempt has been made to model FSS

framework where qualitative and quantitative factors are

combined by an integrated approach of AHP, IRP and

MINLP. AHP and IRP are applied to rank the suppliers

whereas MINLP is applied to solve the supplier selection

problem optimally. The proposed FSS framework

integrates the suppliers ranking and the optimal solution

obtained from MINLP to provide flexibility in generating

FSS where a least preferred suppliers are omitted from final

supplier assignment. To demonstrate the proposed FSS

framework, a randomly generated problem is considered

and tested. To show practical relevance of the proposed

FSS framework, a real case of the firm can also be taken for

study. In addition, the proposed FSS framework can be also

applied for stochastic supplier selection problem. In future,

other qualitative and quantitative models such as ANP,

PROMETHEE, DEMATEL, ELECTRE etc. can also be

considered as a future scope of the work in the proposed

FSS framework.

Appendix

See the Appendix Tables 10, 11, 12, 13, 14, 15, 16, 17, 18

and 19

Table 10 Comparative judgment matrix of criteria w.r.t overall objective

Goal C1 C2 C3 C4 C5 C6 C7 C8 Weights

C1 1 1/3 1/7 1/5 1/5 1/3 1/3 1/3 0.029

C2 3 1 1/3 1 3 1 3 1/3 0.126

C3 7 3 1 1 1 1 3 3 0.192

C4 5 1 1 1 3 1 5 3 0.194

C5 5 1/3 1 1/3 1 3 1/3 1 0.116

C6 3 1 1 1 1/3 1 1/3 3 0.123

C7 3 1/3 1/3 1/5 3 3 1 1/3 0.106

C8 3 3 1/3 1/3 1 1/3 3 1 0.114

Table 9 Final supplier

selection having deviation up

to 10 %

RESULT P1 P2 P3 Y

T1 S1 0 0 0 0

S2 0 0 0 0

S3 0 450 450 1

S4 0 350 0 1

S5 400 0 0 1

S6 100 0 250 1

T2 S1 0 0 0 0

S2 0 0 0 0

S3 0 300 800 1

S4 500 100 300 1

S5 350 1,000 0 1

S6 350 200 400 1

T3 S1 0 350 300 1

S2 300 200 250 1

S3 0 250 200 1

S4 0 200 0 1

S5 300 0 0 1

S6 200 0 250 1


123

Table 12 Comparative judgment matrix of alternatives w.r.t criteria corporate social responsibility (C2)

C2 S1 S2 S3 S4 S5 S6 Weights

S1 1 1/3 1/3 1 1 1/3 0.072

S2 3 1 1/3 1 1/3 1 0.113

S3 3 3 1 3 3 1/5 0.234

S4 1 1 1/3 1 3 1/3 0.123

S5 1 3 1/3 1/3 1 5 0.206

S6 3 1 5 3 1/5 1 0.252

Table 13 Comparative judgment matrix of alternatives w.r.t criteria environmental management system (C3)


S1 1 3 5 1/3 1 1/3 0.180

S2 1/3 1 1 1/3 1 1/5 0.074

S3 1/5 1 1 1 1 1 0.122

S4 3 3 1 1 5 3 0.311

S5 1 1 1 1/5 1 3 0.135

S6 3 5 1 1/3 1/3 1 0.178

Table 14 Comparative judgment matrix of alternatives w.r.t criteria flexibility and responsiveness (C4)


S1 1 3 5 3 1 3 0.329

S2 1/3 1 1/3 1 1 1 0.105

S3 1/5 3 1 1 1 5 0.192

S4 1/3 1 1 1 1 1 0.118

S5 1 1 1 1 1 3 0.176

S6 1/3 1 1/5 1 1/3 1 0.081

Table 15 Comparative judgment matrix of alternatives w.r.t criteria loyalty and attitude (C5)


S1 1 1/3 1/3 3 1/3 1/3 0.100

S2 3 1 1 1 5 1 0.249

S3 3 1 1 1 1 3 0.201

S4 1/3 1 1 1 1/3 1 0.118

S5 3 1/5 1 3 1 3 0.205

S6 3 1 1/3 1 1/3 1 0.128

Table 11 Comparative judgment matrix of alternatives w.r.t criteria current technology awareness (C1)


S1 1 1 3 3 1 1/5 0.180

S2 1 1 1/3 1 1/3 1/7 0.065

S3 1/3 3 1 5 1 1 0.178

S4 1/3 1 1/5 1 1/5 1/5 0.045

S5 1 3 1 5 1 3 0.251

S6 5 7 1 5 1/3 1 0.281


123

Table 16 Comparative judgment matrix of alternatives w.r.t criteria regulatory measures (Govt. & Legal Regulations) (C6)


S1 1 5 1/3 3 1 1/3 0.169

S2 1/5 1 1/5 1 1 1/5 0.062

S3 3 5 1 1 7 1 0.278

S4 1/3 1 1 1 1 5 0.202

S5 1 1 1/7 1 1 1/3 0.077

S6 3 5 1 1/5 3 1 0.212

Table 17 Comparative judgment matrix of alternatives w.r.t criteria financial status (C7)


S1 1 1/3 1 1/3 3 1/5 0.089

S2 3 1 3 1/5 1 1/3 0.134

S3 1 1/3 1 3 1/3 1 0.165

S4 3 5 1/3 1 7 3 0.319

S5 1/3 1 3 1/7 1 1 0.116

S6 5 3 1 1/3 1 1 0.176

Table 18 Comparative judgment matrix of alternatives w.r.t criteria firm market image (C8)


S1 1 3 1 1/5 1 1/3 0.133

S2 1/3 1 1/3 3 1 1 0.120

S3 1 3 1 5 3 1/3 0.243

S4 5 1/3 1/5 1 1/3 1 0.131

S5 1 1 1/3 3 1 3 0.180

S6 3 1 3 1 1/3 1 0.193

Table 19 Interpretive logic matrix for IRP

Dominating ? S1 S2 S3 S4 S5 S6

Being

dominated ;

S1 – Good flexibility with govt.

environmental norms

makes good image in

market

Available new

environmental

technologies with

flexibility

Follow govt. norms,

ability to handle

new issues,

Financially strong Variety easily

available

S2 Socially and

financially sound

– Financially sound Having good market

value

Reliable to

relationship

–

S3 Follow regulations

and social

responsibility

reliably

Financially weaker and non-

environmental friendly

– Financially good

image in society

and technically

sound

Market respect,

socially good

regulations

Reliable and

responsible for any

change

S4 More expenditure

on green issues

for good image

Green SCM budgets

allocation

– – Awareness

programs for

green technology

for society

Follows govt.

environmental

regulations and

financially good

S5 Positive attitude

for the service

Technology awareness useful

for corporate

Good in financial

ability

High market image

due to new

technologies and

humanity

– Financially week

S6 No customization Financially and legally

strong with emerging

environmental tools

Socially very strong

image

Follows useful

corporate

technologies

Most bounded to

the regulations

–


123

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Key Questions

i. How the flexible supplier selection framework is different

from pure qualitative or quantitative framework?

ii. Can we extend flexible supplier selection framework to a

flexible procurement framework?

iii. Can we extend or link flexible supplier selection framework

to the flexible enterprise?

Mr. Nilesh Rambhau Ware is Research Scholar

in Department of Management Studies (DMS) at

Indian Institute of Technology Delhi (IIT DELHI),

India. Prior to joining at IITD, he worked in

AISSMS’s College of Engineering Pune as

Assistant Professor. He completed his bachelor’s

in Mechanical from Government college of Engi-

neering Aurangabad (MAHARASHTRA) and

received his Master of Technology (M.Tech.) in Industrial Engi-

neering and Management from National Institute of Technology

Calicut (KERALA). He is a professional body member of GLOGIFT

Society, Decision Science Institute (DSI). He has published paper in

Management Science Letters, Expert System with Applications,

Industrial Engineering Journal; published his research work in various

referred proceedings. He has also attended some national and

international well known conferences. His research area is supply

chain management and operations research. Currently his research is

based on mathematical models and heuristic development for supplier

selection and evaluation problem. ([email protected]).

Dr. Surya Prakash Singh is an Associate Pro-

fessor in the DMS, IIT Delhi, India. He holds a

PhD from IIT Kanpur. He is also a Post-Doctoral

Fellow from SMA, NUS, Singapore. His research

interest lies in facility layout problems, heuristics,

and meta-heuristics. His work has been published

in leading international journals such as IJPR,

LNCS, IJAMT, EJM, RBR, IJRTE, and APMR.

He regularly reviews articles for many leading journals. His biogra-

phy appeared in Marquis, USA Who’s Who in Science and Engi-

neering, December 2007, and Who’s Who in the World, November

2010. Recently, he has been awarded Young Outstanding Faculty

Fellowship from IIT Delhi. ([email protected]).

Prof. Devinder Kumar Banwet is Emeritus

Professor, Operations & Supply Chain Manage-

ment at DMS, IIT Delhi, India. He is a graduate

mechanical engineer, a Masters in Industrial

Engineering & a Ph.D. from I.I.T. Delhi. He has

been former Head of the DMS, Dalmia Chair

Professor, Coordinator of ASRP (Applied Systems

Research Programme) and also Coordinator of the

Entrepreneurship Programme, both Interdisciplinary Research Pro-

grammes of IIT, Delhi. He is a member/life member of quite a few

other professional societies like ISTE, ISME, IIIE, GIFT and Systems

Society of India etc. His areas of research interest include Operations

Management, Supply Chain and Logistics Management, Project

Management, IT-enabled DSS, Industrial Systems Engineering,

TQM, Manufacturing Strategy, Technology Management, Materials

Management, Facilities Planning, OR Modelling, Telecom Systems

and Entrepreneurship Management. ([email protected]).


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