In all industrial firms where a large number of parts and components are supplied by different suppliers to the raw materiel stores. So it is necessary to keep a track of their performances and supplier ratings individually. The shortages of many basic raw materials and unfriendly attitudes on the part of suppliers can seriously jeopardize production units. So supplier evaluation comes a necessary task. The supplier selection in the supply chain is a multi-criteria problem for this a problem is formulated. The formulated problem is including three primary objectives such as minimizing the net purchase cost, minimizing the net transportation cost and minimizing the net late deliveries subject to realistic constraints regarding buyer’s demand, vendor’s capacity, budget allocation to individual vendor, Vendor’s quality of the items, vendor’s quota flexibility, purchase value of items etc. This paper consists of allocating the quota to suppliers from a set of pre-selected candidates. In this paper the Lexicographic Method is used and solved above three objectives on the priority basis. A real life example is also presented and solved by the MATLAB and results are shown in the paper.

1. http://www.iaeme.com/IJARET/index.asp 42 editor@iaeme.com
International Journal of Advanced Research in Engineering and Technology
(IJARET)
Volume 6, Issue 11, Nov 2015, pp. 42-49, Article ID: IJARET_06_11_004
Available online at
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=11
ISSN Print: 0976-6480 and ISSN Online: 0976-6499
© IAEME Publication
___________________________________________________________________________
A MULTI-OBJECTIVE APPROACH FOR
OPTIMUM QUOTA ALLOCATION TO
SUPPLIERS IN SUPPLY CHAIN
M. C. S. Reddy
Department of Mechanical Engineering,
University College of Engineering, Osmania University,
Hyderabad – 500 007, India
ABSTRACT
In all industrial firms where a large number of parts and components are
supplied by different suppliers to the raw materiel stores. So it is necessary to
keep a track of their performances and supplier ratings individually. The
shortages of many basic raw materials and unfriendly attitudes on the part of
suppliers can seriously jeopardize production units. So supplier evaluation
comes a necessary task. The supplier selection in the supply chain is a multi-
criteria problem for this a problem is formulated. The formulated problem is
including three primary objectives such as minimizing the net purchase cost,
minimizing the net transportation cost and minimizing the net late deliveries
subject to realistic constraints regarding buyer’s demand, vendor’s capacity,
budget allocation to individual vendor, Vendor’s quality of the items, vendor’s
quota flexibility, purchase value of items etc. This paper consists of allocating
the quota to suppliers from a set of pre-selected candidates. In this paper the
Lexicographic Method is used and solved above three objectives on the
priority basis. A real life example is also presented and solved by the
MATLAB and results are shown in the paper.
Key words: Multi Objective, Optimum Quota, Supply Chain, Linear
Programming.
Cite this Article: M. C. S. Reddy. A Multi-Objective Approach For Optimum
Quota Allocation to Suppliers in Supply Chain. International Journal of
Advanced Research in Engineering and Technology, 6(11), 2015, pp. 42-49.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=6&IType=11
1. INTRODUCTION
A supply chain is a set of facilities, supplies, customers, products and methods of
controlling inventory, purchasing, and distribution. The chain links suppliers and
customers, beginning with the production of raw material by a supplier, and ending

2. A Multi-Objective Approach For Optimum Quota Allocation To Suppliers In Supply Chain
http://www.iaeme.com/IJARET/index.asp 43 editor@iaeme.com
with the consumption of a product by the customer. In a supply chain, the flow of
goods between a supplier and a customer passes through several echelons, and each
echelon may consist of many facilities.
The objective of managing the supply chain is to synchronize the requirements of
the customers with the flow of materials from suppliers in order to strike a balance
between what are often seen as conflicting aims of high customer service, low
inventory, and low unit cost. The supplier selection problem deals with issues related
to the selection of right supplier and their quota allocations. In designing a supply
chain, a decision maker must consider decisions regarding the selection of the right
supplier and their quota allocation. The choice of the right supplier is a crucial
decision with wide ranging implications in a supply chain. Suppliers play an
important role in achieving the objectives of the supply management. Hence, strategic
partnership with better performing supplie should be integrated within the supply
chain for improving the performance in many directions including reducing costs by
eliminating wastages, continuously improving quality to achieve zero defects,
improving flexibility to meet the needs of the end-customers, reducing lead time at
different stages of the supply chain, etc. In designing a supply chain, decision makers
are attempting to involve strategic alliances with the potential supplier. Hence,
Supplier selection is a vital strategic issue for evolving an effective supply chain and
the right supplier play a significant role in deciding the overall performance. The
suppler selection is a complex problem due to several reasons. By nature, the supplier
selection in supply chain is a multi-criterion decision making problem. Individual
supplier may perform differently on different criteria. A supply chain decision faces
many constraints, some of these are related to suuplier’s internal policy and externally
imposed system requirements. (Manoj Kumara, Prem Vratb, R. Shankarc)..
A lot of research has been done on supplier selection problem. (Hongwei Ding,
Lyès Benyoucef, Xiaolan Xie) used discrete-event simulation for performance
evaluation of a supplier portfolio and a genetic algorithm for optimum portfolio
identification based on performance indices estimated by the simulation. This
simulation approach will not give exact optimal solution but only give near to optimal
solution. (Manoj Kumara, Prem Vratb, R. Shankarc) formulated a fuzzy goal
programming for vendor selection problem. This has the capability to handle realistic
situations in a fuzzy environment and provides a better decision tool for the vendor
selection decision in a supply chain. (Chuda Basnet & Janny M.Y.Leung) was given
multi-period inventory lot-sizing model for multiple products and multiple suppliers.
By this the decision maker needs to decide what products to order in what quantities
with which suppliers in which periods.
In this paper a Lexicographic Method is used to solve the multi-objective-
optimization problems. In Lexicographic Method multi-objectives are solved on the
basis of priority of objectives. The vendor selection in the supply chain is a multi-
criteria problem for this a problem has been formulated that includes three primary
objectives shown in the model formulation. A real life example is also presented and
solved by the MATLAB and results are shown in the paper.
2. MATLAB
MATLAB is an integrated technical computing environment that combines numeric
computation, advance graphics and visualization and a high level programming
language. The details of which is available in the web site “www.matworks.com”. In
this software number of toolboxes are available. Among these the SIMULINK tool

3. M. C. S. Reddy
http://www.iaeme.com/IJARET/index.asp 44 editor@iaeme.com
box has built-in simlp (), that implements the solution of a linear programming
problem and the optimization tool box has an almost identical function called linprog
() to solve the problem The simlp () is only suitable for small size problems but the
linprog () is suitable for all small size and complex problems. In this paper linprog ()
is used to solve the supplier selection problem in the supply chain (Brian R. Hunt,
Ronald L. Lipsman & Jonathan M. Rosenberg).
3. MULTI- OBJECTIVE LINEAR PROGRAMMING FOR
SUPPLIER SELECTION PROBLEM
Assumptions
1. Only same type of items is purchased from the vendors.
2. Demand of items is constant and known with certainty.
3. Quantity discounts are not taken into consideration.
4. No Shortage of the item is allowed for any vendor.
Index set
i index for supplier, for all i = 1, 2,..… N
l index for inequality constraints, for all l = 1, 2,..…, L
m index for equality constraints, for all m=1, 2,….., M
k index for objectives, for all k= 1, 2,……K
Decision variable
xi order quantity for the vendor i
Parameters
D = Aggregate demand of the item over a fixed planning period
N = Number of supplier competing for selection
pi = Price of a unit item of the ordered quantity xi from the supplier i
ti = Transportation cost of a unit item of the ordered quantity xi from the supplier i
di = Percentage of the late delivered units by the supplier i
Ci = Upper bound of the quantity available with supplier i
Bi = Budget allocated to each supplier
qi = Percentage of the rejected units delivered by the supplier i
Fi = Supplier quota flexibility for supplier i
F = Lower bound of flexibility in supply quota that a supplier should have
Ri = Supplier rating value for supplier i
PV = Lower bound to total purchasing value that a supplier should have
Model formulation
A multiple-objective linear programming problem can be written as follows:
Maximize / Minimize Zk(xi) = [Z1(xi), Z2(xi), Z3(xi) ….. ZK (xi)] Subject to
Gl(xi)  or  al
Hm(xi) = bm

4. A Multi-Objective Approach For Optimum Quota Allocation To Suppliers In Supply Chain
http://www.iaeme.com/IJARET/index.asp 45 editor@iaeme.com
xi  0
In the above formulation, xi are n decision variables, Z1 (xi), Z2 (xi), Z3 (xi) ZK (xi);
are k distinct objective functions, Gn are the inequality constraints and Hm are the
equality constraints. an and bm are the right hand side constants for inequality and
equality relationships, respectively. The supplier selection problem for three
objectives and to the set of system and policy constraints is formulated as follows:
Min. Z= (Z1, Z2, Z3)
Subjected to
xi =D ------------ (1)
xi  Ci ------------ (2)
xi  Bi ------------ (3)
qixi  Q ------------ (4)
Fixi  F ------------ (5)
Rixi  PV ------------ (6)
xi 0 and ------------ (7)
Z1 =pixi; Z2 =tixi; Z3 =dixi
Objective function (Z1) minimizes the net purchasing cost for all the items.
Objective function (Z2) minimizes the net transportation cost for all the items.
Objective function (Z3) minimizes the net number of late delivered items from the
Suppliers.
Constraint (1) puts restrictions due to the overall demand of items.
Constraint (2) puts restrictions due to the maximum capacity of the supplier.
Constraint (3) puts restrictions on budget amount allocated to the suppliers for
supplying the items
Constraint (4) puts restrictions on number of rejected items from the suppliers for
supplying the items
Constraint (5) incorporates flexibility needed with the suppliers’ quota.
Constraint (6) incorporates total purchase value constraint for all the ordered
quantities.
Constraint (7) puts restrictions on number of items from the supplier purchased should
be greater than zero.
4. LEXICOGRAPHIC METHOD
In the lexicographic method, the objectives are ranked in order of importance by the
problem designer. The optimum solution is then found by minimizing the objective
functions starting with the most important and proceeding according to the order of
importance of the objectives. Let the subscripts of the objectives indicate not only the
objective function number, but also the priorities of the objectives. Thus Z1 and Zk
denotes the most and least important objective functions respectively. The first
problem is formulated as
Minimize Z1 (xi)
Subjected to
Gl(xi)  or al

5. M. C. S. Reddy
http://www.iaeme.com/IJARET/index.asp 46 editor@iaeme.com
Hm(xi) = bm
xi  0
and its solution is X1i and Z1(X1i) = Z1 is obtained. Then second problem is
formulate as
Minimize Z2 (xi)
Subjected to
Gl(xi)  or al
Hm(xi) = bm
Z1 (xi) =Z1
xi  0
The solution of this problem is obtained as X2i and Z2(X2i) = Z2. This procedure
is repeated until all K objectives have been considered. Final values of the xi s will
give the optimum solution of the problem (Singiresu S.Rao)
5. A SAMPLE SUPPLIER SELECTION PROBLEM
Table 1 Supplier source data of the illustrative example
Vendor No. pi Rs. ti Rs di (%) Ci Units Bi Rs qi (%) Fi Ri
1 100 15 2 5500 12,50,000 3 0.05 0.86
2 300 10 3 16000 50,00,000 4 0.02 0.92
3 250 5 9 7000 17,50,000 1 0.07 0.98
4 350 20 5 5000 2,75,500 7 0.03 0.88
The effectiveness of the multi-objective linear approach for the supplier selection
problem presented in this paper is demonstrated through a real life data represented in
Table-1. The data relates to a realistic situation of a manufacturing sector dealing with
any type auto parts (Manoj Kumar, Prem Vrat & R.Shankar). The adopted situation
can easily be extended to any other industry. Those suppliers who successfully passed
the screening processes were eligible for procurement. A multi objective linear
program supplier selection problem is developed for the selection and the quota
allocations of the supplier from a list of four potential suppliers under uncertain
environments. The objective functions and constraint sets reflect the procurement
requirements for a purchased item in the supply chain. The three objectives, viz.
minimizing the net purchasing cost, minimizing the transportation cost and
minimizing the net late deliveries have been considered subject to few practical
constraints regarding demand of the item, suppliers’ capacity limitations, suppliers’
budget allocations, etc. We have considered a sample situation faced by a firm.
The supplier profiles shown in Table 1 represent the data set for the price quoted
(pi in rupees per unit); transportation cost (ti in rupees per unit); the percentage of late
deliveries di; suppliers’ capacities Ci; units; the budget allocations for the suppliers Bi;
the percentage rejections qi; suppliers’ quota flexibility Fi on a scale of 0–1 and
supplier rating Ri on a scale of 0–1. If the purchasers following 95.5% (2 Limits) of
the accepted policy, therefore maximum limit of rejections should not exceed 4.5%of
the demand. Hence the maximum rejections at purchasers are 25000 x 0.045=1125
units The least value of flexibility in suppliers’ quota and least total purchase value of
supplied items are policy decisions and depend on the demand. The least value of
flexibility in suppliers’ quota is given as F = FoD and the least total purchase value of
supplied items is given as PV =RD: If overall flexibility (Fo) is 0.03 on the scale of 0–

6. A Multi-Objective Approach For Optimum Quota Allocation To Suppliers In Supply Chain
http://www.iaeme.com/IJARET/index.asp 47 editor@iaeme.com
1, the overall supplier rating (R) is 0.92 on the scale of 0–1 and the aggregate demand
(D) is 25,000, then the least value of flexibility in suppliers’ quota (F) and the least
total purchase value of supplied items (P) are 750 and 23,000, respectively. Then the
formulated multi-objective linear supplier selection problem can be written as for
supplier source data of the illustrative real life example
Minimize Z1 =100x1+300x2+250x3+350x4
Minimize Z2 = 15x1+10x2+5x3+20x4
Minimize Z3 =0.02x1+0.03x2+0.09x3+0.05x4
Subjected to
x1+x2+x3+x4 =25000
x1  5500
x2  16000
x3  7000
x4  5000
100x1 1250000
300x2  5000000
250x3  1750000
350x4  275000
0.03x1+0.04x2+0.01 x3 +0.07x4  1125
0.05x1+0.02x2+0.07x3+0.03x4 750
0.86x1+0.92x2+0.98x3+0.88x4 23000
x1 , x2 ,x3,x4 0
6. RESULTS & DISCUSSIONS
In the above problem first priority is given to minimization of net purchase cost,
second priority is given to the net transportation cost and final priority is given to net
late deliveries. In lexicographic method objectives are solved on the basis of
priorities. The problem is solved by using MATLAB. The minimized values are
shown in table-2 and vender quota allocation is shown in the table-3.
Table 2 Minimized values of the three objectives
Minimized objectives Minimized values
Net purchased Cost RS. 60,50,000
Net Transportation cost Rs. 2,42,500
Net Late deliveries Items: 1115

7. M. C. S. Reddy
http://www.iaeme.com/IJARET/index.asp 48 editor@iaeme.com
Table 3 Suppliers quota allocation
Suppliers
No
Quota allocated to the
suppliers
Description regarding allocation
1 5500
The supplier-1 has taken quota up to the maximum
capacity because of supplier having less purchase
cost, less percentage of late deliveries, less
percentage of rejections, moderate flexibility,
moderate budget allocation etc.
2 12500
The supplier –2 has taken maximum quota than
other suppliers because of supplier having highest
budget allocation and able to supply maximum
quantity. In addition to this late deliveries with this
supplier is less and percentage of rejection,
transportation cost, percentage of vender purchase
rating are moderate.
3 7000
The supplier –3 has also taken quota up to the
maximum capacity because of supplier having
highest percentage of flexibility, highest vender
purchase rating, less percentage of rejections,
moderate purchasing cost, moderate budget
allocation etc.
4 0.0000
The supplier –4 has losses total his quota due to
highest purchase cost, highest transportation cost,
highest percentage of rejections, moderate
percentage of vender purchase rating , moderate late
deliveries, less budget allocation etc.
7. CONCLUSIONS
1. Supplier selection is an important goal in supply chain management. For this a
multi-objective linear programming problem is successfully formulated and
solved.
2. The Lexicographic Method is applied to solve the multi-objectives on the basis of
priorities.
3. A Sample Supplier Selection Problem is taken for real life problems and which
is solved by built–in function linprog () which is available in optimization tool
box of the MATLAB. Supplier quota allocations according their performances
are shown in table 3 and minimized values of the three objectives are shown in
table 2.
4. Any commercially available software such as MATLAB can also be used to solve
the proposed multi objective supplier selection problem.
REFERENCES
[1] Hongwei Ding, Lyès Benyoucef & Xiaolan Xie, “A Simulation Optimization
Approach Using Genetic Search For Supplier Selection” Proceedings of the
2003 Winter Simulation Conference, FRANCE.
[2] Manoj Kumar, Prem Vrat & R.Shankar “A Fuzzy Goal Programming
Approach For Vendor Selection Problem In A Supply Chain” , Computers &
Industrial engineering, Published by Elsevier Ltd. (www.sciencedirect.com),
September 2003,

8. A Multi-Objective Approach For Optimum Quota Allocation To Suppliers In Supply Chain
http://www.iaeme.com/IJARET/index.asp 49 editor@iaeme.com
[3] Singiresu S. Rao, Engineering optimization Technique: Theory and practice
3rd
Edition, New Age International (p) Ltd & Publishers.
[4] Pradip Kumar Krishnadevarajan Deepak Muthukrishnan S. Balasubramanian
and N. Kannan Supply Chain in India “2011-2015” – A Review: Challenges,
Solution Framework and Key Best Practices. International Journal of
Management, 6(10), 2015, pp. 135-149.
[5] Chuda Basnet & Janny M.Y. Leung, “Inventory Lot Sizing with Supplier
Selection”, June 2002 (white paper).
[6] D. Siva Kumar and Dr. Jayshree Suresh, Optimization of Supply Chain
Logistics Cost. International Journal of Management, 4(1), 2015, pp. 130 -
135.
[7] Brian R. Hunt, Ronald L. Lipsman & Jonathan M. Rosenberg “A guide to
MATLAB for Beginners and Experienced Users”, First South Asian Edition-
2002, CAMBRIDGE University Press.
[8] M. Chandra Sekhar Reddy and Talluri Ravi Teja. New Approach to Casting
Defects Classification and Optimization by Magma Soft. International
Mechanical Engineering and Technology, 5(6), 2014, pp. 25-35.