A weighted additive fuzzy multi-objective model for the supplier selection
problem under price break in a supply chain
Professor
: Chu, Ta Chung
Student
: Nguyen Quang Tung
Student’s ID : M977Z235
1. Introduction
Supplier selection is one of the most critical activities of purchasing management in a supply
chain, because of the key role of supplier’s performance on cost, quality, delivery and service in
achieving the objectives of a supply chain. Supplier selection is a multiple-criteria decisionmaking (MCDM) problem that is affected by several conflicting factors. Depending on the
purchasing situations, criteria have varying importance and there is a need to weight criteria. In
practice, for supplier selection problems, most of the input information is not known precisely. In
these cases, the theory of fuzzy sets is one of the best tools for handling uncertainty. The fuzzy
multiobjective model is formulated in such a way as to simultaneously consider the imprecision
of information and determine the order quantities to each supplier based on price breaks. The
problem includes the three objective functions: minimizing the net cost, minimizing the net
rejected items and minimizing the net late deliveries, while satisfying capacity and demand
requirement constraints. In order to solve the problem, a fuzzy weighted additive and mixed
integer linear programming is developed. The model aggregates weighted membership functions
of objectives to construct the relevant decision functions, in which objectives have different
relative importance.
2. An integrated multiobjective supplier selection model
2.1 Notation definition
xij: the number of units purchased from the ith supplier at price level j
Pij: price of the ith supplier at level j
Vij: maximum purchased volume from the ith supplier at jth price level
D: demand over the period
V*ij: slightly less than Vij
mi: number of price level of the ith supplier
Yij: integer variable for the ith supplier at jth price level
Ci: capacity of the ith supplier
Fi: percentage of items delivered late for the ith supplier
Si: percentage of rejected units for the ith supplier
n: number of suppliers
2.2 Objective function
The model includes three following objective functions
Minimizing total purchasing cost
Minimizing total number of rejected items
Minimizing total number of late delivery items
2.3 Constraints
The above objective functions are subject to the following constraint
Constraint (9) is to make sure that at most one price level per supplier can be chosen, in
which Yij is defined as follow:
3. A fuzzy multiobjective programming
In a real situation, for a supplier selection problem, all objectives might not be achieved
simultaneously under the system constraints; the DM may define a tolerance limit and
membership function m (Zk(x)) for the kth fuzzy goals. The following model is proposed by
Zimmermann, 1978 with the purpose of finding a vector xT = [x1, x2, …, xn] to satisfy
Subject to
For fuzzy constraints
For deterministic constraints
Zimmermann (1978) extended his fuzzy linear programming approach to the fuzzy
multiobjective linear programming problem (10)–(13). He expressed objective functions Zk, k =
1…p and fuzzy constraints by fuzzy sets whose membership functions increase linearly from 0 to
1. In this approach, the membership function of objectives is formulated by separating every
objective function into its maximum and minimum values
The linear membership function for minimization goals is given as follows:
is obtained through solving the multiobjective problem as a single objective using, each time,
only one objective. is the maximum value of negative objective Zk
The linear membership function for the fuzzy constraints is given as
dr denotes subjectively chosen constants expressing the limit of the admissible violation of the
rth inequality constraints. It is assumed that the rth membership function should be 1 if the rth
constraint is well satisfied and 0 if the rth constraint is violated beyond its limit dr
In the supplier selection problem, fuzzy goals and fuzzy constraints have unequal importance to
DM, and the proper fuzzy DM operator should be considered. The weighted additive model can
handle this problem, which is described as follows
Where wk and
are the weighting coefficients that present the relative importance among the
fuzzy goals and fuzzy constraints. The following crisp single objective programming is
equivalent to the above fuzzy model:
Subject to
The linear programming software such as LINDO/LINGO is used to solve this problem.