# Multiple-Objective Fractional TP with Impurities

Preetvanti Singh (Dayalbagh Educational Institute, India)
DOI: 10.4018/978-1-4666-5202-6.ch144

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## Introduction

The main objectives of this chapter are to present

• 1.

Literature review of fractional transportation problems for business optimisation,

• 2.

Multiple-objective fractional transportation problems with impurities,

• 3.

Summarized solution procedure for solving multiple-objective fractional transportation problem with impurities and

• 4.

Point out future research directions in this field.

In various business applications of optimization a function, characterized by one or several ratios like profit/capital, total tax/total public expenditure, may be maximized or minimized. Such optimization problems are commonly called Fractional Programming or Fractional Programs.

Mathematically, fractional program is of the form:

(1) subject to

, i = 1, 2, ..., m(2)
(3)

Here is a (n × 1) vector which is to be determined, and are real valued scalar functions defined on a subset of . Assume for all . For negative ,

Fractional programs arise in various businesses as well as in other areas like:

• 1.

Maximizing relative usage of raw material-stock cutting problem

• 2.

Maximizing return/risk

• 3.

Maximizing return/cost

• 4.

Minimizing expected cost/time

These ratios arise in portfolio selection, resource allocation, the analysis of financial enterprises and undertaking, finance, maintenance, Markov renewal programs, transportation, etc.

Transportation problems with fractional objective functions arise in many real life situations where an individual, or a group, or a commodity is faced with the problem of maintaining good ratios between some very important crucial parameters concerned with the transportation of commodities from certain sources to various destinations. This may be the situation for an enterprise board confronted with optimization of total actual/total standard transportation cost or total return/total investment on machines when acquired from factories i to workshops j. A fractional transportation problem can be formulated as (Swarup, 1966):

(5) subject to

(i=1,…, M)(6)(j=1,…, N)(7)

## Key Terms in this Chapter

Trade-Off: Generally the optimal solution for time transportation problem will not be unique and a large number of transportation solutions will consume the same optimal time of transportation. So out of these solutions, it is quite reasonable to choose the best solution with respect to a second criteria i.e., associated transportation costs. In between the two extremes of minimization of cost and time, there exist a number of situations where the transportation system decision maker would like a partial trade-off on cost to attain a certain degree of time advantage. This gives rise to the problem of obtaining certain trade-off relationships between the transportation cost and the transportation time.

Impurity: Impurity is undesirable substance commonly or naturally contained in something that lowers the quality or value of a commodity mixture, but (depending on its amount) may or may not make it unfit for its intended use.

Lexicographic Order: Lexicographic order is an order function - a way of sorting information. The name comes from the order used in a dictionary, where strings are compared in alphabetical order, from left to right.

Bottleneck: The phenomenon by which the performance or capacity of an entire system is limited by a single or limited number of components or resources is bottleneck. The term bottleneck is taken from the 'assets are water' metaphor. As water is poured out of a bottle, the rate of outflow is limited by the width of the conduit of exit—that is, bottleneck. By increasing the width of the bottleneck one can increase the rate at which the water flows out of the neck at different frequencies.

Fractional Transportation Problem: The fractional transportation problem is to transport various amounts of a single homogeneous commodity, that are initially stored at various origins, to different destinations in such a way that the total fractional transportation cost is minimum.

Multiple-Objective Fractional Transportation Problem: In real-life situations, the transportation problem (TP) usually involves multiple, conflicting, and incommensurate fractional objective functions. This type of problem is called multiple-objective transportation problem.

Fractional Problems: Various applications of optimizing (maximizing or minimizing) a function, characterized by one or several ratios like profit/capital are called Fractional Programming Problems or Fractional Programs.

Optimization: Finding an alternative with the most cost effective or highest achievable performance under the given constraints, by maximizing desired factors and minimizing undesired ones is referred to as business optimization. In comparison, maximization means trying to attain the highest or maximum result or outcome without regard to cost or expanse.

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