# MODM Methods

As discussed earlier, most of the MODM problems start with the following mathematical programming problem:

maximize/minimize Z1 = C1X maximize/minimize Z2 = C2X maximize/minimize Zn = CnX

subjecttoAX = b, X>0 (10)

where the values Z, i = 1, 2, …, n, represent the n objective functions.

1.5.2 Goal programming

Goal programming (GP), likely the oldest school of MCDM approaches, was developed in the 1950s as an extension of linear programming. In its simplest form, the method of GP assigns so-called aspiration levels (also called targets or goals) for the achieve­ment of different objective functions, and minimizes the deviations of actual achievement from the aspiration levels. It is important to stress here that that the term ‘‘goal’’ has the connotation of target when used in goal programming, whereas in the general MCDM context, the term ‘‘goal’’ is taken to represent a generalization of all criteria.

Consider Problem (10). Suppose Ti is the aspira­tion level for the objective function Zi. This means that the decision maker expects to achieve around Ti for the objective function Zi as given by the approxi­mate equation, ZixTi, which can be further written as Zi + d— — df = Ti, where d— and df are devia – tional variables measuring deviations of actual achievement below and above the aspiration level. They are usually called underachievement and overachievement deviational variables, respectively.

Because it is expected that the actual achievement will be as close to the aspiration levels as possible, the deviational variables are minimized in goal programming. Weights, reflecting the relative im­portance of the objective functions, can also be associated with the deviational variables during the minimization. Thus, one simple form of the goal programming objective function is given by the following relationship:

n

min E (wind,- + wipd+), (11)

i=1

where the values are the weight given to minimizing the selected deviational variable from the aspiration level Ti. This objective has to be minimized subject to the original constraints of Problem (1).

Here are two major approaches in goal program­ming: minimizing the weighted function of goals (discussed previously) and preemptive goal pro­gramming that avoids the weighted summation in the goal programming objective function. In pre­emptive GP, the weights used in the objective function are preemptive weights as defined in Section 3.4. Other mathematical forms, such as minimizing the maximum deviation, fractional goal programming, and nonlinear GP, are also available in the literature. GP has been criticized by several authors. For example, it can, in some circumstances, choose the dominated solution. The criticisms can be overcome by careful applications of the method. In one application, GP has been used to provide energy resource allocation for the city of Madras in India. The minimum energy requirements for cooking, water pumping, lighting, and using electrical appliances form the main constraints of the model. The objective functions pertain mainly to the needs for minimizing cost, maximizing effi­ciency, minimizing consumption of petroleum pro­ducts, maximizing employment generation, maximizing the use of locally available resources, minimizing emissions of oxides of carbon, sulfur and nitrogen, and maximizing convenience and safety. The trade-off information about the relative importance of these objectives has been obtained using the AHP.