Goal Programming

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Transcript of Goal Programming



INTRODUCTION The conventional linear programming models are based on the optimization of

single objective function. However , there are situation where multiple (possibly conflicting) objectives may be more appropriate. For example, aspiring politician may promise to reduce the national debt and simultaneously offer income tax relief. In such situations it may be impossible to find a solution that optimizes the

conflicting objectives. This problem can be overcome by a technique known as GOAL PROGRAMMING in which we try to seek a COMPROMISE OR EFFICIENT SOLUTION based on the relative importance of each objective. Goal programming , a powerful and effective methodology for the modeling,

solution and analysis of problems have multiple and conflicting goals and objectives, has often been cited as being the WORKHORSE of multiple objective function

HISTORY AND PHILOSOPHY The roots of GP lie in a paper by Charnes et al. in 1955 in which they deal with executive compensation methods. Recognizing that the method could be extended to a more general class of problems- that is, any quantifiable problems having multiple objective and soft as well as rigid constraint Charnes and Cooper later renamed the method goal programming when describing their classic two volume text Management Models and Industrial Applications of Linear programming. The two philosophical concepts that serve to best distinguish goal programming from conventional (i.e. single objectives) methods of optimization are the incorporation of flexibility in in constraint function and the adherence to the philosophy of satisficing as opposed to optimization. As a consequence of the principle of satisficing, the goodness of any solution to a goal programming problem is represented by achievement function rather than objective function of conventional optimization

Differences Between GP and LPLinear programming LP Goal programming GP

Goal and objectivesTargets or constraints Objective functions

One primary- to be maximized or minimizedInflexible, no deviations are allowed Maximize (minimize) the value of the primary goal

All objectives are ranked each with a targetFlexible, deviations are acceptable, constraints can be relaxed Minimize the sum of the undesirable deviations (weighted by their relative importance) Satisfaction Inefficient, few computer packages Few, but increasing

Theory Computer programs Applications

Optimization Very efficient, many packages Many and varied

CONCEPT OF REAL AND GOAL CONSTRAINTS The real constraints are absolute restrictions on the decision variables, while the goal constraints are the conditions one would like to achieve but are not mandatory. For example- a real constraint is given by

x1 + x2 = 3requires all possible values of x1 + x2 to always equal to 3. As opposed to this, a goal requiring x1+ x2 =3 is not mandatory, and we can choose values of x1 + x2 3 as well as x1 + x2 3. In a goal constraints, positive and negative deviational variables are introduced as follows

x1 + x2 + d1- - d1+ = 3note that if d1- > 0, then x1 + x2 < 3 and if d1+ > 0 then x1 + x2 >3.

Formulation of GP Problems1. Deviations: the amount away from the desired standards or objectives: Overachievement (d+i 0) vs. Underachievement (d-i 0) Desirable vs. Undesirable Deviations: (depend on the objectives)

Max goals () - the more the better - d+i desirable. Min goals () - the less the better - d-i desirable. Exact goals (=) - exactly equal - both d+i and d-i undesirable.

In GP, the objective is to minimize the (weighted) sum of undesirable

deviations (all undesirable d+i and d-i 0 ). For each goal, at least, one of d+i and d-I must be equal to "0". Goals (), or the "surplus" for Max Goals ().

Positive desirable deviations are in fact the indication of "slack" for Min

Formulation of GP Problems2. Ranking Objectives: (Ordinal/Cardinal/Mix of Two) Ordinal: only indicate the importance of goals by ranking order (Pi) - ("Absolute Priorities"). Cardinal: express importance of goals by assigning scaled weights (Wi) - ("Relative Preference").Different objectives may be measured in different scales [($)(%)]. Equal weights can be used if all goals are viewed equal important.

AN EXAMPLE OF FORMULATION OF GP Consider a single point, single pass turning operation in metal cutting wherein an optimum set of cutting speed and feed rate is to be chosen which balances the conflict

between metal removal rate and tool life as well as being within the restriction of horse power , surface finish and other cutting conditions. In developing the mathematical model of this problem, the following constraints will be considered : Constraint 1: maximum permissible feed f f1 where f1 max. permissible feed and f is the given feed in inches per revolution. Constraint 2: maximum cutting speed v v1 where v1 = DN/12 D = mean work diameter in inches N = maximum spindle speed available on the machine, rpm.

vf p1(3300)/ctdc where , and ct are constants. The depth of cut in inches,dc, is fixed at a given value. For a given p1, ct , dc and the right hand side of above constraint is constant. Hence vf constant Constraint 4 : non-negativity restrictions on feed rate and speed v,f 0

in optimizing metal cutting there are a number of optimality criteria which can be considered. Suppose we consider the following objectives in our optimization:i. Maximize metal removal rate ii. Maximize tool life (TL)

The expression for metal removal rate (mrr) isMRR = 12vfdc cu in/min The tool life for a given depth is given as:

TL = A/v1/nf1/n1Now the management considers that a given single point , single pass turning operation will be operating at an acceptable efficiency level if the following goals are met: i. The MRR must be greater than or equal to a given rate M1 ii. The tool life (TL) must equal to T1(min) Let us take MRR goal first, 12vfdc + d1- - d1+ = M1 Where d1- represents the amount by which the MRR goal is underachieved, and d1+ represents any overachievement of the MRR goal. Similarly the TL goal can be expressed as A/v1/nf1/n1 + d2- - d2+ = T1

Since the objective is to have a metal removal rate of atleast M1, the objective function must be set up so that a high penalty will be assigned to the underachievement variable d1-. No penalty will be assigned to d1+. To achieve a tool life of T1 penalties must be assigned to both d2- and d2+ so that both of these are minimized. Accordingly, the goal programming objective function for this problem is: Minimize: Z = P1d1- + P2(d2- + d2+) where P1 and P2 are non-numerical preemptive priority factors such that

P1>>P2 (i.e. P1 is infinitely larger than P2)To express the problem as linear goal programming problem, M1 is replaced by M2, where M2 = M1/12dc

The goal is replaced by T2, whereT2 = A/T1 and then logarithms are taken of the goal and constraints.

Minimize: Z = P1d1- + P2 (d2- + d2+) Subjected to: log v + log f + d1- + d1+ = log M2 1/n log v + 1/n1 log f + d2- + d2+ = log T2 log f log f1 log v log v1 log v + log f log constant log v, log f, d1-, d1+, d2-, d2+ 0 (MRR goal) (TL goal) (f1 constraint) (v1 constraint) ( H.P. constraint)

Here it is important to note that the last three inequalities are real constraint on feed, speed and H.P that must be satisfied all times, whereas the equation for MRR and TL are simply goal constraint

TYPES OF GOAL PROGRAMMING MODEL There are mainly two types of goal programming which are most

commonly used : i. Archimedean or Weighted Goal Programming ii. Non-Archimedean or Lexicographic or Preemptive Goal programming) Here it is important to note that although numerous variant of the GP

model has been developed such as: Fuzzy GP introduced by Zimmermann, MINMAX GP introduced by Flavell, Interactive GP, Imprecise GP, constrained regression GP etc. but Tamiz in his work A Review of Goal Programming and its Application, Annals of Operations Research 58 (1993) 39-53 showed that Weighted GP and the Lexicographical GP are the most popular and used variant of GP model. Between LGP and WGP, LGP is more extensively used.

ARCHIMEDEAN OR WEIGHTED GOAL PROGRAMMING In the weighted goal programming, the single objective function is the

weighted sum of the functions representing the goals of the problem. The general form of weighted goal programming can be expressed as follows:Minimize: Z = Subject to: = + + - =(wi di +wi di )

+ di- - di+ = bi

for all i

xj, di-, di+ 0

for all i and j

The above equation represents the objective function, which minimizes

the weighted sum of the deviational variables and the goal constraints, relating the decision variables (xj) to the targets (bi).

If the relative weights (wi+ and wi-) can be specified by the management, then the goal programming problem reduces to simple linear program. Unfortunately, it is difficult or almost impossible in many cases to secure a numerical approximation to the weights. In reality, goals are usually incompatible (i.e. incommensurable) and some goals can be achieved only at the expense of some other goals and therefore goal programming uses ordinal ranking or preemptive priorities to the goals.

PREEMPTIVE GOAL PROGRAMMING Preemptive goal programming is used when there are such major differences in the importance of the goals that it is not feasible to assign meaningful weights

to these goals to measure their relative importance. Therefore, the goals are listed in the order of their importance. Preemptive GP then begins with by focusing solely on one of the goa