GOAL PROGRAMMING

GOAL PROGRAMMING

INTRODUCTIONGoal programming may be used to solve linear programs with multiple objectives, with each objective viewed as a "goal".

In goal programming, di+ and di- , deviation variables, are the amounts a targeted goal i is overachieved or underachieved, respectively.

The goals themselves are added to the constraint set with di+ and di- acting as the surplus and slack variables.

For each priority level, the objective function is to minimize the (weighted) sum of the goal deviations.

One approach to goal programming is to satisfy goals in a priority sequence. Second-priority goals are pursued without reducing the first-priority goals, etc

APPLICATION AREAS FOR GOAL PROGRAMMING Accounting AgricultureEconomicsEngineeringFinanceGovernmentInternational ContextManagementMarketing

GOAL PROGRAMMING APPROACHStep 1: Decide the priority level of each goal.Step 2: Decide the weight on each goal. If a priority level has more than one goal, for each goal i decide the weight, wi , to be placed on the deviation(s), di+ and/or di-, from the goal.Step 3: Set up the initial linear program.Min w1d1+ + w2d2- s.t. Functional Constraints, and Goal Constraints Step 4: Solve this linear program. If there is a lower priority level, go to step 5. Otherwise, a final optimal solution has been reached

Step 5: Set up the new linear program.Consider the next-lower priority level goals and formulate a new objective function based on these goals. Add a constraint requiring the achievement of the next-higher priority level goals to be maintained. The new linear program might be: Min w3d3+ + w4d4- s.t. Functional Constraints, Goal Constraints, and w1d1+ + w2d2- = kGo to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.)

EXAMPLE 1

Goal Programming Model Formulation

Beaver Creek Pottery Company Example:

Minimize Z = $40x1 + 50x2subject to:1x1 + 2x2 40 hours of labor4x2 + 3x2 120 pounds of clayx1, x2 0where: x1 = number of bowls produced x2 = number of mugs produced

Adding objectives (goals) in order of importance, the company:

Does not want to use fewer than 40 hours of labor per day. Would like to achieve a satisfactory profit level of $1,600 per day. Prefers not to keep more than 120 pounds of clay on hand each day. Would like to minimize the amount of overtime.

Goal Constraint Requirements

All goal constraints are equalities that include deviational variables d- and d+.A positive deviational variable (d+) is the amount by which a goal level is exceeded.A negative deviation variable (d-) is the amount by which a goal level is underachieved.At least one or both deviational variables in a goal constraint must equal zero.The objective function in a goal programming model seeks to minimize the deviation from goals in the order of the goal priorities.

Goal Constraints and Objective FunctionLabor goals constraint (1, less than 40 hours labor; 4, minimum overtime): Minimize P1d1-, P4d1+Add profit goal constraint (2, achieve profit of $1,600): Minimize P1d1-, P2d2-, P4d1+Add material goal constraint (3, avoid keeping more than 120 pounds of clay on hand): Minimize P1d1-, P2d2-, P3d3+, P4d1+

Complete Goal Programming Model:

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Graphical InterpretationMinimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Figure 1Goal Constraints

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Figure 2The First-Priority Goal: Minimize

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Figure 3The Second-Priority Goal: Minimize

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Figure 4The Third-Priority Goal: Minimize

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

Figure 5The Fourth-Priority Goal: Minimize

Goal programming solutions do not always achieve all goals and they are not optimal, they achieve the best or most satisfactory solution possible.

Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to:

x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 x1 = 15 bowls x2 = 20 mugs d1 + = 15 hours

Goal ProgrammingComputer Solution Using QM for Windows Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0

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EXAMPLE 2The production manager of a company wants to schedule a weeks production run for two product P1 and P2, each of which requires the labor and materials as shown below:ProductP1 P2Labour Hours24Material M1 (kg)45Material M2 (kg)54

The weekly availability of resources is limited to 600 labor hours, 1000 kg of material M1 and 1200kg of M2 . The unit profit for P1 and P2 is Rs 20 and Rs 32, respectively.

Product P1 and P2 are, in fact, new models and are replacemets of the older ones which have been discontinued very recently. The manager would like to maximise profit but he is equally concerned with maintaining workforce of the division at nearly constant level in the interest of employee morale.

The workforce which consists of people engaged in production, sales, distribution, peons and other general staff, consisted of 108 persons in all. From a detailed study, it is known that production of one unit of P1 would maintain 0.3 person in the workforce, while one unit of P2 would maintain 0.75 person.

Had the production manager been considering only maximising profit, without regard to maintaining the workforce, he would do so by producing 167.67 units of P1 and 66.67 units of P2 (this can be checked by solving the problem as an LPP). On the basis of the available capacity, this would yield a profit equal to 167.67*20 + 66.67*32 or Rs 5486.67.

However, this would maintain 100.3 people in the workforce. The manager feels that probably he could increase the workforce requirement to desired level by accepting somewhat lower profit. In keeping with this, the following two goals are established: a profit of Rs 5400 per work and a workforce of 108 persons. Formulate and solve this as a goal programming problem.

Goal ProgrammingComputer Solution Using QM for Windows

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