Chapter 8 Goal Programming

Multi-Criteria Decision Making Tools LP models presented have always assumed a single objective Numerous managerial problems have more than one objectives In addition to minimizing total costs, a company might want to maximize market share Objectives may not be commensurate Objectives may be conflicting with each other Called multi-criteria decision making or simply MCDM technique Several approaches to MCDM models including multi-criteria linear programming, analytical hierarchy process or AHP, and goal programming (GP) technique We present goal programming technique as it builds on the development of LP

Goal Programming GP is an extension of LP Objectives function consists of various goals GP models are constructed the same way and solved graphically We begin with two decision variables in order to see the main differences between two models

A GP Example Furniture Company example will be used to illustrate the way a GP model is formulated This model was formulated as: Maximize Z = 40x1+50x2 subject to: x1+2x2 ≤ 40 hrs 4x1+3x2 ≤ 120 lb x1,x2 ≥ 0 Z represents total profit made from tables and chairs, given that each chair and table contribute $40 and $50 in profit, respectively Standard LP that has a single objective function

The Company’s Goals Developed the following goals in order of importance. 1. Does not want to use fewer than 40 hours of labor 2. Established a satisfactory profit level of $1,600 per day 3. Prefers not to keep more than 120 pounds of wood on hand each day 4. Would like to minimize the amount of overtime

The GP Formulation: Step 1 First step is to transform the linear programming model constraints into goals First goal: “not to use fewer than 40 hours of labor each day”. That is to avoid underutilization of labor To represent this goal, x1+2x2≤40, is reformulated as: Two new variables are called deviational variables. represents the amount of labor hours less than 40 and represents the amount of labor hours exceeding 40 shows labor underutilization and shows overtime x1 = 5, x2 = 10, a total of 25 hours of labor has been expended for production Substituting these values gives =15 hours and =0 Resulted in labor underutilization and obviously there is no overtime If one of the devotional variables is positive, the other one has to be zero

The GP Formulation: Step1-Cont. Suppose that x1= 10, x2= 20 Indicates a total of 50 hours have been used for production, or 10 hours is overtime = 0 and = 10 hours Impossible to use fewer than 40 hours of labor and more than 40 hours of labor at the same time Illustrate one of the fundamental characteristics of goal programming, which at least one or both of the deviational variables in a goal constraint must equal zero Under one condition, both deviational variables would equal zero if exactly 40 hours are used in production

GP formulation: Step 2 Reflect goal in our objective function. Let the objective function minimize labor underutilization Tends to minimize the value of as the first step before addressing any other goal Accomplished by creating a new form of objective function as to minimize P1 , where the symbol P1 designates as the priority one goal Objective function tends to make equal to zero or to the minimum possible amount

Formulating the 4th Goal Fourth-priority goal is also related to the labor constraint Forth goal states that management of the company desires to minimize overtime Overtime is represented by , we let objective minimize this deviational variable Objective function is to minimize P4 , where P4 designates as the priority four goal This goal will not be achieved until goals one, two, and three have been considered

Formulating 2nd Goal Second goal is to achieve a profit of $1,600 Original linear programming objective function was Z = 40x1+ 50x2 Above objective function is reformulated as: and represent the amount of profit less than $1,600 and higher than $1,600, respectively Let objective function minimize Logical to assume that the company would be willing to accept all profits in excess of $1,600 Minimizing means that the company hopes that will equal zero, which will result in at least $1,600 in profit Since this goal is at the second-priority level, the objective function is to minimize P2

Formulating 3rd Goal Third goal: “not to keep more than 120 pounds of wood on hand each day” Original LP wood constraint would become as: and represent the amount of wood on land less and higher than 120 pounds, respectively Let the objective function minimize the Minimize P3 , where P3 designates as the priority three goal

Complete Model Complete GP model is: Basic difference between this model and the original linear programming model is the objective function Terms in the objective function are not summed to a single value, z Because each term has a different unit of measure One of the terms minimizes deviation from the goal of the profit level and another from the goal of the labor hours level

The Solution Approach of GP Problems Solution involves solving a sequence of linear programs with different objective functions P1 goals are considered first, P2 goals second, P3 goals third, and so on Number of LP is determined by the number of priority levels Once a solution for the first formulation is achieved, the value of the deviational variable is added to the model as a constraint and the second-priority deviational variable becomes the new objective The solution process continues until all the priorities are implemented or we reach at this conclusion that a better solution cannot be reached This solution process can exactly be followed to solve a goal programming problem using Excel