1. Chapter 8
Goal Programming

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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 P , where P4 designates as the priority four goal
This goal will not be achieved until goals one, two, and three have been considered

10. 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

11. 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

12. 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

13. 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