Goal Programming Model
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Goal ProgrammingGoal programming which reflects the Simon's theory of satisficing is widely applied techniques for modeling modern decision-making problems. The advantage of using goal programming over other techniques is with dealing with real-world decision problems is that it reflects the way manages actually make decisions.Goal programming allows decision maker to incorporate environmental, organizational, and managerial consideration into model through goal levels and priorities.
Goal Programming originally introduced by A. Charnes and W.W. Cooper and further developed by Y.Ijiri, S. M. Lee, and others is similar to linear programming concept. Goal programming can be employed in decision problems with a single goal (objective) and multiple sub goals, as well as in cases having multiple goals and sub goals. With in goal programming model, goals may be achieved only at the expense of other goals.
The goal programming necessitates the establishment of a weighting system for the goals such that lower-ranked (or weighted) goals are considered only after higher-ranked goals have been satisfied or have reached the point beyond which no further improvement is desirable. This weights can be ordinal or cardinal.
Goal programming is a form of linear programming, goal programming models must be formulated under the same limitations, assumptions, and conditions as linear programming models (linearity, divisibility, determinism, etc.).
Goal programming problems can also be solved by using the simplex method (in a modified form)
Goal Programming has been widely applied to decision problems in business organizations, government agencies, and nonprofit institutions. Examples include the following:
Academic administration planningManpower planningAccounting analysisMarketing logisticsAdvertising media schedulingMilitary strategiesBlood bank logisticsOrganizational analysisCapital budgetingPersonnel administrationComputer resource allocationPolicy analysisDecision support system planningPortfolio managementEconomic policy analysisProduction schedulingEducational system planningProject managementEnergy resources planningQuality controlEnvironmental protectionResearch and developmentFacilities layout and location decisionsTransportation logisticsFinancial analysisUrban planningHealth care delivery planningWater resources planningInventory management
Example 13.1 Product Mix ProblemA manufacturing company produces three products, 1, 2, and 3. The three products have resource requirements as follows:At present the firm has a normal production capacity of 240 hours of labor available daily and a daily supply of 400 pounds of material.Maximize Z= 3X1+5X2+2X3SUBJECT TO5X1+6X2+3X32404X1+6X2+3X3400X1,X2,X30
Labor (hr/unit)Materials (lb/unit)Profit ($/unit)Product 1543Product 2265Product 3432
This model has a single objective, profit maximization. Now considering the management developed set of goals, arranged in order of their importance to the firm.Because of labor relations difficulties, management desires to avoid underutilization of normal production capacity (i.e., no layoffs of workers).Management has established a satisfactory profit level of $500 per day.Overtime is to be minimized as much as possible.Management wants to minimize the purchase of additional materials because of handling and storage problems.The goal constraints developed are as follows:
Labor UtilizationIn order to reflect the possibility of underutilization of labor (as well as overtime), the original linear programming constraint is reformulated as The variable are referred to as deviational variables. They represent the number of hours less than (underutilization) and the number of hours exceeding (overtime) for the amount of production determined by the values of X1,X2,X3.
In the analysis, one of the deviational variable, must always be zero in the solution.
It is not possible to physically have both underutilization and over utilization at the same time.
A constraint in which we attempt to minimize or both is referred to as a goal constraint.
The objective function for underutilization is specified as follows:Minimize P1 is the preemptive priority designation for this goal.
The term reflects the fact that the first priority goal of the firm is to minimize , the underutilization of labor.
The first goal is to minimize ( drive it as close to zero as possible)
The minimization of overtimethe fact that management has ranked third is reflected in the objective function as follows:Minimize
P3 designates minimization of overtime, as the third priority goal.
Z represents a multidimensional function composed of various priority factors and associated income immensurable objective criteria.
Profit Level Managements second goal is to achieve the satisfactory profit level of $500. This goal constraint is formulated as
Where is underachievement of the profit goal and is the overachievement of the profit goal. The goal is reflected in the objective function by minimizing at the second priority level.
Purchase of Materials Managements final goal is that daily material purchases in excess of 400 pounds be minimized. Formulating, the goal constraintwhere is the over utilization of normal material requirement and is the purchase of extra materials. The objective function at the fourth priority level The last term reflects managements desire to minimize the Purchase of extra materials at a level of priority below those of the other three goals.
The goal programming model for the problem can be summarized as:
The last term reflects the managements desire to minimize the purchase of extra material at a level of priority below those of the other three goals.goal programming model can be summarized as follows:subject to
Solution of this problem requires that the deviations from the goals specified in the objective function be minimized. The value of the deviational variable associated with the highest preemptive priority (P1) must be first minimized to the fullest possible extent. when no further improvement is possible or desired for this goal, the value of the deviational variable ( ) associated with the next highest priority factor, P2 is minimized, and so on.The solution procedures is a modified simplex approach. Z represents the sum of unattained portions of each of the goals at different priority levels.
Example 13.2 Weighted GoalsA small manufacturing firm produces washers and dryers. Production of either product requires 1 hour of production time. The plant has a normal production capacity of 40 hours per week. A maximum of 24 washers and 30 dryers can be stored per week. The profit margin is $80 for a washer and $40 for a dryer. The manager has established the following goals, arranged in order of their priority.
P1: Avoid underutilization of normal production capacity.P2: Produce as many washers and dryers as possible. However, since the profit margin for a washer is twice that for a dryer, the manager has twice as much desire to achieve the production of washers as to achieve the production of dryers.P3: Minimize overtime as much as possible.
Production CapacityThe first goal constraint reflects the production time requirements for both products.where X1 and X2 are the respective numbers of washers and dryers produced. The deviational variable, , reflects underutilization of the normal production capacity of 40 hours per week, while overtime. Priority goals 1 and 3 can be reflected as
Storage constraintThe production goal constraints are:
The first goal constraint represents the underachievement of the production goal for washers.The second goal constraint is the underachievement of the production goal for dryers.The production goals have been eliminated, because these goal levels represent absolute maximum values (i.e., storage capacities) not to be exceeded.
This type of constraint is referred to as system constraint because deviation in the positive and/or negative direction is prohibited. Second priority goal is reflected in the objective function as follows:
The goal programming model is formulated as:
Example 13.3 Deviational Variable Goal ConstraintExtending from the previous problem, the added goal that overtime not exceed 10 hours per week, if possible. The priority level of this new goal places it between the old P1 and P2 levels.
The production goal constraint:
Our new goal is that overtime be restricted to 10 hours, which is formulated as
Another way to formulate the same goal constraint in terms of decision variables is adding the allowed overtime of 10 hours to the original production requirement goal as follows:The new second priority goal specifies that the amount of overtime in excess of 10 hours is to be minimized. This goal is not incompatible with the goal of minimizing overtime.
The new goal programming model is Minimize
Example 13.4 Recreational Facility FundingA city parks and recreational authority has been given a federal grant of $600,000 to expand its public recreational facilities. Four different types of facilities have been requested by city council members speaking for their constitutes: gymnasiums, athletic fields, tennis courts, and swimming pools. The total demand by various neighborhoods has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools.
Each facility costs a certain amount, requires a certain number of acres, and has an expected usage. These paramete