Goal Programming. Goal programming which reflects the Simon's theory of “satisficing” is widely applied techniques for modeling modern decision-making problems.
Goal 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.
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Financial analysis Urban planning
Health care delivery planning Water resources planning
A 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+2X3
The goal constraints developed are as follows:
In order to reflect the possibility of underutilization of labor (as well as overtime), the original linear programming constraint is reformulated as
The variableare 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.
must always be zero in the solution.
or both is referred to as agoal constraint.
P1is the preemptive priority designation for this goal.
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.
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.
Formulating, the goal constraint
where 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 management’s desire to minimize the
Purchase of extra materials at a level of priority below those of the other three goals.
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:
associated with the next highest priority factor, P2 is minimized, and so on.
A 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.
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.
where X1 and X2 are the respective numbers of washers and dryers produced.
The production goal constraints are:
Extending 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
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.
A 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.
Eachfacility costs a certain amount, requires a certain number of acres, and has an expected usage. These parameters are summarized in the following table:
The authority has established the following list of prioritized goals:
P1:The authority must spend the total grant (otherwise the amount not spent will be returned to the federal government).
P2: The park authority desires that the facilities be used weekly by 20,000 or more people.
P3: If more land is acquired, the additional amount should be limited to 10 acres.
P5: The park authority wants to avoid securing land beyond the 50 acres presently available.
The cost requirement for the various facilities are shown in goal constraint:
where X1,X2,X3,X4 are number of facilities of each type to be constructed.
The deviational variables are the amounts of weekly underutilization or over utilization of the facilities. The priority 2 goal of minimizing under utilization is shown in objective function as
The deviational variables represent the amount by which the number of acres used is less than 50, , and the excess above 50 acres, . The park authority desires that the amount of land in excess of 50 acres be limited, to 10 acres.
This goal is reflected in the objective function by minimization of
at the priority 3 level. This goal and the priority 5 goal are shown in objective function as
The demand for facilities is shown in four goal constraint.
A investment firm has $1,000,000 to invest in four alternatives: stocks, bonds, savings certificates, and real estate. The firm wishes to determine the mix of investments that will maximize the cash value at the end of 6 years. Investment opportunities in stocks and bonds are available at the beginning of each of the next 6 year. Each dollar invested in stocks at the beginning of each year will return $1.20 ( a profitof $0.20) 2 years later, which can be immediately reinvested in any alternative. Each dollar invested in bonds at the beginning of each year will return $1.40 3 years later, which can be reinvested immediately.
The management of the firm wishes to determine the optimal mix of investments in the various alternatives that will achieve the following goals, listed in the order of their importance.
P1:: In order to maximize risk, the total amount invested in stocks and bonds should be limited to 40% of the total investment.
P3: Real estate is expected to be very attractive in the future. Thus, management would like to invest at least $300,000 in real estate.
P4: The total cash value by the end of the sixth year should be maximized.
Si= amount of money invested in stocks at the beginning of year i; i=1,2,3,4,5
Bi=amount of money invested in bonds
C2= amount of money invested in saving certificates in year 2
Ri=amount of money invested in real estate
Ii= amount of money held idle and not invested during year i;
Year 1: S1 +B1+I1=1,000,000
I1 is the amount of money not invested at the beginning of year 1.
Investment Opportunities Amounts Available
Year 2: S2+B2+C2+I2 =I1
Year 3: S3+B3+I3 =I2+1.2S1
Year 4: S4+B4+I4 =I3+1.2S2+1.4B1
Year 5: S5+R5+I5 =I4+1.2S3+1.4B2
Year 6: R6+I6=I5+1.2S4+1.4B3+1.8C2+1.1R5
We can formulate the four goal constraint as follows:
P1: The total amount invested in stocks and bonds,
should not exceed 40% of the total investment in all the alternatives,
P3: For the real estate investment, we should minimize
in the following goal constraint:
The complete goal programming model can be summarized as:
The general goal programming model can be formulated as follows:
where Pk is the preemptive priority weight (Pk>>>Pk+1) assigned to goal k (k=0 is reserved for system constraint),
are the numerical( differential) weights assigned to the deviational variables of goal i at a given priority level k, represent the negative and positive deviations, aij is the technological coefficients of xj in goal i, and bi, is the ith goal level.