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### Goal Programming

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.

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 planning Manpower planning

Accounting analysis Marketing logistics

Advertising media scheduling Military strategies

Blood bank logistics Organizational analysis

Capital budgeting Personnel administration

Computer resource allocation Policy analysis

Decision support system planning Portfolio management

Economic policy analysis Production scheduling

Educational system planning Project management

Energy resources planning Quality control

Environmental protection Research and development

Facilities layout and location decisions Transportation logistics

Financial analysis Urban planning

Health care delivery planning Water resources planning

Inventory management

Example 13.1 Product Mix Problem

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

SUBJECT TO

5X1+6X2+3X3≤240

4X1+6X2+3X3≤400

X1,X2,X3≥0

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 Utilization

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.

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 agoal constraint.

The objective function for underutilization is specified as follows:

Minimize

P1is 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 overtime the 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

- Management’s 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.

Minimize

Purchase of Materials

- Managements final goal is that daily material purchases in excess of 400 pounds be minimized.

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 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 Goals

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.

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 Capacity

- The 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 constraint

The 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:

subject to:

Example 13.3 Deviational Variable Goal Constraint

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

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 Funding

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 park authority has located 50 acres of land for construction (although more land could be located if necessary).

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.

P4: The authority would like to meet the demands of the city council members for new facilities. However, this priority should be weighted according to the number of people expected to use each facility.

P5: The park authority wants to avoid securing land beyond the 50 acres presently available.

Funding Constraint

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 variable is the portion of the grant not spent.
- The deviational variable has been eliminated, the first priority goal is reflected in the objective function as follows:

Facility Use

- The Expected total weekly usage for all the facilities is formulated as

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

Land Requirements

- The land requirements for the various facility types are reflected in the equation 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

Facility Demand

The demand for facilities is shown in four goal constraint.

Example 13.5 Multiperiod Investment Problem

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.

Investment opportunities in savings certificates are available only once, at the beginning of the second year. Each dollar invested in certificates at the beginning of the second year will return $1.80 4 years later. Investment opportunities in real estate are available at the beginning of the fifth and sixth years. Each dollar invested in real estate will return $1.10 a year later.

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.

P2: The amount invested in savings certificates should be at least 25% 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;

System constraints

- Investment opportunity constraint in first year

Year 1: S1 +B1+I1=1,000,000

I1 is the amount of money not invested at the beginning of year 1.

- Second year, the investment will be S2,,B2, and C2.

Investment Opportunities Amounts Available

Year 2: S2+B2+C2+I2 =I1

Year 3: S3+B3+I3 =I2+1.2S1

Investment Opportunities Amounts Available

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

Goal Constraint

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,

P2: Since the amount invested in savings certificates should be at least 25% of the total investment, we should minimize from the following goal constraint:

P3: For the real estate investment, we should minimize

in the following goal constraint:

P4: Our goal is to maximize the total cash value by the end of the sixth year. The investments alternatives are S5,B4, and R6. By setting a cash value at an arbitrarily large number M ($500,000,00) and minimize , we will be maximizing the cash value. We can formulate as follows:

The complete goal programming model can be summarized as:

General Goal Programming Model

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.

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