Linear Goal Programming• Form of Multi-Objective optimization

Terminology• Objective - general statement of the desire of the decision

maker- Minimize Cost- Maximize Profit- Maximize Effectiveness- Minimize Loss

• Aspiration Level - Specific value associated with the desired or acceptable level of the objective- Used to measure achievement of the objective

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Goal - an objective in conjunction with an aspiration

level- Achieve at least $20,000 in profit- Reduce emissions by 50%

Goal Deviation - difference between what we aspire to and what we accomplish with respect to objective

- Can be high or low

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• Goals act as constraints in the GPAdvantages- Allows multiple objectives- Allows slack in the constraint–Disadvantages- Complexity of the “overall objective”- Must elicit goal values from Decision Maker- Often must elicit weights as well- Must find a way to homogenize these values

“Overall Objective” called the Achievement Function

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Achievement Function

• This is the objective of the LP we finally solve Forms(1) Minimize weighted sum of goal deviations(2) Minimize some function of the goal deviations

Often sum of weighted percent of goal levels(3) Minimize the worst deviation

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Goal Programming Formulation

• A company produces two products. Relevant information for each product is shown below:

Product 1 Product 2Labor required 4 hours 2 hoursContribution to profit $4 $2

The company has a goal of $48 in profits and incurs a $1 penalty for each dollar it falls short of this goal. A total of 32 hours of labor are available. A $2 penalty is incurred for each hour of overtime (labor over 32 hours) used, and a $1 penalty is incurred for each hour of available labor that is unused. Marketing considerations require that at least 7 units of product 1 be produced and at least 10 units of product 2 be produced. For each unit (of either product) by which production falls short of demand, a penalty of $5 is assessed.Formulate an LP that can be used to minimize the total penalty incurred by the company.

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• A company manufactures two products, A and B which must be processed through assembly and finishing departments. Assembly department has 90 hours available, finishing department can handle up to 72 hours of work. Manufacturing A require 6 hours in assembly and 3 hours in finishing department. Each B requires 3 hours in assembly and 6 hours in finishing department. If profit is Rs. 120 per product A and Rs 90 per product B, determine the best combination of A and B to realize a profit of Rs. 2100.

• If the company sets two equally ranked goals, one to reach a profit of Rs. 1500 and the other to meet a product A goal of 10. Find the optimal solution.

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• The manufacturing plants of an firm produces two products A and B. According to the past experience, production of either A or B requires an average of one hour in the plant. The plant has a normal production capacity of 40 hours a week. The marketing department reports that, because of limited sales opportunity, the maximum no. of A and B that can be sold are 24 and 30 respectively for the week. The gross margin from the sale of A is Rs. 80 whereas it is Rs. 40 from B.

The chairman of the company has set the following goals as arranged in order of their importance to the organization:

1) First, he wants to avoid any underutilization of normal production capacity (no layoffs of production workers).

2) Second, he wants to sell as many products as possible. Since the gross margin from the sale of A is twice the amount from B, he has twice as much desire to achieve sales for A as for B.

3) Third, the chairman wants to minimize the overtime operation of the plants much as possible.

Formulate this as a goal programming problem.

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Application Example

• A textile company produces two types of materials A and B. The material A is produced according to direct orders from furniture manufacturers. The material B, is distributed to retail stores. The average rate of production for material A and B are identical at 1000 metres/hour. By running two shifts the operational capacity of the plant is 80 hours per week.

• The marketing department reports that the maximum estimated sales for the following week is 70000 metres of material A and 45000 metres of material B. According to the accounting department the profit from a metre of material A is Rs. 2.50 and from a metre of material B is Rs. 1.50.

• The management of the company decides that a stable employment level is a primary goal for the firm. Therefore, whenever there is a demand exceeding normal production capacity, the management simply expands production capacity by providing overtime. However, management feels that overtime operation of the plant of more than 10 hours per week should be avoided because of the accelerating costs.

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The management has the following goals in the order of importance:1. The first goal is to avoid any underutilization of production capacity

(i. e. to maintain stable employment at normal capacity).

2. The second goal is to limit the overtime operation of the plant to 10 hours.

3. The third goal is to achieve the sales goals of 70000 metres of material A and 45000 metres of material B.

Formulate this as a goal programming problem to help the management for the best solution.

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The ABC advertising agency is trying to determine a TV advertising schedule for Priceler Auto Company. Priceler has three goals: – Goal 1: Its ads should be seen by at least 40 million high income men (HIM).– Goal 2: Its ads should be seen by at least 60 million low income people (LIP).– Goal 3: Its ads should be seen by at least 35 million high income women

(HIW).ABC can purchase two types of ads: those shown during cricket games and those during soap operas. At most, $ 600,000 can be spent on ads. The advertising costs and potential audiences of a one minute ad for each type are shown below. ABC must determine how many cricket ads and soap opera ads to purchase for Priceler.

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Millions of Viewers

Ad HIM LIP HIW Cost ($)

Cricket 7 10 5 100000

Soap opera 3 5 4 60000

• Each million exposure by which Priceler falls short of the HIM goal costs Priceler a $200,000 penalty because of lost sales.

• Each million exposure by which Priceler falls short of the LIP goal costs Priceler a $100,000 penalty because of lost sales.

• Each million exposure by which Priceler falls short of the HIM goal costs Priceler a $50,000 penalty because of lost sales.

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Suppose we modify the Priceler example by deciding that the budget restriction of $600,000 is a goal. If we decide that a penalty is assessed for each dollar by which this goal is unmet, then formulate the goal programming problem.

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