When a preemptive goal programming problem involves only two decision variables, the optimal solution can be found graphically. For example, suppose HIW is the highest-priority goal, LIP is the second-highest, and HIM is the lowest. From Figure 14, we find that the set of points satisfying the highest-priority goal (HIW) and the budget constraint is bounded by the triangle ABC. Among these points, we now try to come as close as we can to satisfying the second-highest-priority goal (LIP). Unfortunately, no point in trian-gle ABC satisfies the LIP goal. We see from the figure, however, that among all points sat-isfying the highest-priority goal, point C (C is where the HIW goal is exactly met and the budget constraint is binding) is the unique point that comes the closest to satisfying the LIP goal. Simultaneously solving the equations
5x1 4x2 35 (HIW goal exactly met) 100x1 60x2 600 (Budget constraint binding)
we find that point C (3, 5). Thus, for this set of priorities, the preemptive goal pro-gramming solution is to purchase 3 football game ads and 5 soap opera ads.
Goal programming is not the only approach used to analyze multiple objective decision-making problems under certainty. See Steuer (1985) and Zionts and Wallenius (1976) for other approaches to multiple objective decision making under certainty.
Using LINDO or LINGO to Solve Preemptive
Goal 1 (HIM) can be met, so LINDO reports an optimal z-value of 0. We now want to come as close as possible to meeting goal 2 while ensuring that the deviation from goal 1 remains at its current level (0). Using an objective function of s2(to minimize goal 2) we add the constraint s1 0 (to ensure that goal 1 is still met) and ask LINDO to solve
min z s2
s.t. 7x1 3x2 s1 s1 40 (HIM constraint) s.t. 10x1 5x2 s2 s2 60 (LIP constraint) s.t. 5x1 4x2 s3 s3 35 (HIW constraint) s.t. 100x1 60x2 s3 s3 600 (Budget constraint)
s1 0 All variables nonnegative
Because goals 1 and 2 can be simultaneously met, this LP will also yield an optimal z-value of 0. We now come as close as possible to meeting goal 3 (HIW) while keeping the deviations from goals 1 and 2 at their current levels. This requires LINDO to solve the following LP:
min z s3
s.t. 7x1 3x2 s1 s1 40 (HIM constraint) s.t. 10x1 5x2 s2 s2 60 (LIP constraint) s.t. 5x1 4x2 s3 s3 35 (HIW constraint) s.t. 100x1 60x2 s3 s3 600 (Budget constraint)
s1 0 s2 0 All variables nonnegative
Of course, the LINDO (or LINGO) full-screen editor makes it easy to go from one step of the goal programming problem to the next. To go from step i to step i 1, simply mod-ify your objective function to minimize the deviation from the i 1 highest-priority goal and add a constraint that ensures that the deviation from the ith highest-priority goal remains at its current level.
R E M A R K S 1 The optimal solution to this LP is z 5, x1 6, x2 0, s1 0, s2 0, s3 5, s1 2, s2 0, s3 0, which agrees with the solution obtained by the preemptive goal programming method. The z-value of 5 indicates that if goals 1 and 2 are met, then the best that Priceler can do is to come within 5 million exposures of meeting goal 3.
2 By the way, suppose we could only have come within two units of meeting goal 1. When solv-ing our second LP, we would have added the constraint s1 2 (instead of s1 0).
3 The goal programming methodology of this section can be applied without any changes when some or all of the decision variables are restricted to be integer or 0–1 variables (see Problems 11, 12, and 14).
4 Using LINGO, the goal programming methodology of this section can be applied without any changes even if the objective function or some of the constraints are nonlinear.
P R O B L E M S
Group A
1 Graphically determine the preemptive goal progamming solution to the Priceler example for the following priorities:
a LIP is highest-priority goal, followed by HIW and then HIM.
b HIM is highest-priority goal, followed by LIP and then HIW.
c HIM is highest-priority goal, followed by HIW and then LIP.
d HIW is highest-priority goal, followed by HIM and then LIP.
2 Fruit Computer Company is ready to make its annual purchase of computer chips. Fruit can purchase chips (in lots of 100) from three suppliers. Each chip is rated as being of excellent, good, or mediocre quality. During the coming year, Fruit will need 5,000 excellent chips, 3,000 good chips, and 1,000 mediocre chips. The characteristics of the chips purchased from each supplier are shown in Table 57. Each year, Fruit has budgeted $28,000 to spend on chips. If Fruit does not obtain enough chips of a given quality, then the company may special-order additional chips at $10 per excellent chip, $6 per good chip, and $4 per mediocre chip.
Fruit assesses a penalty of $1 for each dollar by which the amount paid to suppliers 1–3 exceeds the annual budget.
Formulate and solve an LP to help Fruit minimize the penalty associated with meeting the annual chip requirements. Also use preemptive goal programming to determine a purchasing strategy. Let the budget constraint have the highest priority, followed in order by the restrictions on excellent, good, and mediocre chips.
3 Highland Appliance must determine how many color TVs and VCRs should be stocked. It costs Highland $300 to purchase a color TV and $200 to purchase a VCR. A color TV requires 3 sq yd of storage space, and a VCR requires 1 sq yd of storage space. The sale of a color TV earns Highland a profit of $150, and the sale of a VCR earns Highland a profit of $100. Highland has set the following goals (listed in order of importance):
Goal 1 A maximum of $20,000 can be spent on purchas-ing color TVs and VCRs.
Goal 2 Highland should earn at least $11,000 in profits from the sale of color TVs and VCRs.
Goal 3 Color TVs and VCRs should use no more than 200 sq yd of storage space.
Formulate a preemptive goal programming model that High-land could use to determine how many color TVs and VCRs to order. How would the preemptive goal formulation be modified if Highland’s goal were to have a profit of exactly $11,000?
4 A company produces two products. Relevant information for each product is shown in Table 58. 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 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.
a Formulate an LP that can be used to minimize the penalty incurred by the company.
b Suppose the company sets (in order of importance) the following goals:
Goal 1 Avoid underutilization of labor.
Goal 2 Meet demand for product 1.
Goal 3 Meet demand for product 2.
Goal 4 Do not use any overtime.
Formulate and solve a preemptive goal programming model for this situation.
5† Deancorp produces sausage by blending together beef head, pork chuck, mutton, and water. The cost per pound, fat per pound, and protein per pound for these ingredients is given in Table 59. Deancorp needs to produce 100 lb of sausage and has set the following goals, listed in order of priority:
Goal 1 Sausage should consist of at least 15% protein.
Goal 2 Sausage should consist of at most 8% fat.
Goal 3 Cost per pound of sausage should not exceed 8¢.
Formulate a preemptive goal programming model for Deancorp.
6‡ The Touche Young accounting firm must complete three jobs during the next month. Job 1 will require 500 hours of work, job 2 will require 300 hours of work, and job 3 will require 100 hours of work. Currently, the firm consists of 5 partners, 5 senior employees, and 5 junior employees, each of whom can work up to 40 hours per month. The dollar amount (per hour) that the company can bill depends on the type of accountant who is assigned to each job, as shown in Table 60. (The X indicates that a junior employee does not have enough experience to work on job 1.) All jobs must be completed. Touche Young has also set the following goals, listed in order of priority:
Goal 1 Monthly billings should exceed $68,000.
Goal 2 At most, 1 partner should be hired.
Goal 3 At most, 3 senior employees should be hired.
Goal 4 At most, 5 junior employees should be hired.
T A B L E 57
Characteristics of a
Price Lot of 100 Chips
Per 100
Supplier Excellent Good Mediocre Chips ($)
1 60 20 20 400
2 50 35 15 300
3 40 20 40 250
T A B L E 58
Product 1 Product 2 Labor required 4 hours 2 hours
Contribution to profit $4 $2
†Based on Steuer (1984).
‡Based on Welling (1977).
T A B L E 59
Head Chuck Mutton Moisture
Fat (per lb) .05 .24 .11 0
Protein (per lb) .20 .26 .08 0
Cost (in ¢) .12 .96 8 0