Goal programming

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Goal Programming and

Multiple Objective Optimization Goal programming involves solving problems containing not one specific objective function, but rather a collection of goals. In linear and integer programming methods the objective function is measured in one dimension only but in goal programming, conflicting goals or goals with different priorities and weights can be combined with each other and solved by using simplex algorithm and programs such as QM solver or Excel. Goal programming yields only an efficient and satisfactory result rather than optimum, solution to the problem.

1. Goal Programming An important technique that has been developed to supplement LP is called goal programming. Typically, goals set by management can be achieved only at the expense of other goals. A hierarchy of importance needs to be established so that higher-priority goals are satisfied before lower-priority goals are addressed. It is not always possible to satisfy every goal so goal programming attempts to reach a satisfactory level of multiple objectives. The main difference is in the objective function where goal programming tries to minimize the deviations between goals and what we can actually achieve within the given constraints.

1.1 A Goal Programming Example (Harrison Electric Revisited)

The LP formulation for the Harrison Electric problem is Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 ≤ 12 (wiring hours) 6X1 + 5X2 ≤ 30 (assembly hours) X1, X2 ≥ 0 where X1 = number of chandeliers produced X2 = number of ceiling fans produced Harrison is moving to a new location and feels that maximizing profit is not a realistic objective. Management sets a profit level of $30 that would be satisfactory during this period. The goal programming problem is to find the production mix that achieves this goal as closely as possible given the production time constraints. We need to define two deviational variables d1

– = underachievement of the profit target

d1+ = overachievement of the profit target

We can now state the Harrison Electric problem as a single-goal programming model.

Minimize under or overachievement of profit target = d1– + d1

+

subject to $7X1 + $6X2 + d1–

– d1+ = $30 (profit goal constraint)

2X1 + 3X2 ≤ 12 (wiring hours) 6X1 + 5X2 ≤ 30 (assembly hours) X1, X2, d1

–, d1

+ ≥ 0

2

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Now Harrison’s management wants to achieve several goals of equal in priority

Goal 1: to produce a profit of $30 if possible during the production period, Goal 2: to fully utilize the available wiring department hours, Goal 3: to avoid overtime in the assembly department, Goal 4: to meet a contract requirement to produce at least seven ceiling fans.

The deviational variables are; d1

– = underachievement of the profit target

d1+ = overachievement of the profit target

d2– = idle time in the wiring department (underutilization)

d2+ = overtime in the wiring department (overutilization)

d3– = idle time in the assembly department (underutilization)

d3+ = overtime in the assembly department (overutilization)

d4– = underachievement of the ceiling fan goal

d4+ = overachievement of the ceiling fan goal

Because management is unconcerned about d1

+, d2

+, d3

–, and d4

+ these may be omitted from the objective

function. The new objective function and constraints are,

Minimize total deviation = d1– + d2

– + d3+ + d4

–

subject to 7X1 + 6X2 + d1– – d1

+ = 30 (profit constraint)

2X1 + 3X2 + d2– – d2

+ = 12 (wiring hours)

6X1 + 5X2 + d3– – d3

+ = 30 (assembly hours)

X2 + d4– – d4

+ = 7 (ceiling fan constraint)

All Xi, di variables ≥ 0

1.2 Ranking Goals with Priority Levels In most goal programming problems, one goal will be more important than another, which will in turn be more important than a third. Goals can be ranked with respect to their importance in management’s eyes. Priorities (Pi’s) are assigned to each deviational variable with the ranking so that P1 is the most important goal, P2 the next most important, P3 the third, and so on. The deviation from the high-priority goal must be minimized to the greatest extent possible before the next-highest-priority goal is considered. Harrison Electric has set the following priorities for their four goals;

GOAL PRIORITY

Reach a profit as much above $30 as possible P1

Fully use wiring department hours available P2

Avoid assembly department overtime P3

Produce at least seven ceiling fans P4

This effectively means that each goal is infinitely more important than the next lower goal. With the ranking of goals considered, the new objective function is

Minimize total deviation = P1d1– + P2d2

– + P3d3+ + P4d4

– The constraints remain identical to the previous ones.

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1.3 Solving Goal Programming Problems Graphically Recall the Harrison Electric goal programming model.

Minimize total deviation = P1d1– + P2d2

– + P3d3

+ + P4d4

–

subject to 7X1 + 6X2 + d1– – d1

+ = 30 (profit )

2X1 + 3X2 + d2– – d2

+ = 12 (wiring)

6X1 + 5X2 + d3– – d3

+ = 30 (assembly)

X2 + d4– – d4

+ = 7 (ceiling fans)

All Xi, di variables ≥ 0 (non-negativity) where

X1 = number of chandeliers produced, X2 = number of ceiling fans produced

Analysis of the first goal To solve this we graph one constraint at a time starting with the constraint with the highest-priority deviational variables. In this case we start with the profit constraint as it has the variable d1

– with

a priority of P1 . Note that in graphing this constraint the deviational variables are ignored. To minimize d1

– the feasible area is the shaded region.

Analysis of first and second goals

The next graph is of the second priority goal of minimizing d2

–.

The region below the constraint line 2X1 + 3X2 = 12 represents the values for d2

– while the region above the line stands for d2

+.

To avoid underutilizing wiring department hours the area below the line is eliminated. This goal must be attained within the feasible region already defined by satisfying the first goal.

Analysis of all four priority goals

The third goal is to avoid overtime in the assembly department. We want d3

+ to be as close to zero as possible.

Any point inside the feasible region bounded by the first three constraints will meet the three most critical goals. The fourth constraint seeks to minimize d4

–.

To do this requires eliminating the area below the constraint line X2 = 7 which is not possible given the previous, higher priority, constraints.

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The optimal solution must satisfy the first three goals and come as close as possible to satisfying the fourth goal. This would be point A on the graph with coordinates of X1 = 0 and X2 = 6. Substituting into the constraints we find, d1

– = $0 d1

+ = $6

d2– = 0 hours d2

+ = 6 hours

d3– = 0 hours d3

+ = 0 hours

d4– = 1 ceiling fan d4

+ = 0 ceiling fans

A profit of $36 was achieved exceeding the goal.

1.4 Modified Simplex Method for Goal Programming The modified simplex method can be used to solve problems with more than two real variables. Recall the Harrison Electric model.

Minimize = P1d1– + P2d2

– + P3d3

+ + P4d4

–

subject to 7X1 + 6X2 + d1– – d1

+ = 30

2X1 + 3X2 + d2– – d2

+ = 12

6X1 + 5X2 + d3– – d3

+ = 30

X2 + d4– – d4

+ = 7

All Xi, di variables ≥ 0

There are four features of the modified simplex tableau that differ from earlier simplex tableaus.

1. The variables in the problem are listed at the top, with the decision variables (X1 and X2) first, then the negative deviational variables and, finally, the positive deviational variables. The priority level of each variable is assigned on the very top row.

2. The negative deviational variables for each constraint provide the initial basic solution. This is analogous to the use of slack variables in the earlier simplex tableaus. The priority level of each variable in the current solution mix is entered in the Cj column.

3. There is a separate Zj and Cj – Zj row for each of the Pi priorities because different units of measurement

are used for each goal. The bottom row of the tableau contains the highest ranked (P1) goal; the next row has the P2 goal, and so forth. The rows are computed exactly as in the regular simplex method, but they are done for each priority level.

4. In selecting the variable to enter the solution mix, we start with the highest-priority row, P1, and select

the most negative Cj – Zj value in it. If there was no negative number for P1, we would move on to priority P2’s Cj – Zj row and select the largest negative number there. A negative Cj – Zj that has a positive number

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in the P row underneath it, however, is ignored. This means that deviations from a more important goal (one in a lower row) would be increased if that variable were brought into the solution.

We move towards the optimal solution just as with the regular minimization simplex method. We find the pivot row by dividing the quantity values by their corresponding pivot column (X1) values and picking the one with the smallest positive ratio. In this case, d1

– leaves the basis and is replaced by X1 .

We continue this process until an optimal solution is reached.

In the final solution the first three goals have been fully achieved with no negative entries in their Cj – Zj rows A negative value appears in the d3

+ column in the priority 4 row indicating this goal has not been fully attained

But the positive number in the d3+ at the P3 priority level (shaded cell) tells us that if we try to force d3

+ into the

solution mix, it will be at the expense of the P3 goal which has already been satisfied.

The final solution is: X1 = 0 chandeliers produced X2 = 6 ceiling fans produced d1

+ = $6 over the profit goal

d2+ = 6 wiring hours over the minimum set

d4– = 1 fewer fan than desired

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1.4 Using EXCEL for Windows to Solve Harrison’s Problem

We transfer the objective function and constraints into the Excel spreadsheet as follows.

Minimize = P1d1– + P2d2

– + P3d3

+ + P4d4

–

subject to 7X1 + 6X2 + d1– – d1

+ = 30

2X1 + 3X2 + d2–

– d2+ = 12

6X1 + 5X2 + d3–

– d3+ = 30

X2 + d4–

– d4+ = 7

All Xi, di variables ≥ 0

The cell L7 is the sum of the products B4:C4 with B7:C7, plus D7 minus H7, The cell L8 is the sum of the products B4:C4 with B8:C8 plus E8 minus I8, The cell L9 is the sum of the products B4:C4 with B9:C9 plus F9 minus J9, The cell L10 is the sum of the products B4:C4 with B10:C10 plus G10 minus K10. Objective function is the minimization of d1

– + d2

– + d3

+ + d4

– , which are D7+E8+J9+G10.

The column L is the summation of X1, X2 and deviations so L7:L10 values are equal to N7:N10. We insert the constraints into Excel solver.

The results are as follows.

=SUMPRODUCT(B4:C4;B7:C7)+D7-H7

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The final solution is as found in the graphical method. X1 = 0 chandeliers produced X2 = 6 ceiling fans produced d1

+ = $6 over the profit goal

d2+ = 6 wiring hours over the minimum set

d4– = 1 fewer fan than desired

1.5 Goal Programming with Weighted Goals in Excel The deviations measure different variables, such as cost, time, material etc. In order to combine the deviations in a single objective function, the percentage of each deviation from target value must be calculated.

Minimizing the summation of the percentage deviations Min: 𝟏

𝒕𝒊(𝒅𝒊

− + 𝒅𝒊+)

Here ti is target value and di is the deviation. If the decision maker has some priorities, the weight of each deviation can be included in the objective function.

Minimizing the weighted sum of the percentage deviations Min: 𝟏

𝒕𝒊(𝒘𝒊

−𝒅𝒊− +𝒘𝒊

+𝒅𝒊+)

WORKED-OUT PROBLEM 1 Davis McKeown is the owner of a resort hotel and convention center in Myrtle Beach, South Carolina. Although his business is profitable, it is also highly seasonal; the summer months are the most profitable time of year. To increase profits during the rest of the year, Davis wants to expand his convention business but, to do so, he needs to expand his conference facilities. Davis hired a marketing research firm to determine the number and sizes of conference rooms that would be required by the conventions he wants to attract. The results of this study indicated that Davis’s facilities should include at least 5 small (400 square foot) conference rooms, 10 medium (750 square foot) conference rooms, and 15 large (1,050 square foot) conference rooms. Additionally, the marketing research firm indicated that if the expansion consisted of a total of 25,000 square feet, Davis would have the largest convention center among his competitors—which would be desirable for advertising purposes. While discussing his expansion plans with an architect, Davis learned that he can expect to pay $18,000 for each small conference room in the expansion, $33,000 for each medium conference room, and $45,150 for each large conference room. Davis wants to limit his expenditures on the convention center expansion to approximately $1,000,000.

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Solution:

Goal 1: The expansion should include approximately 5 small conference rooms. Goal 2: The expansion should include approximately 10 medium conference rooms. Goal 3: The expansion should include approximately 15 large conference rooms. Goal 4: The expansion should consist of approximately 25,000 square feet. Goal 5: The expansion should cost approximately $1,000,000.

Notice that the word “approximately” appears in each goal. This word reinforces that these are soft goals rather than hard constraints. The goals are X1= 5, X2=10, X3=15 . The goals for the room numbers with deviational variables are;

X1 + d1–

– d1+ = 5 (small rooms )

X2 + d2–

– d2+ = 10 (medium rooms)

X3 + d3–

– d3+ = 15 (large rooms)

We can formulate the goal constraints for the remaining goals in the problem in a similar manner. Because each small, medium, and large conference room requires 400, 750,and 1,050 square feet, respectively, and the hotel owner wants the total square footage of the expansion to be 25,000, the constraint representing this goal is

400X1+ 750X2 + 1,050 X3 + d4

–– d4

+=25,000 (square footage)

Because each small, medium, and large conference room results in building costs of $18,000, $33,000, and $45,150 , respectively, and the hotel owner wants to keep the cost of the expansion at approximately $1,000,000, the constraint representing this goal is:

18,000X1 + 33,000X2 + 45,150 X3 + d5–– d5

+=1,000,000 (building cost)

Objective function is ;

Minimizing the summation of the deviations Min: 𝒅𝒊− + 𝒅𝒊

+

The deviations in this problem measure different units (number of rooms, area and cost), we have to modify the objective function and measure the sum of the percentage deviations.

Minimizing the summation of the percentage deviations Min: 𝟏

𝒕𝒊(𝒅𝒊

− + 𝒅𝒊+)

Here “d” represents deviation and “t” represents the target value.

One solution to the previous criticisms is to allow the decision maker to assign weights to the deviational variables in the objective function of a GP problem, to better reflect the importance and desirability of deviations from the various goals. So, a more useful type of objective function for a GP problem is:

Minimizing the weighted sum of the percentage deviations Min: 𝟏

𝒕𝒊(𝒘𝒊

−𝒅𝒊− +𝒘𝒊

+𝒅𝒊+)

Here 𝒘𝒊− and 𝒘𝒊

+ represent numeric constants that can be assigned to values to weight the various deviational variables in the problem by the decision maker. Such as if all 𝒘𝒊

−and 𝒘𝒊+’s are equal to 1 then they have equal

importance. If one weight is 10, then it is much more important than the others and so on.

Min: 𝒘𝟏− 𝒅𝟏

−

𝟓+𝒘𝟐

− 𝒅𝟐−

𝟏𝟎+𝒘𝟑

− 𝒅𝟑−

𝟏𝟓+𝒘𝟒

− 𝒅𝟒−

𝟐𝟓,𝟎𝟎𝟎+𝒘𝟒

+ 𝒅𝟒+

𝟐𝟓,𝟎𝟎𝟎+𝒘𝟓

+ 𝒅𝟓+

𝟏,𝟎𝟎𝟎,𝟎𝟎𝟎

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To begin our analysis of this problem, we will assume that all weights are equal to 1.

We transfer the data into an Excel spreadsheet as follows.

The target cell is O7= SUMPRODUCT (B7:D7; B4:D4) +E7-J7 (copy through O7 to O11)

Objective cell is B13=E7/Q7+F8/Q8+G9/Q9+O10/Q10+M10/Q10+N11/Q11 (summation of percentage deviations)

Variable cells are B4:D4 and E7:N11

The model can be solved using the solver parameters. The solution obtained using these settings are given below.

X1=5 , X2=10 , X3=15, d4

+=250, d5+=97250

So the result is 5 small rooms , 10 medium rooms, and 15 small rooms will be constructed.

The area constraint will exceed by 250 square feet and the cost will exceed $97250.

Variable cells

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PROBLEMS

(Goal Programming formulations) 1. (8.1-1/Taha) (Tax Planning)Fairville is a small city with a population of about 20,000 residents. The city council is in the process of developing an equitable city tax rate table. The annual taxation base for real estate property is $550 million. The annual taxation bases for food and drugs and for general sales are $35 million and $55 million, respectively. Annual local gasoline consumption is estimated at 7.5 million gallons. The city council wants to develop the tax rates based on four main goals.

1.Tax revenues must be at least $16 million to meet the city's financial commitments. 2. Food and drug taxes cannot exceed 10% of all taxes collected. 3. General sales taxes cannot exceed 20% of all taxes collected. 4. Gasoline tax cannot exceed 2 cents per gallon.

Let the variables Xp , Xf and Xs represent the tax rates (expressed as proportions of taxation bases) for property, food and drug, and general sales, and the variable Xg as the gasoline tax in cents per gallon. Write the goals of the city council. 2. (8.1A-2/Taha) The NW Shopping Mall conducts special events to attract potential patrons. Among the events that seem to attract teenagers, the young/middle-aged group, and senior citizens,the two most popular are band concerts and art shows. Their costs per presentation are $1500 and $3000, respectively. The total (strict) annual budget allocated to the two events is $15,000. The mall manager estimates the attendance as follows: Number attending per presentation

Event Teenagers Young/middle age Seniors

Band concert 200 100 0 Art show 0 400 250

The manager has set minimum goals of 1000, 1200, and 800 for the attendance of teenagers, the young/middle-aged group, and seniors, respectively. Formulate the problem as a goal programming model. 3. (8.1A-3/Taha) Ozark University admission office is processing freshman applications for the upcoming academic year. The applications fall into three categories: in-state, out-of-state, and international. The male-female ratios for in-state and out-of-state applicants are 1:1 and 3:2,respectively. For international students, the corresponding ratio is 8:1.The American College Test (ACT) score is an important factor in accepting new students. The

statistics gathered by the university indicate that the average ACT scores for in-state, out-of-state, and international students are 27,26, and 23, respectively. The committee on admissions has established the following desirable goals for the new freshman class: (a) The incoming class is at least 1200 freshmen. (b) The average ACT score for all incoming students is at least 25. (c) International students constitute at least 10% of the incoming class. (d) The female-male ratio is at least 3:4. (e) Out-of-state students constitute at least 20% of the incoming class. Formulate the problem as a goal programming model. 4. (8.1A-4/Taha)Circle K farms consumes 3 tons of special feed daily. The feed -a mixture of limestone, corn, and soybean meal-must satisfy the following nutritional requirements:

Calcium. At least 0.8% but not more than 1.2%. Protein. At least 22%. Fiber. At most 5%.

The following table gives the nutritional content of the feed ingredients.

Lb per Lb of ingredient

Ingredient Calcium Protein Fiber

Limestone .380 .00 .00 Corn .001 .09 .02 Soybean meal .002 .50 .08

Formulate the problem as a goal programming model. (Goal Programming Algorithms) 5. (8.2-1/Taha) TopAd, a new advertising agency with 10 employees, has received a contract to promote a new product. The agency can advertise by radio and television. The following table gives the number of people reached by each type of advertisement and the cost and labor requirements.

Data/min advertisement

Radio Television

Exposure (in millions of persons) 4 8 Cost (in thousands of dollars) 8 24 Assigned employees 1 2

The contract prohibits TopAd from using more than 6 minutes of radio advertisement. Additionally, radio and television advertisements need to reach at least 45 million people. TopAd has a budget goal of $100,000 for the project and it's management assumes that the exposure goal is twice as important as the budget goal.How many minutes of radio and television advertisement should TopAd use?

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6. An electronics company produces two types of television sets, color and black-and-white. The production of a color set requires 10 hours of skilled and 100 hours of unskilled labor. The production of a black-and-white set requires 5 hours of skilled and 150 hours of unskilled labor. The company has 100 hours of skilled labor and 1,500 hours of unskilled labor normally available per month for the production of television sets. The maximum number black-and-white and color sets that can be sold each month are 45 and 70, respectively. The profit margin from the sale of a color set is $20, whereas it is $15 from a black-and-white set. The company has set the following goals:

1. Avoid the over utilization of skilled labor since it is hard to obtain in the labor market. 2. Minimize the under utilization of unskilled labor. 3. Meet the demand as much as possible. 4. Limit over utilization of unskilled labor to 100 hours.

Formulate the above as a goal programming problem and solve using Excel.

7. A department store plans to schedule its annual advertising. The total budget is set at $200,000. The store can purchase local radio spots at $100 per spot, local television spots at $500 per spot and local newspaper advertising at $200 per ad. The payoff from each advertising medium is a function of its audience size and audience characteristics. The generally accepted objective criterion for advertising is audience points, reflected in the following table:

Medium Points

Radio 30 per spot

Television 150 per spot

Newspaper 150 per ad

The president of the firm has established the following goals for the campaign:

1. The total budget should not exceed $200,000. 2. Meet the contract with the local television station that requires that the firm spend at least $30,000. 3. The corporate advertising policy prohibits annual newspaper ad expenditures in excess of $50,000. 4. Maximize the audience points for the advertising campaign.

The president has established unit weights on the goals of 10, 6, 3 and 1 for the goals 1 through 4, respectively. Formulate the above as a goal programming problem and solve using Excel.

8. The Midtown City Council is reviewing housing proposals for a new development area. There is some dispute among various interest groups as to what goals should be sought. The zoning committee has recommended three types of housing: one-family houses, deluxe condominiums and apartments. The zoning committee has compiled the following data for each type of housing:

One-

family Condominiums Apartments

Acres per unit .25 .20 .125

Families housed per unit

1 4 6

Tax base generated per unit

$50,000 $100,000 $150,000

Taxes required for city services

$4,000 $8,000 $10,000

There are 50 acres available for zoning. The League for Better Housing has conducted a campaign to gain housing for at least 500 families. The Taxpayers’ Union has strongly lobbied for an added tax base of $5,000,000. The Gray Panthers have disrupted the city council meetings to demand that taxes for city services be no more than $250,000. The city council hired a public opinion survey company to assess the priorities of the citizens. The poll results are as follows:

Priority 1 Priority 2 Priority 3

Housing for 500 families 55% 35% 10%

Tax base of $5,000,000 40 30 30

Taxes for services of $250,000

15 20 65

Based on this survey the city council has established the following goals:

1. Provide housing for at least 500 families. 2. Establish at least $5,000,000 worth of new tax base. 3. Limit taxes for city services to $250,000. 4. Reserve at least 5 acres for a neighborhood park area.

It is assumed that the first goal is met fully before the second, the second met fully before the third and the third met fully before the fourth. Formulate the above as a goal programming problem and solve using Excel.

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9. Acme Sawmill can produce plywood, chipboard and pulp for sale, realizing profit margins of $10, $9 and $6 per ton, respectively. The mill can run any number of operations at the same time, but the setup costs for each of the operations differ. While the pulp production only costs $2,000 ,to set up the plywood production costs $50,000 and the chipboard production costs $25,000. Plywood consists of 95% wood and 5% resin glue. Chipboard consists of 91% wood, 5% resin glue and 4% other additives. Pulp consists of 86% wood and 14% other additives. Demand limits the amount of plywood produced to 10,000 tons, while as much as 5,000 tons each may be produced of chipboard and pulp. For the next month of production there are 15,000 tons of wood and 500 tons each of resin glue and other additives available. Finally, Acme’s customers always place orders for whole tons of plywood and chipboard. No partial tons may be accepted.

The operations manager at Acme has set the following goals for the upcoming month:

a. Achieve at least $70,000 profit. b. Avoid having to special order more glue and additives.

Although it is an inconvenience to have to reorder glue

and additives, the target profit is a more important

consideration. Accordingly, the operations manager

has placed subjective weights of 5 on profit deviations

and 1 on resource deviations. Formulate the above as

a goal programming problem and solve using Excel.

10. (10/Ragsdale)The CFO for the Shelton Corporation has $1.2 million to allocate to the following budget requests from 5 departments: Dept 1 Dept 2 Dept 3 Dept 4 Dept 5 $450,000 $310,000 $275,000 $187,500 $135,000

Because the total budget requests exceed the available $1.2 million, not all the requests can be satisfied. Suppose that the CFO considers the requests for departments 2 and 3 to be twice as important as those from departments 4 and 5, and the request from department 1 to be twice as important as those from departments 2 and 3. Further suppose that the CFO wants to make sure each department receives at least 70% of the requested amount. a. Formulate a GP model for this problem. b. Implement your model and solve it. What is the optimal solution? c. Suppose the CFO is willing to allocate more than $1.2 million to these budgets but regards exceeding the $1.2 million figure as being twice as undesirable as not

meeting the budget request of department 1. What is the optimal solution? d. Suppose the CFO regards all deviations from the original budget amounts (including the $1.2 million available) to be equally undesirable. What solution minimizes the maximum percentage deviation from the budgeted amounts? 11. (14/Ragsdale) A new Italian restaurant called the Olive Grove is opening in several locations in the Memphis area. The marketing manager for these stores has a budget of $150,000 to use in advertising and promotions for the new stores. The manager can run magazine ads at a cost of $2,000 each that result in 250,000 exposures each. TV ads result in approximately 1,200,000 exposures each, but cost $12,000 each. The manager wants to run at least five TV ads and ten magazine ads, while maximizing the number of exposures generated by the advertising campaign. But the manager also wants to spend no more than $120,000 on magazine and TV advertising so that the remaining $30,000 could be used for other promotional purposes. However, the manager would spend more than $120,000 on advertising if it resulted in a substantial increase in advertising coverage. a. Formulate a GP model for this problem assuming that the marketing manager has the following goals: Goal 1: Exposures should be maximized. Goal 2: No more than $120,000 should be spent on advertising. (Note that you will have to determine an appropriate target value for the first goal.) Assume that the marketing manager wants to minimize the maximum percentage deviation from either goal. b. Implement your model in a spreadsheet and solve it. 12. (15/Ragsdale)The city of Abingdon is determining its tax rate structure for the coming year. The city needs to generate $6 million in tax revenue via taxes of property, sales, prepared food, and utilities. The following table summarizes how much tax revenue would be generated from each segment of the population by the 1% increase in each tax category. (For instance, a 2% tax on prepared food would generate $240,000 in tax revenue from upper income residents.)

Revenues (in $1000s) per 1% Tax Rate Income Group

Sales

Property

Food

Utility

Low $200 $600 $50 $80 Middle $250 $ 800 $100 $100 Upper $400 $1200 $120 $120

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City commissioners have specified that the tax rate for each revenue category must be between 1% and 3% and that the tax rate on prepared food cannot exceed half the sales tax rate. Ideally, the commissioners have a goal of making up the $6 million tax budget with $1.5 million from low income residents, $2.1 million from middle income residents, and $2.4 million from high income residents. If that is not possible, the commissioners would like a solution that minimizes the maximum percentage deviation from these tax revenue goals for each income group. a. Create a spreadsheet model for this problem. b. What is the optimal solution?

13. (17/Ragsdale) Virginia Tech operates its own power generating plant. The electricity generated by this plant supplies power to the university and to local businesses and residences in the Blacksburg area. The plant burns three types of coal, which produces steam that drives the turbines that generate the electricity. The Environmental Protection Agency (EPA) requires that for each ton of coal burned, the emissions from the coal furnace smokestacks contain no more than 2,500 parts per million (ppm) of sulfur and no more than 2.8 kilograms (kg) of coal dust. However, the managers of the plant are concerned about the environment and want to keep these emissions to a minimum. The following table summarizes the amounts of sulfur, coal dust, and steam that result from burning a ton of each type of coal.

Coal Sulfur (in

ppm)

Coal Dust (in

kg)

Pounds of Steam

Produced 1 1,100 1.7 24,000 2 3,500 3.2 36,000 3 1,300 2.4 28,000

The three types of coal can be mixed and burned in any combination. The resulting emission of sulfur or coal dust and the pounds of steam produced by any mixture are given as the weighted average of the values shown in the table for each type of coal. The manager of this facility wants to select a blend of coal to burn while considering the following objectives: Objective 1: Maximize the pounds of steam produced. Objective 2: Minimize sulfur emissions. Objective 3: Minimize coal dust emissions. a. Formulate an MOLP model for this problem and implement your model in a spreadsheet. b. Determine the best possible value for each objective in the problem. c. Determine the solution that minimizes the maximum percentage deviation from the optimal

objective function values. What solution do you obtain? d. Suppose management considers maximizing the amount of steam produced five times as important as achieving the best possible values for the other objectives. What solution does this suggest?

14. (11.22/Render)

15. (11.24/Render)

Goal programming

14

16. (11.28/Render)

17. (11.29/Render)

a) Formulate this as a goal programming

problem,

b) Solve the problem using computer software.

18. (11.30/Render)

19. (7.5-2./Hillier ) Management of the Albert Franko Co. has established goals for the market share it wants each of the company’s two new products to capture in their respective markets. Specifically, management wants Product 1 to capture at least 15 percent of its market and Product 2 to capture at least 10 percent of its market. Three advertising campaigns are being planned to try to achieve these market shares. One is targeted directly on the first product. The second targets the second product. The third is intended to enhance the general reputation of the company and its products. Letting x1, x2, and x3 be the amount of money allocated (in millions of dollars) to these respective campaigns, the resulting market share (expressed as a percentage) for the two products are estimated to be Market share for Product 1 = 0.5x1 + 0.2x3, Market share for Product 2 = 0.3x2 + 0.2x3. A total of $55 million is available for the three advertising campaigns, but management wants at least $10 million devoted to the third campaign. If both market share goals cannot be achieved, management considers each 1 percent decrease in the market share from the goal to be equally serious for the two products. In this light, management wants to know how to most effectively allocate the available money to the three campaigns. (a) Formulate a goal programming model for this problem. (b) Reformulate this model as a linear programming model. C (c) Use the simplex method to solve this model.

Goal programming

15

20. (7.5-5./Hillier) Montega is a developing country which has 15,000,000 acres of publicly controlled agricultural land in active use. Its government currently is planning a way to divide this land among three basic crops (labeled 1, 2, and 3) next year. A certain percentage of each of these crops is exported to obtain badly needed foreign capital (dollars), and the rest of each of these crops is used to feed the populace. Raising these crops also provides employment for a significant proportion of the population. Therefore, the main factors to be considered in allocating the land to these crops are (1) the amount of foreign capital generated, (2) the number of citizens fed, and (3) the number of citizens employed in raising these crops. The following table shows how much each 1,000 acres of each crop contributes toward these factors, and the last column gives the goal established by the government for each of these factors.

Contribution per 1,000 Acres

Crop: Goal Factor 1 2 3

Foreign capital

$3,000 $5,000 $4,000 ≥ $70,000,000

Citizens fed 150 75 100 ≥ 1,750,000

Citizens employed

10 15 12 = 200,000

In evaluating the relative seriousness of not achieving these goals, the government has concluded that the following deviations from the goals should be considered equally undesirable: (1) each $100 under the foreign-capital goal, (2) each person under the citizens-fed goal, and (3) each deviation of one (in either direction) from the citizens-employed goal. (a) Formulate a goal programming model for this problem and solve in Excel, (b) Now suppose that the government concludes that the importance of the various goals differs greatly so that a preemptive goal programming approach should be used. In particular, the first-priority goal is citizens fed ≥ 1,750,000, the second priority goal is foreign capital ≥ $70,000,000, and the third priority goal is citizens employed = 200,000. Use the goal programming technique to formulate one complete linear programming model for this problem.

21. (7.5-Example/Hillier) The DEWRIGHT COMPANY is considering three new products to replace current models that are being discontinued, so their OR department has been assigned the task of determining which mix of these products should be produced. Management wants primary consideration given to three factors: long-run profit, stability in the workforce, and the level of capital investment that would be required now for new equipment. In particular, management has established the goals of (1) achieving a long-run profit (net present value) of at least $125 million from these products, (2) maintaining the current employment level of 4,000 employees, and (3) holding the capital investment to less than $55 million. However, management realizes that it probably will not be possible to attain all these goals simultaneously, so it has discussed priorities with the OR department. This discussion has led to setting penalty weights of 5 for missing the profit goal (per $1 million under), 2 for going over the employment goal (per 100 employees), 4 for going under this same goal, and 3 for exceeding the capital investment goal (per $1 million over).Each new product’s contribution to profit, employment level, and capital investment level is proportional to the rate of production. These contributions per unit rate of production are shown in the following table, along with the goals and penalty weights.

Unit Contribution

Product Goal Penalty Factor 1 2 3 weight Long-run profit

12 9 15 ≥ 125 (million$) 5

Employment level

5 3 4 = 40 (100 employees) 2(+), 4(-)

Capital investment

5 7 8 ≤ 55 (million$) 3

Formulate a goal programming model for this problem by using weights and percentage deviations and solve in Excel.

Goal programming

16

Answers

1. (8.1-1/Taha) 550xp + 35xf + 55xs + .075xg 16 (Tax revenue)

35xf .1(550xp + 35xf + 55x3 + .075xg ) (Food/drug tax)

55xs .2(550xp + 35xf + 55xs + .075xg ) (General tax)

xg 2 (Gasoline tax) These constraints are then simplified as

550xp + 35xf + 55xs + .075xg 16

55xp - 31.5xf + 5.5xs + .0075xg 0

110xp + 7xf - 44xs + .015xg 0

x g 2

xp ,xf ,xs ,xg 0 We convert each inequality into a flexible goal. 550xp + 35xf + 55xs + .075xg + d1

- - d1

+ = 16

55xp - 31.5xf + 5.5xs + .0075xg + d2- - d2

+= 0

110xp + 7xf - 44xs + .015xg + d3- -d3

+ = 0

xg + d4- -d4

+ = 2

di- ,di

+ 0, i = 1,2,3,4.

the compromise solution tries to satisfy the following four objectives as much as possible

Min: d1-+ d2

-+ d3

-+ d4

+

2. (8.1A-2/Taha) ….. 3. (8.1A-3/Taha) Let x1 = No. of in-state freshmen,

x2 = No. of out-of-state freshmen, -x3 = No.of international freshmen.

Gi : Minimize di

-, i = 1,2, ... ,5, subject to

x1 + x2 + x3 + d1- - d1

+ = 1200,

2x1+ x2- 2x3 + d2- - d2

+ = 0,

- .1x1 - .1x2 + .9x3 + d3- - d3

+ = 0,

.125 x1- .05 x2 - .556 x3 + d4- - d4

+ = 0,

-.2x1+ .8x2- .2x3 + d5- - d5

+ = 0

All variables are nonnegative

4. (8.1A-4/Taha) …. 5. (8.2-1/Taha) Minimize z = d1

- + d2

- + d3

- + d4

++ d5

+

xp = .0201, xf = .0457, xs = .0582, xg = 2 cents, d5+= 1.45

Gasoline tax is $1.45 million short of goal.

6. Note that since the demand goal stated "meet the

demand as much as possible" production deviation either above or below demand is considered undesirable for both types of televisions. Also note that all apparent "hard" constraints are superseded by goal constraints.

Let: C = # of color televisions produced

B = # of black and white televisions produced

ds+ = deviation above skilled labor utilization target

ds- = deviation below skilled labor utilization target

du+ = deviation above unskilled labor utilization target

du- = deviation below unskilled labor utilization target

dc+ = deviation above color television demand target

dc- = deviation below color television demand target

db+ = deviation above black and white television demand

target

db- = deviation below black and white television demand

target

do+ = deviation above unskilled labor overutilization target

do- = deviation below unskilled labor overutilization target

Minimize Z = ds

+ + du

- + dc

+ + dc

- + db

+ + db

- + do

+

s.t. 10C + 5B + ds-- ds

+ =100 (Skilled use goal)

100C + 150B + du

-- du

+ =1,500 (Unskilled use goal)

C + dc-- dc

+ =70 (Color demand goal)

B + db-- db

+ =45 (B&W demand goal)

du+ + do

-- do

+ =100 (Unskilled overuse goal)

where: C, B, ds

+, ds

-, du

+, du

-, dc

+, dc

-, db

+, db

-, do

+, do

- > 0 and

C, B are integer

7. Note that an arbitrarily high number has been chosen for

the RHS of the final goal constraint to "maximize the audience points." Also note that all apparent "hard" constraints are superseded by goal constraints.

Let: R = # of local radio spots purchased

T = # of local television spots purchased

N = # of newspaper ads purchased

db

+ = deviation above budget target

db

- = deviation below budget target

dc

+ = deviation above television contract target

dc

- = deviation below television contract target

dp

+ = deviation above newspaper policy target

dp

- = deviation below newspaper policy target

da

+ = deviation above audience points target

Goal programming

17

da

- = deviation below audience points target

Minimize Z = 10 db+ + 6 dc

- + 3 dp

+ + da

-

s.t. 100R + 500T + 200N - db+ + db

-=200,000 (Budget goal)

500T - dc

+ + dc

- = 30,000 (TV contract goal)

200N - dp+ + dp

- = 50,000 (Newspaper goal)

30R + 150T + 150N - da+ + da

-=1,000,000 (Audience goal)

where: R, T, N, db

+, db

-, dc

+, dc

-, dp

+, dp

-, da

+, da

- > 0 and

R, T, N are integer

8. Note that a heuristic is employed in assigning objective

coefficients in order to prioritize goal compliance. The variable representing the deviation with the lowest priority is assigned a weight of "1" and the weights are increased by a factor of 10 for each successively more important goal. The resulting solution, however, does not allow compliance with goal 3 before that for goal 4. There is no feasible solution that allows compliance with goal 3 if goals 1 and 2 are both satisfied. Also note that only the total acres available for building remains as a "hard" constraint.

Let: S = # of single-family homes built

C = # of deluxe condominiums built

A = # of apartments built

D = # of acres of land not used for building

df

+ = deviation above families housed target

df

- = deviation below families housed target

dt

+ = deviation above tax base target

dt

- = deviation below tax base target

ds

+ = deviation above city services tax target

ds

- = deviation below city services tax target

dp

+ = deviation above park set-aside target

dp

- = deviation below park set-aside target

Minimize Z = 1,000 df- + 100 dt

- + 10 ds

+ + dp

-

s.t. .25 S + .2 C + .125 A + D=50(Acres available)

S + 4 C + 6 A - df

+ + df

-=500(Families goal)

5 S + 10 C + 15 A - dt+ + dt-=500(Tax base goal)

4 S + 8 C + 10 A - ds+ + ds

-=250(City service tax goal)

D - dp+ + dp

-=5(Park goal)

where: S, C, A, D, df

+, df

-, dt

+, dt

-, ds

+, ds

-, dp

+, dp

- > 0 and

S, C, A are integer

9.

Let: X1 = # of tons of plywood produced

X2 = # of tons of chipboard produced

X3 = # of tons of pulp produced

X4 = { 1 if plywood is produced 0 if otherwise

X5 = { 1 if chipboard is produced 0 if otherwise

X6 = { 1 if pulp is produced

0 if otherwise

dp

+ = deviation above profit target

dp

- = deviation below profit target

dg

+ = deviation above glue requirement target

dg

- = deviation below glue requirement target

da

+ = deviation above additive requirement target

da

- = deviation below additive requirement target

Minimize Z = 5 dp- + dg

+ + da

+

s.t. .95 X1 + .91X2 + .86X3 < 15,000 (Wood available)

X1 - 10,000 X4 < 0 (Ply demand)

X2 - 5,000 X5 < 0 (Chip demand)

X3 - 5,000 X6 < 0 (Pulp demand)

10 X1 + 9 X2 + 6 X3 - 50,000 X4 - 25,000 X5 - 2,000 X6 - dp

+ + dp

-

= 70,000 (Profit goal)

.05 X1 + .05 X2 - dg

+ + dg

- = 500 (Glue goal)

.04 X2 + .14 X3 - da

+ + da

- = 500 (Additives goal)

where: X1, X2, X3, X4, X5, X6, dp

+, dp

-, dg

+, dg

-, da

+, da

- > 0,

X1, X2 are integer, and X4, X5, X6 are binary

10.