**Analysis**

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Consider the following linear programming problem: Maximize Z = 400 x + 100y Subject to 8 x + 10y ? 80 2 x + 6y ? 36 x? 6 x, y ? 0 BSTA 450 Find the optimal solution using the graphical method (use graph paper). Identify the feasible region and the optimal solution on the graph.

How much is the maximum profit? Consider the following linear programming problem: Minimize Z = 3 x + 5 y (cost, $) subject to 10 x + 2 y ? 20 6 x + 6 y ? 36 y ? 2 x, y ? 0 Find the optimal solution using the graphical method (use graph paper).

Identify the feasible region and the optimal solution on the graph. How much is the minimum cost? 2. The Turner-Laberge Brokerage firm has just been instructed by one of its clients to invest $250 000 for her, money obtained recently through the sale of land holdings in British Columbia. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. She requests that the firm select whatever stocks and bonds they believe are well rated but within the following guidelines: 1.

At least 20% of the investment should be in accounts with only Canadian content. 2. At least 40% of the investment should be placed in a combination of U. S. electronics firms, aerospace firms, and pharmaceutical companies. 3. No more than 50% of the invested amount should be in precious metals. 4. Ratio of aerospace to pharmaceutical investment should be at least 2 : 1 . Subject to these restrains, the client’s goal is to maximize projected return on investments. The analysts at Turner-Laberge, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return.

Projected Rate of Return (%) Investment 3. Canadian RRSP Thompson Electronics, Inc. (USA) United Aerospace Corp. (USA) Palmer Pharmaceuticals (USA) Alberta Gold Mines (Canada) Formulate this portfolio selection problem using LP. 5. 3 6. 8 4. 9 8. 4 11. 8 2 4. The manager of a department store in Seattle is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences.

The television station representative indicates that a TV commercial, which costs $15 000, would reach 25 000 potential customers. The breakdown of the audience is as follows. Male Female Senior 5 000 5 000 Young 5 000 10 000 The newspaper representative claims to be able to provide an audience of 10 000 potential customers at a cost of $4 000 per ad. The breakdown of the audience is as follows Male Female Senior 4 000 3 000 Young 2 000 1 000 The radio station representative says that the audience for one of the station’s commercials, which costs $6 000, is 15 000 customers.

The breakdown of the audience is as follows Male Female Senior 1 500 1 500 Young 4 500 7 500 The store has the following advertising policy: Use at least twice as many radio commercials as newspaper ads Reach at least 100 000 customers Reach at least twice as many young people as senior citizens Make sure that at least 30% of the audience is female Available space limits the number of newspaper ads to seven. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost. Formulate a linear programming model for this problem.

The Pyrotec Company produces three electrical products – clocks, radios, and toasters. These products have the following resource requirements. Resource Requirements Cost/Unit Labor Hours/Unit Clock Radio Toaster $7 $10 $5 2 3 2 5. The manufacturer has a daily production budget of $2 000 and a maximum of 660 hours of labor. Maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters. Clocks sell for $15, radios for $20, and toasters for $12. The company desires to know the optimal product mix that will maximize profit. Formulate a linear programming model for this problem. 3 6.

The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. One gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50.

The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. Formulate a linear programming model for this problem. Consider the following linear programming problem: Maximize Z = 300 x1 + 500 x2 7. subject to : 3×1 + 5 x2 ? 30 x1 + x2 ? 18 x1 , x2 ? 0 Why this problem has no solution? Definitions and concepts to know 1. 2. 3. 4. 5. 6. 7. 8. 9. Decision variable Objective function Constraint Feasible region Isoprofit line Isocost line Unboundedness Infeasibility Maximization problem 10.

Minimization problem 11. Redundant constraint 12. Alternate optimal solutions 13. Binding constraint 14. Slack value 15. Corner point method 16. Classical problems: a. the marketing problem b. the investment problem c. the blend problem d. the product mix problem e. the transportation problem 4 BSTA 450 – Review Sheet – Test 2 Solutions BSTA 450 1. Consider the following linear programming problem: Maximize Z = 400 x + 100y (profit $) Subject to 8 x + 10y ? 80 2 x + 6y ? 36 x? 6 x, y ? 0 Find the optimal solution using the graphical method (use graph paper).

Identify the feasible region and the optimal solution on the graph. How much is the maximum profit? Solution: 36 ? 6 y 30 = = 4. 29 x= 2 x + 6 y = 36 ? 4 2 x + 6 y = 36 2 7 B: ? v? ? B (4. 29 , 4. 57) 32 8 x + 10 y = 80 14 y = 64 y= = 4. 57 7 x = 6 x = 6 80 ? 8 x 80 ? 48 32 C: ? y = = = = 3. 2 C (6 , 3. 2) 8 x + 10 y = 80 10 10 10 y 8 A 6 4 Z 2 B C 8 x + 10 y = 80 x=6 A: (0 , 6) Z = 600 B: (4. 29 , 4. 57) Z = 2 171 2 x + 6 y = 36 * C: 0 2 D: (6 , 0) Z = 2 400 Answer: Point C (6 , 3. 2) is optimal; maximum profit is $2 720 4 6 D 8 10 12 14 16 18 (6 , 3. 2) Z = 2 720 x 5 2.

Consider the following linear programming problem: Minimize Z = 3 x + 5 y (cost, $) subject to 10 x + 2 y ? 20 6 x + 6 y ? 36 y ? 2 x, y ? 0 Find the optimal solution using the graphical method (use graph paper). Identify the feasible region and the optimal solution on the graph. How much is the minimum cost? Solution: 20 ? 2 y 20 ? 10 x= = =1 10 x + 2 y = 20 ? 6 10 x + 2 y = 20 10 10 B: B (1 , 5) ? v? ? 240 ? 48 y = ? 240 6 x + 6 y = 36 ? 10 y= =5 48 36 ? 6 y 24 6 x + 6 y = 36 = =4 x= C (4 , 2) C: ? 6 6 y = 2 y=2 x2 14 12 *A: x1 = 0 x2 = 10 Z = 50 A 10x + 2y =20 10 8 6 B: x1 = x2 Z = 1 5 28 B 4 C 2 Z 0 2 4 6 8 10 6x + 6y = 36 y=2 *C: = 4 x2 = 2 Z = 22 x1 14 12 x Answer: Point C (4 , 2) is optimal: minimum cost is $22 6 3. The Turner-Laberge Brokerage firm has just been instructed by one of its clients to invest $250 000 for her, money obtained recently through the sale of land holdings in British Columbia. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. She requests that the firm select whatever stocks and bonds they believe are well rated but within the following guidelines: 1.

At least 20% of the investment should be in accounts with only Canadian content. 2. At least 40% of the investment should be placed in a combination of U. S. electronics firms, aerospace firms, and pharmaceutical companies. 3. No more than 50% of the invested amount should be in precious metals. 4. Ratio of aerospace to pharmaceutical investment should be at least 2 : 1 . Subject to these restrains, the client’s goal is to maximize projected return on investments. The analysts at Turner-Laberge, aware of these guidelines, prepare a list of high-quality tocks and bonds and their corresponding rates of return. Investment Canadian RRSP Thompson Electronics, Inc. (USA) United Aerospace Corp. (USA) Palmer Pharmaceuticals (USA) Alberta Gold Mines (Canada) Projected Rate of Return (%) 5. 3 6. 8 4. 9 8. 4 11. 8 Formulate this portfolio selection problem using LP. Solution: Four decision variables represent the amount invested in each investment alternative. x1 = amount ($) invested in Canadian RRSP x2 = amount ($) invested in Thompson Electronics x3 = amount ($) invested in United Aerospace Corp. 4 = amount ($) invested in Palmer Pharmaceuticals x5 = amount ($) invested in Alberta Gold Mines The objective of the investor is to maximize the total return from the investment in the five alternatives. The total return is the sum of the individual returns from each alternative. Thus, the objective function is expressed as maximize Z = 0. 053 x1 + 0. 068 x2 + 0. 049 x3 + 0. 089 x4 + 0. 118 x5 where Z = total return from all investments In this problem the constraints are the guidelines established for diversifying the total investment.

Each guideline is transformed into a mathematical constraint separately. The first guideline states that at least 20% of the total investment should be in accounts with only Canadian content. The total investment is $250 000; 20% of $250 000 is $50 000 (0. 2 ? 250 000 = 50 000 ) . Thus, this constraint is x1 + x5 ? $50 000 . The second guideline indicates that at least 40% of the investment should be placed in a combination of U. S. electronics firms, aerospace firms, and pharmaceutical companies. 40% of $250 000 is $100 000 . Thus, this constraint is x2 + x3 + x4 ? 100 000 The third guideline states that no more than 50% of the total investment should be in precious metals bonds. 50% of $250 000 is $125 000. Thus, this constraint is x5 ? $125 000 . 7 The fourth guideline states that the ratio of the amount invested in aerospace firms to the amount x 2 invested in pharmaceutical companies should be at least 2 to1: 3 ? . This constraint is not in x4 1 standard linear programming form because of the fractional relationship of the decision variables, x3 . It is converted as follows: x3 ? 2 x4 or x3 ? 2 x4 ? 0 x4 Finally, the client wants to invest the entire $250 000 in the five alternatives.

Thus, the sum of all the investments in the five alternatives must equal $250 000: x1 + x2 + x3 + x4 + x5 = $250 000 Answer: Maximize Z = 0. 053 x1 + 0. 068 x2 + 0. 049 x3 + 0. 089 x4 + 0. 118 x5 subject to x1 + x5 ? 50 000 x2 + x3 + x4 ? 100 000 x5 ? 125 000 x3 ? 2 x 4 ? 0 x1 + x2 + x3 + x4 + x5 = 250 000 x1 , x2 , x3 , x4 x5 ? 0 4. The manager of a department store in Seattle is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences.

The television station representative indicates that a TV commercial, which costs $15 000, would reach 25 000 potential customers. The breakdown of the audience is as follows. Male Senior Young 5 000 5 000 Female 5 000 10 000 The newspaper representative claims to be able to provide an audience of 10 000 potential customers at a cost of $4 000 per ad. The breakdown of the audience is as follows Male Senior Young 4 000 2 000 Female 3 000 1 000 The radio station representative says that the audience for one of the station’s commercials, which costs $6 000, is 15 000 customers.

The breakdown of the audience is as follows Male Senior Young 1 500 4 500 Female 1 500 7 500 8 The store has the following advertising policy: Use at least twice as many radio commercials as newspaper ads Reach at least 100 000 customers Reach at least twice as many young people as senior citizens Make sure that at least 30% of the audience is female Available space limits the number of newspaper ads to seven. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost. Formulate a linear programming model for this problem.

Solution: This model consists of three decision variables representing the number of each type of advertising produced: x1 = number of television commercials x2 = number of newspaper ads x3 = number of radio commercials The manager’s objective is to minimize the total cost of advertising. The total cost is the sum of the individual costs of each type of advertising purchased. The objective function that represents total cost is expressed as minimize Z = $15 000 x1 + $4 000 x2 + $6 000 x3 , where $15 000 x1 = cost of television commercials $4 000 x2 = cost of newspaper ads $6 000 x3 = cost of radio commercials

The first constraint specifies that the ratio of the number of radio commercials to the number of newspaper ads should be at least 2 to1. x3 2 ? (using at least twice as many radio commercials as newspaper ads) x2 1 This constraint is not in standard linear programming form because of the fractional relationship x of the decision variables, 3 . It is converted as follows: x3 ? 2 x2 or x3 ? 2 x2 ? 0 . x2 The second constraint represents the fact that advertising should reach at least 100 000 customers: 25 000 x1 + 10 000 x2 + 15 000 x3 ? 00 000 (Reach at least 100 000 customers), where 25 000 x1 + 10 000 x2 + 15 000 x3 = total number of people reached by advertising The third constraint specifies that advertising should reach at least twice as many young people as senior citizens: 15 000 x1 + 3 000 x2 + 12 000 x3 2 ? (Reach at least twice as many young people as senior citizens) 10 000 x1 + 7 000 x2 + 3 000 x3 1 where 15 000 x1 + 3 000 x2 + 12 000 x3 = estimated number of young people reached by advertising 10 000 x1 + 7 000 x2 + 3 000 x3 = estimated number of senior citizens reached by advertising This constraint is not in standard linear programming form.

It is converted as follows: 15 000 x1 + 3 000 x2 + 12 000 x3 ? 2(10 000 x1 + 7 000 x2 + 3 000 x3 ) or ? 5 000 x1 ? 11 000 x2 + 6 000 x3 ? 0 9 The fourth constraint in this model represents the fact that at least 30% of the audience is female. 15 000 x1 + 4 000 x2 + 9 000 x3 ? 0. 30 (at least 30% of the audience is female) 25 000 x1 + 10 000 x2 + 15 000 x3 where 15 000 x1 + 4 000 x2 + 9 000 x3 = estimated number of females reached by advertising. This constraint is not in standard linear programming form. It is converted as follows: 15 000 x1 + 4 000 x2 + 9 000 x3 ? 0. 30(25 000 x1 + 10 000 x2 + 15 000 x3 ) or 7 500 x1 + 1 000 x2 + 4 500 x3 ? The final constraint represents the fact that newspaper ads are limited to seven. x2 ? 7 (limit the number of newspaper ads) Answer: Minimize Z = 15 000 x1 + 4 000 x2 + 6 000 x3 subject to x3 ? 2 x2 ? 0 25 000 x1 + 10 000 x2 + 15 000 x3 ? 100 000 ? 5 000 x1 ? 11 000 x2 + 6 000 x3 ? 0 7 500 x1 + 1 000 x2 + 4 500 x3 ? 0 x2 ? 7 x1 , x2 , x3 ? 0 5. The Pyrotec Company produces three electrical products – clocks, radios, and toasters. These products have the following resource requirements. Resource Requirements Cost/Unit Labor Hours/Unit Clock Radio Toaster $7 $10 $5 2 3 2

The manufacturer has a daily production budget of $2 000 and a maximum of 660 hours of labor. Maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters. Clocks sell for $15, radios for $20, and toasters for $12. The company desires to know the optimal product mix that will maximize profit. Formulate a linear programming model for this problem. Solution: x1 = number of clocks x2 = number of radios x1 = number of toasters The company’s objective is to maximize profit. The total profit is the sum of the individual profit (Profit = Selling Price – Cost) gained from each type of electrical products.

The objective function is expressed as maximize Z = ($15 ? $7 )x1 + ($20 ? $10)x 2 + ($12 ? $5)x3 or maximize Z = 8 x1 + 10 x2 + 7 x3 The first constraint is for daily production budget. The total budget available for production is $2 000: $7 x1 + $10 x2 + $5 x3 ? $2 000 . The second constraint is for time of labor. The maximum of available time is 660 hours. 10 2 x1 + 3 x2 + 2 x3 ? 660 hr The last three constraints reflect the customer demand for clocks, radios, and toasters. x1 ? 200 (clocks) x2 ? 300 (radios) x3 ? 150 (toasters) Answer: Maximize Z = 8 x1 + 10 x2 + 7 x3 subject to 7 x1 + 10 x2 + 5 x3 ? 2 000 2 x1 + 3 x2 + 2 x3 ? 660 x1 ? 00 x2 ? 300 x3 ? 150 x1 , x 2 , x 3 ? 0 The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. One gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50.

The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. Formulate a linear programming model for this problem. Solution: x1 = number of grams of Ingredient 1 x2 = number of grams of Ingredient 2 The company’s objective is to minimize cost. The total cost is the sum of the individual costs of each type of ingredient ? minimize Z = $80 x1 + $50 x2 . The requirements for antibiotics (1 , 2 , 3) represent the constraints of the model: 3×1 + x2 ? (antibiotic 1 , units ) 6. x1 + x2 2 x1 + 6 x2 Answer: Minimize Z = 80 x1 + 50 x2 subject to 3 x1 + x2 ? 6 ? 4 x1 + x2 2 x1 + 6 x2 ? 12 x1 , x2 ? 0 ? 4 (antibiotic 2 , units ) ? 12 (antibiotic 3 , units ) 11 Consider the following linear programming problem: 7. Maximize Z = 300 x1 + 500 x2 subject to : 3×1 + 5 x2 ? 30 x1 + x2 ? 18 x1 , x2 ? 0 Why this problem has no solution? Solution: 3 x1 + 5 x2 = 30 or 30 ? 3 x1 x2 = ? 5 x2 = 6 ? 0. 6 x1 x1 0 10 x2 6 0 x1 + x2 = 18 or x2 = 18 ? x1 x1 0 18 x2 18 0 x2 18 15 12 9 3×1 + 5 x2 =30 6 3 0 3 6 9 12 15 x1 + x2 = 18 18 x1 Answer: Infeasible region; No solution

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