1. The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently promotional plans are under way for this year's program. Advertising alternatives include television, radio and newspaper. Audience estimates, costs, and maximum media Audience per advertisement 100,000 18,000 40,000 Cost per advertisement $2,000 5300 $600 Maximum media usage 20 To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of a. If the promotional budget is limited to $18.200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached? b. By how much would audience contact increase if an extra 5 S100 were allocated to the advertising budget? 2. The Hartman Company is trying to determine how much of each of two products to produce over the coming planning period. The following information concerns labor availability labor utilization, and product profitability Profit contribution unit a. Develop a linear programming model of the Hartman Company problem. Solve the model to determine the optimal production quantities of products 1 and 2. b. In computing the profit contribution per unit, Harman doesn't deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? c. Suppose that 10, 6, and 8 hours of overtime may be scheduled in departments A, B, and C, respectively. The cost per hour of overtime is S18 in department A, $22.50 in department B, and $12 in department C. Formulate a linear programming model that can be used to determine the optimal production quantities and what is the revised total contribution to profit? How much overtime do you recommend using in each department? What is the increase in the total contribution to profit if overtime is used? recommend using in each department? What is the in the total contribution to profit if overtime is used? 3. The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue - producing investments together with annual rates of return are as follows Type of loan / Imvestwest Annual Rate of Return(%) Automobile loans secured loanms Signature loans 12 Risk-free securities The credit union will have S2,000,000 available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments. Risk-free securities may not exceed 30% of the total funds available for investments Signature loans may not exceed 10% ofthe funds invested in all loans (automobile, furniture, and other secured and signature loans). Furniture loans plus other secured loans may not exceed the automobile loans. Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the S2,000,000 be allocated to each of the loan/investments alternatives to maximize total annual return? What is the projected total annual return? 4. Hilltop coffee manufactures a coffee product by blending three types of coffee beans. The cost per pound and the available pounds of each bean are as follows. 5045 Consumer tests with coffee products were used to provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product quality standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings for the aroma and taste for coffee made from 100% of each bean are as follows. Bean Aroma Rating Taste Rating 75 83 Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend. a. What is the minimum-cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee product? 14:39 6%EC%9E%A5%20%EC%84% 1] three types of coffee beans. The cost per pound and the available pounds of each bean are as follows. 50,70 Consumer tests with coffee products were used to provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product quality standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings for the aroma and taste for coffee made from 100% ofeach bean are as follows. Bean Aroma Rating Taste Rating 75 75 Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend a. What is the minimum-cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee product? b. What is the cost per pound for the coffee blend? c. Determined the aroma and taste ratings for the coffee blend. d. If additional coffee were to be produced, what would be the expected cost per pound? 5. Ajax Fuels, Inc, is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B and C For proper performance, the total amount of additive (amount of A+ amount of B + amount of C) must be at least 10 ounces per gallon for fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical. At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be greater than one-half the amount of ingredient A. If the costs per ounce for ingredients A, B and C are %0.10, S0.30, and S0.90, respectively, find the minimum-cost mixture of A, B and C for each gallon of airplane fuel. 6. G. Kunz and Sons, Inc., manufactures two products used in the heavy equipment industry. Both products require manufacturing operations in two departments. The following are the production time (in hours) and profit contribution figures for the two products. Product Profit Per Unit Labor-lears Dept A Laber-Hours Dept $20 Year 1 2 3 45 6 Paymenn 190213 240 283 315 460 The annual payments must be made at the beginning of each year. The judge will approve an amount that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to savings (at 4% annually) and govermment securities, at prices and rates currently quoted in The Wall Street Journal Hoxworth wants to develop a plan for making the annual payments by investing in the following securities (par value- 1,000). Funds not invested in these securities will be placed in savings. Security Current Price Rate (%) Year to Maturity SI0S5 6.70 $1000 5.125 Assume that interest is paid annually. The plan will be submitted to the judge and, if approved, Hoxworth will be required to pay a trustee the amount that will be required to fund the plan. a. Use linear programming to find the minimum cash settlement necessary to fund the annual payments. b. Use the dual price to determine how much more much Hoxworth should be willing to pay now to reduce the payment at the beginning of year 6 to $400,000 c. Use the dual price to determine how much more Hoxworth should be willing to pay to reduce the year 1 payment to 150,000? d. Suppose that the annual payments are to be made at the end of each year. Reformulate the model to accommodate this change. How much would Hoxworth save if this change could be negotiated? 8. The Clark County Sheriff's Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 AM.. noon, 4:00 PM.. 8:00 P.M., midnight, and 4:00 A.M. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty: Time of Day Minim Officers on Duty 4:00 p.m-8.00 PM :00 am-800am Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times (8:00 A.M., noon, 4:00 P.M., 8:00 PM., midnight, and 4:00 A.M) to minimize the total number of officers required. (Hint: Let x 1 the number of officers beginning work at 8:00 A.M., x2 -the number of officers beginning work at noon, and so on.) 1. The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently promotional plans are under way for this year's program. Advertising alternatives include television, radio and newspaper. Audience estimates, costs, and maximum media Audience per advertisement 100,000 18,000 40,000 Cost per advertisement $2,000 5300 $600 Maximum media usage 20 To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of a. If the promotional budget is limited to $18.200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached? b. By how much would audience contact increase if an extra 5 S100 were allocated to the advertising budget? 2. The Hartman Company is trying to determine how much of each of two products to produce over the coming planning period. The following information concerns labor availability labor utilization, and product profitability Profit contribution unit a. Develop a linear programming model of the Hartman Company problem. Solve the model to determine the optimal production quantities of products 1 and 2. b. In computing the profit contribution per unit, Harman doesn't deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? c. Suppose that 10, 6, and 8 hours of overtime may be scheduled in departments A, B, and C, respectively. The cost per hour of overtime is S18 in department A, $22.50 in department B, and $12 in department C. Formulate a linear programming model that can be used to determine the optimal production quantities and what is the revised total contribution to profit? How much overtime do you recommend using in each department? What is the increase in the total contribution to profit if overtime is used? recommend using in each department? What is the in the total contribution to profit if overtime is used? 3. The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue - producing investments together with annual rates of return are as follows Type of loan / Imvestwest Annual Rate of Return(%) Automobile loans secured loanms Signature loans 12 Risk-free securities The credit union will have S2,000,000 available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments. Risk-free securities may not exceed 30% of the total funds available for investments Signature loans may not exceed 10% ofthe funds invested in all loans (automobile, furniture, and other secured and signature loans). Furniture loans plus other secured loans may not exceed the automobile loans. Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the S2,000,000 be allocated to each of the loan/investments alternatives to maximize total annual return? What is the projected total annual return? 4. Hilltop coffee manufactures a coffee product by blending three types of coffee beans. The cost per pound and the available pounds of each bean are as follows. 5045 Consumer tests with coffee products were used to provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product quality standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings for the aroma and taste for coffee made from 100% of each bean are as follows. Bean Aroma Rating Taste Rating 75 83 Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend. a. What is the minimum-cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee product? 14:39 6%EC%9E%A5%20%EC%84% 1] three types of coffee beans. The cost per pound and the available pounds of each bean are as follows. 50,70 Consumer tests with coffee products were used to provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product quality standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings for the aroma and taste for coffee made from 100% ofeach bean are as follows. Bean Aroma Rating Taste Rating 75 75 Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend a. What is the minimum-cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee product? b. What is the cost per pound for the coffee blend? c. Determined the aroma and taste ratings for the coffee blend. d. If additional coffee were to be produced, what would be the expected cost per pound? 5. Ajax Fuels, Inc, is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B and C For proper performance, the total amount of additive (amount of A+ amount of B + amount of C) must be at least 10 ounces per gallon for fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical. At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be greater than one-half the amount of ingredient A. If the costs per ounce for ingredients A, B and C are %0.10, S0.30, and S0.90, respectively, find the minimum-cost mixture of A, B and C for each gallon of airplane fuel. 6. G. Kunz and Sons, Inc., manufactures two products used in the heavy equipment industry. Both products require manufacturing operations in two departments. The following are the production time (in hours) and profit contribution figures for the two products. Product Profit Per Unit Labor-lears Dept A Laber-Hours Dept $20 Year 1 2 3 45 6 Paymenn 190213 240 283 315 460 The annual payments must be made at the beginning of each year. The judge will approve an amount that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to savings (at 4% annually) and govermment securities, at prices and rates currently quoted in The Wall Street Journal Hoxworth wants to develop a plan for making the annual payments by investing in the following securities (par value- 1,000). Funds not invested in these securities will be placed in savings. Security Current Price Rate (%) Year to Maturity SI0S5 6.70 $1000 5.125 Assume that interest is paid annually. The plan will be submitted to the judge and, if approved, Hoxworth will be required to pay a trustee the amount that will be required to fund the plan. a. Use linear programming to find the minimum cash settlement necessary to fund the annual payments. b. Use the dual price to determine how much more much Hoxworth should be willing to pay now to reduce the payment at the beginning of year 6 to $400,000 c. Use the dual price to determine how much more Hoxworth should be willing to pay to reduce the year 1 payment to 150,000? d. Suppose that the annual payments are to be made at the end of each year. Reformulate the model to accommodate this change. How much would Hoxworth save if this change could be negotiated? 8. The Clark County Sheriff's Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 AM.. noon, 4:00 PM.. 8:00 P.M., midnight, and 4:00 A.M. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty: Time of Day Minim Officers on Duty 4:00 p.m-8.00 PM :00 am-800am Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times (8:00 A.M., noon, 4:00 P.M., 8:00 PM., midnight, and 4:00 A.M) to minimize the total number of officers required. (Hint: Let x 1 the number of officers beginning work at 8:00 A.M., x2 -the number of officers beginning work at noon, and so on.)