Q&A Hero

Q: NARRBEGIN: SA_117_120 A meat market manager for a large grocery store is preparing a processing plan to stock the shelves with sausage, ground meat, and jerky, which he can prepare from beef, pork and venison. Sausage and ground meat can be made of any mix of the beef, pork and venison, as long at the fat contents are below 15% for sausage and 10% for ground meat. Sausage sells for $5/pound and ground meat sells for $3/pound. Jerky, which sells or $10/pound, is made in a drying process from beef or venison. In the drying process, there is a 50% loss in weight for jerky made from beef (e.g., one pound of beef yields 0.5 pounds of beef jerky) and a 30% loss in weight for jerky made from venison. The market can sell at most 500 pounds of sausage, 1000 pounds of ground meat, and 100 pounds of jerky before their expiration dates. There are currently 1,000 pounds of beef (10% fat content), 500 pounds of pork (8% fat content), and 200 pounds of venison (2% fat content) available for processing. NARREND Determine the optimal processing plan for the meat market.

Q: The production manager believes the cost of the contract employees, who are currently in high demand, could be somewhat higher " perhaps as high as $10,000 per month. For the model in Question 113, perform a sensitivity analysis to determine the effect on the number of full-time employees that will be needed for the project.

Q: NARRBEGIN: SA_113_116 A construction company is preparing for a nine-month project, and will need to develop a staffing plan. The company can assign up to 30 of its own full-time employees to the project, and will hire short-term contract employees to make up any shortage in meeting the personnel requirements. Company employees earn $6,000 per month, while short-term contract employees make $8,600/month. Contract employees can be assigned to the project beginning in any month, and their contract period is two months. The number of workers required for the project by month is shown below: NARREND Suppose the bonus for completing the project three months early is $250,000. What would be the net bonus to the company, after adjusting for any difference in personnel costs under the accelerated schedule?

Q: NARRBEGIN: SA_113_116 A construction company is preparing for a nine-month project, and will need to develop a staffing plan. The company can assign up to 30 of its own full-time employees to the project, and will hire short-term contract employees to make up any shortage in meeting the personnel requirements. Company employees earn $6,000 per month, while short-term contract employees make $8,600/month. Contract employees can be assigned to the project beginning in any month, and their contract period is two months. The number of workers required for the project by month is shown below: NARREND The project manager is evaluating options to complete the project early so that the company can earn a bonus. He has determined that the project schedule can be compressed into a six-month schedule, with the same total number of worker-months. In that case, the staffing requirements are as shown below. Develop an optimal staffing plan for the project under the accelerated schedule.

Q: NARRBEGIN: SA_113_116 A construction company is preparing for a nine-month project, and will need to develop a staffing plan. The company can assign up to 30 of its own full-time employees to the project, and will hire short-term contract employees to make up any shortage in meeting the personnel requirements. Company employees earn $6,000 per month, while short-term contract employees make $8,600/month. Contract employees can be assigned to the project beginning in any month, and their contract period is two months. The number of workers required for the project by month is shown below: NARREND Determine the optimal staffing plan for the project.

Q: My friend Lee Meadow moved from Ferris State University in Michigan to Indiana University East in Indiana and has rented a truck that can haul up to 1100 cubic feet of furniture. The volume and value of each item he considered moving on the truck are giving below. Which items did he take with him to Indiana? What is the value of these items?

Q: A total of 160 hours of labor are available each week at $15 per hour. Additional labor can be purchased at $25 per hour. Capital can be purchased in unlimited quantities at a cost of $45 per unit. If Kunits of capital and Lunits of labor are available during a week, then machines can be produced. Each machine sells for $270. How can the firm maximize its weekly profit?

Q: A product can be produced on four different machines. Each machine has a fixed setup cost, variable production cost per unit processed, and a production capacity, as shown below: A total of 2000 units of the product must be produced. Determine how to minimize the total cost.

Q: A company produces three types of glue on two different production lines. Each line can be utilized by up to six workers at a time. Workers are paid $550 per week on production line 1 and $950 per week on production line 2. For a week of production it costs $1,050 to set up production line 1 and $2,050 to set up production line 2. During a week on a production line each worker produces the number of units of glue shown in the table below: Each week at least 120 units of glue 1, at least 150 units of glue 2, and at least 200 units of glue 3 must be produced. Determine how to minimize the total cost of meeting weekly demands.

Q: The financial CEO is given a group of possible investments projects for his company's capital. For each project, he is given the NPV that each project would add to the firm, as well as the cash outflow required by each project during each year as shown in the table below: Determine the investments that maximize the firm's NPV. The firm has 30 million dollars available during each of the next 5 years. All numbers are in millions of dollars.

Q: Laila, an Egyptian Broker, is currently trying to maximize her profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the table be below: Laila can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price. During each of the next 3 years, the person who sells a bond will pay the owner of the bond the cash payments shown in the table below: Laila's goal is to maximize her revenue from selling bonds less her payment for buying bonds, subject to the constraint that after each year's payments are received, her current cash position (due only to cash payments from bonds and not purchases or sales of bonds) is nonnegative. Note that her current cash position can depend on past coupons and that cash accumulated at the end of each year earns 13% annual interest. Determine how to maximize net profit from buying and selling bonds, subject to the constraints previously described. Why do you think we limit the number of units of each bond that can be bought or sold?

Q: NARRBEGIN: SA_104_105 A motorcycle company is determining its production schedule for the next four quarters. Demands for motorcycles are forecasted to be 40 in quarter 1; 70 in quarter 2; 50 in quarter 3; 20 in quarter 4. The company incurs four types of costs: It costs the company $400 to manufacture each motorcycle A holding cost of $100 per motorcycle left in inventory is incurred at the end of each quarter. Increasing production from one quarter to the next incurs costs for training employees. It is estimated that a cost of $700 per motorcycle is incurred if production is increased from one quarter to the next. Decreasing production from one quarter to the next incurs costs for severance pay, decreasing morale, and so forth. It is estimated that a cost of $600 per motorcycle is incurred if production is decreased from one quarter to the next. All demands must be met on time, and a quarter's production can be used to meet demand for the current quarter (as well as future quarters). During the quarter immediately preceding quarter1, 50 motorcycles were produced. Assume that at the beginning of quarter 1, no motorcycles are in inventory. NARREND Discuss how the company's optimal production schedule would be affected by a change in the cost of decreasing production from one quarter to the next.

Q: NARRBEGIN: SA_104_105 A motorcycle company is determining its production schedule for the next four quarters. Demands for motorcycles are forecasted to be 40 in quarter 1; 70 in quarter 2; 50 in quarter 3; 20 in quarter 4. The company incurs four types of costs: It costs the company $400 to manufacture each motorcycle A holding cost of $100 per motorcycle left in inventory is incurred at the end of each quarter. Increasing production from one quarter to the next incurs costs for training employees. It is estimated that a cost of $700 per motorcycle is incurred if production is increased from one quarter to the next. Decreasing production from one quarter to the next incurs costs for severance pay, decreasing morale, and so forth. It is estimated that a cost of $600 per motorcycle is incurred if production is decreased from one quarter to the next. All demands must be met on time, and a quarter's production can be used to meet demand for the current quarter (as well as future quarters). During the quarter immediately preceding quarter1, 50 motorcycles were produced. Assume that at the beginning of quarter 1, no motorcycles are in inventory. NARREND Discuss how the company's optimal production schedule would be affected by a change in the cost of increasing production from one quarter to the next.

Q: NARRBEGIN: SA_104_105 A motorcycle company is determining its production schedule for the next four quarters. Demands for motorcycles are forecasted to be 40 in quarter 1; 70 in quarter 2; 50 in quarter 3; 20 in quarter 4. The company incurs four types of costs: It costs the company $400 to manufacture each motorcycle A holding cost of $100 per motorcycle left in inventory is incurred at the end of each quarter. Increasing production from one quarter to the next incurs costs for training employees. It is estimated that a cost of $700 per motorcycle is incurred if production is increased from one quarter to the next. Decreasing production from one quarter to the next incurs costs for severance pay, decreasing morale, and so forth. It is estimated that a cost of $600 per motorcycle is incurred if production is decreased from one quarter to the next. All demands must be met on time, and a quarter's production can be used to meet demand for the current quarter (as well as future quarters). During the quarter immediately preceding quarter1, 50 motorcycles were produced. Assume that at the beginning of quarter 1, no motorcycles are in inventory. NARREND Determine how to minimize the company's total cost during the next four quarters.

Q: A large accounting firm has three auditors. Each can work up to 180 hours during the next month, during which time three projects must be completed. Project 1 takes 140 hours, project 2 takes 150 hours, and project 3 takes 170 hours. The amount per hour that can be billed for assigning each auditor to each project is given in the table below: Determine how to maximize total billings during the next month by formulating the company's problem as a transportation model.

Q: The risk index of an investment can be obtained by taking the absolute values of percentage changes in the value of the investment for each year and averaging them. Suppose you are trying to determine what percentage of your money you should invest in T-bills, gold, and stocks. The table below lists the annual returns (percentage changes in value) for these investments for the years 1968-1988. Let the risk index of a portfolio be the weighted average of the risk indexes of these investments, where the weights are the fractions of your money assigned to the investments. Suppose that the amount of each investment must be between 20% and 50% of the total invested. You would like the risk index of your portfolio to equal 0.15, and your goal is to maximize the expected return on your portfolio. Determine the maximum expected return on your portfolio, subject to the stated constraints. Use the average return earned by each investment during the years 1968-1988 as your estimate of expected return.

Q: A company blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $75 per pound. Fertilizer 2 must be at least 70% silicon and sells for $45 per pound. The company can purchase up to 9000 pounds of nitrogen at $16 per pound and up to 12,000 pounds of silicon at $12 per pound. Assuming that all fertilizer produced can be sold, determine how the company can maximize its profit.

Q: You have decided to enter the candy business. You are considering producing two types of candies: A and B, both of which consist solely of sugar, nuts, and chocolate. At present you have in stock 12,000 ounces of sugar, 3000 ounces of nuts, and 3000 ounces of chocolate. The mixture used to make candy A must contain at least 10% nuts and 10% chocolate. The mixture used to make candy B must contain at least 20% nuts. Each ounce of candy A can be sold for $0.40 and each ounce of candy for $0.50. Determine how you can maximize your revenues from candy sales.

Q: You are given the following means, standard deviations, and correlations for the annual return on three stocks. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20, 0.30, and 0.40. The correlation between stocks 1 and 2 is 0.65, between stocks 1 and 3 is 0.75, and between stocks 2 and 3 is 0.41. You have $13,000 to invest and can invest no more than half of your money in any single stock. Determine the minimum variance portfolio that yields an expected annual return of at least 0.14.

Q: A company manufactures two products. If it charges price for product I, it can sell units of product I, where and . It costs $25 to produce a unit of product 1 and $72 to produce a unit of product 2. How many units of each product should the company produce, and what prices should it charge, to maximize its profit?

Q: The cost per day of running a hospital is 250,000 + dollars, where x is the number of patients served per day. What number of patients served per day minimizes the cost per patient of running the hospital?

Q: A manufacturer can sell product 1 at a profit of $2 per unit and product 2 at a profit of $6 per unit. Three units of raw material are needed to manufacture one unit of product 1, and 6 units of raw material are needed to manufacture unit of product 2. A total of 120 units of raw material are available. If any of product 1 is produced, a setup cost of $10 is incurred, and if any of product 2 is produced, a setup cost of $20 is incurred. Determine how to maximize the manufacturer's profit.

Q: During the next 4 quarters, an automobile company must meet (on time) the following demands for cars: 4000 in quarter 1; 2000 in quarter 2; 5000 in quarter 3; 1000 in quarter 4. At the beginning of quarter 1, there are 300 autos in stock. The company has the capacity to produce at most 3000 cars per quarter. At the beginning of each quarter, the company can change production capacity. It costs $100 to increase quarterly production capacity by 1 unit. For example, it would cost $20,000 to increase capacity from 3000 to 3200. It also costs $60 per quarter to maintain each unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is $2200. A holding cost of $160 per car is assessed against each quarter's ending inventory. It is required that at the end of quarter 4, plant capacity must be at least 4000 cars. Determine how to minimize the total cost incurred during the next 4 quarters.

Q: NARRBEGIN: SA_93_94 An oil company has oil fields in San Diego and Los Angeles. The San Diego field can produce up to 500,000 barrels per day, and the Los Angeles field can produce up to 400,000 barrels per day. Oil is sent from the fields to a refinery, either in Dallas or in Houston. Assume that each refinery has unlimited capacity. To refine 100,000 barrels costs $725 at Dallas and $950 at Houston. Refined oil is shipped to customers in Chicago and New York. Chicago customers require 400,000 barrels per day, and New York customers require 300,000 barrels per day. The costs of shipping 100,000 barrels of oil (refined or unrefined) between cities are shown in the table below: NARREND (A) Determine how to minimize the total cost of meeting all demands. (B) If each refinery had a capacity of 380,000 barrels per day, how would you modify the model in (A)?

Q: An oil company controls two oil fields. Field 1 can produce up to 45 million barrels of oil per day, and field 2 can produce up to 55 million barrels of oil per day. At field1, it costs $3 to extract and refine a barrel of oil; at field 2, the cost is $2. The company sells oil to two countries: France and Japan. The shipping costs per barrel are shown below. Each day, France is willing to buy up to 45 million barrels (at $6 per barrel), and Japan is willing to buy up to 35 million barrels (at $6.50 per barrel). Determine how to maximize the company's profit.

Q: Linear programming models are used by many financial firms to select a desirable bond portfolio. The following is a simplified version of such a model. Abby is considering investing in four bonds; $1.5 million is available for investment. The expected annual return, the worst-case annual return on each bond, and the "duration" of each bond are given below (The duration of a bond is a measure of the bond's sensitivity to interest rates.) Abby wants to maximize the expected return from its bond investments, subject to the following three constraints: The worst-case return of the bond portfolio must be at least 8%. The average duration of the portfolio must be at most 6. For example, a portfolio that invests $600,000 in bond 1 and $400,000 in bond 4 has an average duration of [600,000(3) + 400,000 (9)]/1,000,000 = 5.4 Because of diversification requirements, at most 40% of the total amount invested can be invested in a single bond. Determine how Abby can maximize the expected return on her investment.

Q: A pharmaceutical company produces a drug from four chemicals. Today the company must produce 1000 pounds of the drug. The three active ingredients in the drug are labeled A, B, and C. By weight, at least 8% of the drug must consist of A, at least 4% must consist of B, and at least 2% must consist of C. The cost per pound of each chemical and the amount of each active ingredient in 1 pound of each chemical are given below. It is necessary that at least 100 pounds of chemical 2 be used. Determine the cheapest way of producing today's batch of this drug.

Q: During each 4-hour period, the police force in a small town in Ohio requires the following number of on-duty police officers: 8 from midnight to 4 A.M.; 7 from 4 A.M. to 8 A.M.; 6 from 8 A.M. to noon; 6 from noon to 4 P.M.; 5 from 4 P.M. to 8 P.M.; and 4 from 8 P.M. to midnight. Each police officer works two consecutive 4-hour shifts. Determine how to minimize the number of police officers needed to meet the town's daily requirements.

Q: The cost per day running a hotel is 200,000 + 0.002 dollars, where xis the number of customers served per day. What number of customers served per day minimizes the cost per customer of running the hotel?

Q: At time 0, you have $10,000. Investments A and B are available; their cash flows are shown in the table below: Assume that any money not invested in A or B earns interest at an annual rate of 8%. Determine how to maximize your cash on hand at time 3.InvestmentTime 0Time 1Time 2Time 3A-$1.00 $0.20$1.50$0.00B $0.00-$1.00$0.00$1.90

Q: NARRBEGIN: SA_84_86An oil company produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to customers in Los Angeles and New York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs (in dollars) of shipping 1000 barrels between various locations are shown below:NARREND(A) Assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at either Mobile or Galveston. To refine 10000 barrels of oil costs $12 at Mobile and $10 at Galveston. Assuming that both Mobile and Galveston have infinite refinery capacity, determine how to minimize the daily cost of transporting and refining the oil requirements of Los Angeles and New York.(B) Rework (A) under the assumption that Galveston has a refinery capacity of 150,000 barrels per day, and Mobile has a refinery capacity of 180,000 barrels per day.

Q: NARRBEGIN: SA_84_86An oil company produces oil at two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to customers in Los Angeles and New York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs (in dollars) of shipping 1000 barrels between various locations are shown below:FromWell 1Well 2MobileGalvestonNew YorkLos AngelesWell 1$10,000$10,000$10$13$25$28Well 2$10,000$10,000$15$12$26$25Mobile$10,000$10,000$10,000$6$16$17Galveston$10,000$10,000$6$10,000$14$16New York$10,000$10,000$10,000$10,000$10,000$15Los Angeles$10,000$10,000$10,000$10,000$15$10,000NARRENDDetermine how to minimize the transportation cost in meeting the oil demands of Los Angeles and New York.

Q: NARRBEGIN: SA_82_83Each year, a computer company produces up to 600 computers in New York and up to 500 computers in Memphis. Los Angeles customers must receive 600 computers, and 500 computers must be supplies to Oklahoma City customers. Producing a computer costs $850 in New York and $950 in Memphis. Computers are transported by plane and can be sent through Chicago. The costs of shipping a computer between cities are shown below.ToFromChicagoOklahoma CityL.A.New York$95$245$295Memphis$115$155$195ChicagoNA$60$65NARREND(A) Determine how to minimize the total (production plus distribution) costs of meeting the company's annual demand.(B) How would you modify the model in (A) if at most 300 units can be shipped through Chicago?

Q: NARRBEGIN: SA_80_81An auto company produces cars at Los Angeles and Detroit, and has a warehouse in Atlanta. The company supplies cars to dealers in Dallas and Orlando. The costs of shipping a car between various points are shown in the table below, where "NA" means that a shipment is not allowed. Los Angeles can produce up to 1400 cars, and Detroit can produce up to 3200 cars. Dallas must receive 2800 cars, and Orlando must receive 1800 cars. ToFromLADetroitAtlantaDallasOrlandoLA$12,000$190$150$140$275Detroit$195$12,000$160$160$160Atlanta$155$165$12,000$160$130Dallas$140$160$170$12,000$12,000Orlando$260$170$130$12,000$12,000NARREND(A) Determine how to minimize the cost of meeting demands at Dallas and Orlando(B) Modify the answer to (A) if shipments between Los Angeles and Detroit are not allowed.

Q: NARRBEGIN: SA_78_79A company supplies goods to three customers, each of whom requires 50 units. The company has two warehouses. In warehouse 1, 75 units are available, and in warehouse 2, 55 units are available. The costs of shipping one unit from each warehouse to each customer are shown in the table below.ToFromCustomer 1Customer 2Customer 3Warehouse 1$20$40$30Warehouse 2$15$35$45Not shipped (shortage)$100$90$115NARRENDSuppose that the company can purchase and ship extra units to either warehouse for a total cost of $125 per unit and that all customer demand must be met. Determine how to minimize the sum of purchasing and shipping costs.

Q: NARRBEGIN: SA_78_79A company supplies goods to three customers, each of whom requires 50 units. The company has two warehouses. In warehouse 1, 75 units are available, and in warehouse 2, 55 units are available. The costs of shipping one unit from each warehouse to each customer are shown in the table below.ToFromCustomer 1Customer 2Customer 3Warehouse 1$20$40$30Warehouse 2$15$35$45Not shipped (shortage)$100$90$115NARRENDDetermine how to minimize the sum of shortage and shipping costs.

Q: NARRBEGIN: SA_76_77A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required each day is given in the table below.MonTueWedThuFriSatSun20161822171914Union rules state that each full-time employee must work five consecutive days and then receive two days off. The post office wants to meet its daily requirements using only full-time employees. Its objective is to minimize the number of full-time employees that must be hired.NARREND(A) Use Solver to formulate and solve the post office's problem.(B) Suppose the post office has 30 full-time employees and is not allowed to hire or fire any employees. Determine a schedule that maximizes the number of weekend days off received by the employees.

Q: Suppose that on Monday morning you have $5000 in cash on hand. For the following seven days, the following cash requirements must be met: Monday, $6,000; Tuesday, $7,000; Wednesday, $10,000; Thursday, $3,000; Friday, $8,000; Saturday, $3,000; Sunday, $4,000. At the beginning of each day, you must decide how much money (if any) to withdraw from the bank. It costs $8 to make a withdrawal of any size. You believe that the opportunity cost of having $1 of cash on hand for a year is $0.25. Assume that opportunity costs are incurred on each day's ending balance. Determine how much money you should withdraw from the bank during each of the next 7 days.

Q: An oil delivery truck contains five compartments, holding up to 2800, 2900, 1200, 1800, and 3200 gallons of fuel, respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands, penalty per gallon short, and the maximum allowed shortage are shown in the table below. Each compartment of the truck can carry only one type of gasoline. Determine how to load the truck in a way that minimizes shortage costs. DemandCost (Gallon short)Maximum shortage allowedSuper3000$9400Regular4000$7400Unleaded5000$5400

Q: A company is considering investing a total amount of $2.50 million in four bonds. The expected annual return, the worst-case annual return on each bond, and the "duration" of each bond are given in the table below. Bond 1Bond 2Bond 3Bond 4Expected16%11%14%19%Worst case7%9%11%10%Duration45810The duration of a bond is a measure of the bond's sensitively to interest rates. The company wants to maximize the expected return from its bond investments, subject to the following constraints:The worst-case return of the bond portfolio must be at least 90%.The average duration of the portfolio must be at most 7Because of diversification requirements, at most 35% of the total amount invested in a single bond.Determine how the company can maximize the expected return on its investment.

Q: An electronic company is considering opening warehouses in New York, Los Angeles, Madison, and Tampa. Each warehouse can ship 125 units per week. The weekly fixed cost for keeping each warehouse open is $500 for New York, $600 for Los Angeles, $400 for Madison, and $200 for Tampa. Region 1 of the country requires 90 units per week, region 2 requires 80 units per week, and region 3 requires 50 units per week. The costs (including production and shipping costs) of sending one unit from a plant to a region are shown in the table below. ToFromRegion 1Region 2Region 3NY$25$45$55LA$52$19$30Madison$29$38$21Tampa$28$54$39Show how the company can meet weekly demands at a minimum cost, subject to the above information and the following restrictions:If the New York warehouse is opened, then Los Angeles must be opened.At most two warehouses can be opened.Either the Tampa or the Los Angeles warehouses must be opened.

Q: Assume that you are given the following means, standard deviations, and correlations for the annual return on three stocks. Stock 1Stock 2Stock 3Mean return0.150.180.23Stdev. of return0.180.280.38 Correlation matrix Stock 1Stock 2Stock 3Stock 11.000.620.72Stock 20.621.000.39Stock 30.720.391.00The correlation between stocks 1 and 2 is 0.62, between stocks 1 and 3 is 0.72, and between stocks 2 and 3 is 0.39. You have $12,000 to invest and can invest no more than 55% of your money in any single stock. Determine the minimum variance portfolio that yields an expected annual return of at least 0.15

Q: A statistician is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the table below. The statistician can buy up to 1300 units of each bond at the ask price or sell up to 1300 units of each bond at the bid price. During each of the next three years the person who sells a bond will pay the owner of the bond the cash payments that are also shown in the table below. The statistician's goal is to maximize his revenue from selling bonds less his payments for buying bonds, subject to the constraint that after each year's payments are received, his current cash position (due only to cash payments from bonds and not purchases or sales of bonds) is nonnegative. His current cash position can depend on past coupons and that cash accumulated at the end of each year earns 12% annual interest. Determine how to maximize net profit from buying and selling bonds, subject to the constraints previously described.Bid (for selling) and ask (for buying) prices of bonds Bond 1Bond 2Bond 3Bond 4Bid$1,000$99$980$960Ask$1,020$1,015$1,002$184Cash payments from seller to buyer Bond 1Bond 2Bond 3Bond 4Year 1$120$100$90$70Year 2$140$130$110$90Year 3$1,300$1,320$1,290$1,310

Q: A company has daily staffing requirements for two types of jobs, cleaning and customer service persons. The minimum numbers of workers required each day for each type of job are shown in the table below. To meet these requirements, the company can employ three types of workers: those who clean only, those who can perform customer service only, and those who are able to do both. In each of these three categories, the company wants to meet its daily requirements using only full-time workers. A full-time worker must work five consecutive days with two days off. Workers who are able to perform only one type of work (cleaning or customer service) earn $50 per day. Those who are able to perform both types of work earn $60 per day. As a matter of policy, the company wants to ensure that at least 20% of its total hours are staffed by "swing workers"; those who can do both types of jobs. The company wants to find a staffing policy that covers the daily worker requirements at minimum total costs per week. Use solver to formulate and solve the company's problem.Type of job Mon Tue Wed Thu Fri Sat SunCleaning 8 7 7 10 9 15 11Customer service 13 13 9 14 18 20 19

Q: Chemical Bank is attempting to determine where its assets should be invested during the current year. At present, $800,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be the following: bonds, 12%, home loans, 18%, auto loans, 15%, personal loans, 22%. To ensure that the banks portfolio is not too risky, the bank's investment manager has placed the following restrictions on the bank portfolio:No more than 30% of the total amount invested may be in personal loansThe amount invested in home loans cannot exceed the amount invested in auto loansThe amount invested in personal loans cannot exceed the amount invested in bonds.Determine how the bank can maximize the annual return on its investment portfolio.

Q: A company manufactures two products. If it charges price for product , it can sell units of product , where and . It costs the company $20 to produce a unit of product 1 and $65 to produce a unit of product 2. How many units of each product should the company produce, and what prices should it charge, to maximize its profit?

Q: An auto company must meet (on time) the following demands for cars: 5000 in quarter 1; 3000 in quarter 2; 6000 in quarter 3; 2000 in quarter 4. At the beginning of quarter 1, there are 500 autos in stock. The company has the capacity to produce at most 3600 cars per quarter. At the beginning of each quarter, the company can change production capacity. It costs $125 to increase quarterly production capacity by one unit. It also costs $60 per quarter to maintain each unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is $2400. A holding cost of $200 per car is assessed against each quarter's ending inventory. It is required that at the end of quarter 4, plant capacity must be at least 5000 cars. Determine how to minimize the total cost incurred during the next four quarters.

Q: A Michigan company consists of three subsidiaries. Each has the respective average payroll, unemployment reserve fund, and estimated payroll shown in the table below (all figures are in millions of dollars). Any employer in the state of Michigan whose reserve to average payroll ratio is less than 1 must pay 25% of its estimated payroll in unemployment insurance premiums. Otherwise, if the ratio is at least one, the employer pays 13%. The company can aggregate its subsidiaries and label them as separate employers. For example, if subsidiaries 1 and 2 are aggregated, they must pay 25% of their combined payroll in unemployment insurance premiums. Determine which subsidiaries should be aggregated.SubsidiaryAve. PayrollReserveEst. Payroll135045040026505604503850650550

Q: A global optimal solution is not necessarily the best solution overall.

Q: A local optimal solution is better than all nearby solutions, but a solution far away might be better than it.

Q: For some types of integer programming problems, their LP relaxation solutions are optimal.

Q: When we solve a nonlinear programming problem (NLP), it is very possible that Solver will obtain the wrong answer.

Q: A nonlinear programming problem (NLP) is an optimization problem in which the objective function and/or the constraints are not linear functions of the decision variables.

Q: In a set-covering model, each member of a given set (set 1) must be "covered" by an acceptable member of another set (set 2). The objective of such problems is to minimize the number of elements in set 2 that are needed to cover all the elements in set 1.

Q: Any integer programming problem involving 0-1 variables with only one constraint is called a knapsack problem.

Q: A 0-1 variable, also called a binary variable, is a variable that must equal 0 or 1.

Q: In aggregate planning models, we can model backlogging of demand by allowing a month's inventory to be negative.

Q: Aggregate planning models are usually implemented through a rolling planning horizon.

Q: In aggregate planning models, the number of workers available influences the possible production levels.

Q: In an optimized network flow model (MCNFM), all the available capacity will be used.

Q: The flows in a general minimum cost network flow model (MCNFM) do all necessarily have to be from "left to right"; that is, from supply points to demand points.

Q: Transshipment points are locations where goods neither originate nor end up, but goods are allowed to enter such points to be shipped out to their eventual destinations.

Q: A good shipping plan uses as many cheap routes as possible, but ultimately is constrained by capacities and demands.

Q: In network models of transportation problems, arcs represent the routes for getting a product from one node to another.

Q: In transportation problems, the three sets of input numbers that are required are capacities, demands and flows.

Q: In transportation problems, shipments between supply points or between demand points are possible.

Q: NARRBEGIN: SA_114_120A chemical manufacturer produces two products, chemical X and chemical Y. Each product is manufactured by a two-step process that involves blending and mixing in machine A and packaging on machine B. Chemical X provides a $60/unit contribution to profit, while Chemical Y provides a $50 contribution to profit. The processing times for the two products on the mixing machine (A) and the packaging machine (B) are as follows:ProductMachine A (hours)Machine B (hours)Chemical X23Chemical Y42For the upcoming two-week period, machine A has available 80 hours and machine B has available 60 hours of processing time. Forecasts of the markets indicate that the manufacturer can expect to sell a maximum of 16 units of chemical X and 18 units of chemical Y.NARREND(A) Write out algebraic expressions for all of the constraints in this problem.(B) Construct a graph of the feasible region for this problem, given the constraints you identified in (A).(C) Describe how you would find the location of the optimal solution in the feasible region you graphed in (B).(D) Use the procedure you described in (C) to identify the optimal production plan. Confirm your solution using Solver. What is the maximized profit?(E) What constraints are binding on the optimal solution? Use your graphical solution to explain your answer.

Q: NARRBEGIN: SA_114_120A chemical manufacturer produces two products, chemical X and chemical Y. Each product is manufactured by a two-step process that involves blending and mixing in machine A and packaging on machine B. Chemical X provides a $60/unit contribution to profit, while Chemical Y provides a $50 contribution to profit. The processing times for the two products on the mixing machine (A) and the packaging machine (B) are as follows:ProductMachine A (hours)Machine B (hours)Chemical X23Chemical Y42For the upcoming two-week period, machine A has available 80 hours and machine B has available 60 hours of processing time. Forecasts of the markets indicate that the manufacturer can expect to sell a maximum of 16 units of chemical X and 18 units of chemical Y.NARRENDWrite out an algebraic expression for the objective function in this problem.

Q: NARRBEGIN: SA_114_120A chemical manufacturer produces two products, chemical X and chemical Y. Each product is manufactured by a two-step process that involves blending and mixing in machine A and packaging on machine B. Chemical X provides a $60/unit contribution to profit, while Chemical Y provides a $50 contribution to profit. The processing times for the two products on the mixing machine (A) and the packaging machine (B) are as follows:ProductMachine A (hours)Machine B (hours)Chemical X23Chemical Y42For the upcoming two-week period, machine A has available 80 hours and machine B has available 60 hours of processing time. Forecasts of the markets indicate that the manufacturer can expect to sell a maximum of 16 units of chemical X and 18 units of chemical Y.NARRENDWhat are the decision variables in this problem?

Q: NARRBEGIN: SA_112_113Adam Enterprises manufactures two products. Each product can be produced on either of two machines. The time (in hours) required to make each product on each machine is shown below:Each month, 500 hours of time are available on each machine, and also customers are willing to buy up to the quantities of each product at the prices shown below:The company's goal is to maximize the revenue obtained from selling units during the next two months.NARREND(A) Determine how the company can meet its goal. Assume that Adam will not produce any units in either month that it cannot sell in that month.(B) Referring to (A), suppose Adam wants to see what will happen if customer demands for each product in each month simultaneously change by a factor 1 + k. Revise the model so that you can use the SolverTable add-in to investigate the effect of this change on total revenue as k varies from -0.3 to 0.3 in increments of 0.1. Does revenue change in a linear manner over this range? Can you explain intuitively why it changes in the way it does?

Q: NARRBEGIN: SA_109_111Mary has a limited food budget, but still wants to make sure her family members meet their daily nutritional requirements. Mary can buy two foods. Food 1 sells for $7 per pound, and each pound contains 3 units of vitamin A and 1 unit of vitamin C. Food 2 sells for $1 per pound, and each pound contains 1 unit of each vitamin. Each day, the family needs at least 12 units of vitamin A and 6 units of vitamin C.NARREND(A) Verify that Mary should purchase 12 units of food 2 each day and thus oversatisfy the vitamin C requirement by 6 units.(B) Mary's husband has put his foot down and demanded that Mary fulfill the family's daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C. The optimal solution to the new problem will involve ingesting less vitamin C, but it will be more expensive, why?(C) Starting with the optimal solution to (B), use the SloverTable add-in to see what happens to the total cost when the vitamin A and vitamin C requirements both vary (independently) from 3 to 18 in 3-unit increments. That is, from a two-way table. Describe the behavior you observe. In particular, are the changes in total cost the same as you look across each row of the table? Are they the same as you look across each column of the table?

Q: NARRBEGIN: SA_107_108A company must meet (on time) the following demands: quarter 1, 3000 units; quarter 2, 2000 units; quarter 3, 4000 units. Each quarter, up to 2700 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 20% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be use used to meet any future demands. After each quarter's demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarter's ending inventory.NARREND(A) Determine how to minimize the total cost of meeting the next 3 quarters' demand. Assume that 1000 usable units are available at the beginning of quarter 1.(B) Referring to (A), the company wants to know how much money it would be worth to decrease the percentage of unsuitable items and/or the percentage of items that spoil. Write a short report that provides relevant information. Base your report on two uses of the SolverTable add-in: one where the percentage of unsuitable items decreases and the percentage of items that spoil stays at 10%; and one where the percentage of unsuitable items stays at 20% and the percentage of items that spoil decreases.

Q: NARRBEGIN: SA_105_106A company faces the following demands during the next 3 weeks: week 1, 20 units; week 2, 10 units; week 3, 15 units. The unit production cost during each week is as follows: week 1, $13; week 2, $14; week 3, $15. A holding cost of $2 per unit is assessed against each week's ending inventory. At the beginning of week 1, the company has 5 units on hand. Since not all goods produced during a month can be used to meet the current month's demand, assume that half of the goods produced during a week can be used to meet the current week's demands.NARREND(A) Determine how to minimize the cost of meeting the demand for the next three weeks.(B) Revise the model in Question 106 so that the demands are of the form, where is the original demand in month t, k is a factor, and is an amount of change in month t. Formulate the model in such away that you can use the SolverTable add-in to analyze changes in the amounts produced and the total cost when k varies from 0 to 10 in 1-unit increments, for any fixed values of the 's. For example, try this when = 2, = 5, and = 3. Describe the behavior you observe in the table. Can you find any "reasonable" 's that induce positive production levels in week 3?

Q: NARRBEGIN: SA_103_104A customer requires 50, 65, 100, and 70 units of a commodity during the next 4 months, respectively, and no backlogging is allowed; that is, the customer's requirements must be met on time. Production costs $5, $8, $4, and $7 per unit during these months. The storage cost from one month to the next is $2 per unit (assessed on ending inventory). It is estimated that each unit on hand at the end of month 4 could be sold for $6.NARREND(A) Determine how to minimize the net cost incurred in meeting the demands for the next four months.(B) Starting with the optimal solution to (A), use SolverTable add-in to see what happens to the decision variables and the total cost when the initial inventory varies from 0 (the implied value in (A)) to 100 in 10-units increments. How much lower would the total cost be if the company started with 10 units in inventory, rather than none? Would the same cost decrease occur for every 10-init increase in initial inventory?

Q: NARRBEGIN: SA_91_102 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold. NARREND (A) Find an optimal solution to the problem. What is the production plan, and what is the total revenue? (B) Obtain a sensitivity report for the solution reported in (A). Which constraints are binding? (C) What is the incremental contribution associated with adding an hour of assembly time? Over what range of increase is the marginal value valid? (D) What is the value of additional capacity on the polisher? How much increase and decrease in this capacity is possible before a change occurs in the optimal production schedule? (E) An advertising agency has devised a marketing plan for the Valley Chassis Company that will increase the market for Deluxe chassis. The plan will increase demand by 75 Deluxe chassis per month at a cost of $100 per month. Should Valley adopt the plan? Briefly explain why. (F) Suppose that four more hours of chassis assembly time could be made available. How much would profit change? (G) Suppose next that Valley's marketing department proposes lowering the price for a standard chassis from $12 to $11.50 so that more can be sold (since there is slack under the demand constraint). Would the optimal solution change? Explain why, or why not. (H) If Valley could obtain 1,000 pounds more of raw material (steel or aluminum), which should it procure? How much should they be willing to pay per pound for the steel or aluminum? Explain your answer. (I) In doing some contingency planning, Valley thinks that the aging stamping machine will soon need to be taken down for repairs that could last 2 months and will cost $10,000. During that time, they can continue to operate by outsourcing the stamping at $2.50 per chassis (deluxe or standard), although the capacity will be reduced from 2,500 to 1,500. What will be the total cost to repair the stamping machine?

Q: NARRBEGIN: SA_91_102 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold. NARREND What are the constraints in this problem?

Q: NARRBEGIN: SA_91_102 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold. NARREND What is the objective function in this problem?

Q: NARRBEGIN: SA_91_102 Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers, modems, and other telecommunications equipment. The contribution margin (contribution toward profit) for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound. Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack. During the next production cycle, Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis. Similarly, 20,400 pounds of aluminum are available. One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum. A Deluxe chassis rack requires 12 pounds of each metal. The output of metal sheeting is restricted only by the capacity of the polisher. For the next production cycle, the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting. Chassis manufacture can be restricted by either metal stamping or assembly operations; no polishing is required. During the cycle no more than 2,500 total chassis can be stamped, and there will be 920 hours of assembly time available. The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack. Finally, market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe. Any quantities of metal sheeting can be sold. NARREND What are the decision variables in this problem?

Q: NARRBEGIN: SA_87_90Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which produces 5000 tons per month and the Lakeview plant which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:Grand RapidsBlue RidgeSunsetOhio Valley5040100Lakeview755075NARRENDWhat is the optimal shipping plan? What are the total costs in that case?

Q: NARRBEGIN: SA_87_90Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which produces 5000 tons per month and the Lakeview plant which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:Grand RapidsBlue RidgeSunsetOhio Valley5040100Lakeview755075NARRENDWhat are the constraints in this problem?