Grand Canyon University BUS 660. Chapters 7-8 Homework. Problems and Solutions- All Worked Solutions

BUS-660 Chapters 7-8 Homework Problem 7-15 Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard ...

BUS-660 Chapters 7-8 Homework Problem 7-15 Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard model, and a high-priced golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can make the bags at a competitive price, the distributor will purchase all the bags that Par can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows the production time requirements for the four required manufacturing operations and the accounting department’s estimate of the profit contribution per bag: The director of manufacturing estimates that 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time, and 135 hours of inspection and packaging time will be available for the production of golf bags during the next three months. A. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag. Standard = 300, Deluxe = 420 and Profit = 10,560 b. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.) then S = 708, D = 0 , Z = 14160 c. New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 10A + 9B is the appropriate objective function.) S = 540 ,D = 252 , Z = 7668 Problem 7-25 (Algorithmic) George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 9% for the bond fund and 20% for the stock fund. Whatever portion of the inheritance George finally decides to commit to the trust fund, he wants to invest at least 60% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 8.5%. a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. If required, round your answers to three decimal places. Let B = percentage of funds invested in the bond fund S = percentage of funds invested in the stock fund b. Solve the problem using the graphical solution procedure. If required, round the answers to one decimal place. Problem 7-41 Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil's distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons. a. Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. If required, round your answers to two decimal places. Let R = number of gallons of regular gasoline produced P = number of gallons of premium gasoline produced b. What is the optimal solution? R = 40000 P = 10000 Profit = $17000 c. What are the values and interpretations of the slack variables? d. What are the binding constraints? Grade A crude oil and Productive capacity are binding constriants Problem 7-49 (Algorithmic) PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 100 full-time-equivalent pharmacists (combination of full time and part time) and 175 full-time-equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 280 full-timeequivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management states that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $35 per hour and the average salary for technicians is $15 per hour. a. Determine a minimum-cost staffing plan for PharmaPlus. How many pharmacists and technicians are needed? 105*$35 =3675, 175*15 = 2625, 3675+2625 = $6300 a. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)? New Hires Required Pharmacists 15 Technicians 30 What will be the impact on the payroll? The payroll cost will Go up by $ per hour. Problem 8-19 Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows: Two operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced. Let P1 = units of product 1 P2 = units of product 2 P3 = units of product 3 0.00000 15.00000 Infinite a. How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit associated with your solution? $ 1250 b. How many hours of production time will be scheduled on each machine? If required, round your answers to two decimal places. Machine Hours Schedule: Machine 1: Hours; Machine 2: Hours c. What is the value of an additional hour of labor? If required, round your answers to two decimal places. $ d. Assume that labor capacity can be increased to 120 hours. Develop the optimal product mix, assuming that the extra hours are made available. 24,8,16 Profit: $ Would you be interested in using the additional 20 hours available for this resource? Y/N Problem 8-21 (Algorithmic) Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows: Type I rooms do not have wireless Internet access and are not available for the Business rental class. Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 120 rentals in the Super Saver class, 65 rentals in the Deluxe class, and 40 rentals in the Business class. Round Tree has 105 Type I rooms and 115 Type II rooms. he decision variable in Green The constrains are in Blue And the objective function is in Orange 31.25 37.50 12.50 1440 a. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. No the demand of each rental class is satisfied b. How many reservations can be accommodated in each rental class? c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered? As per current capacity this is not possible as the Deluxe class capacity is full. But if its allowed. then: Super Saver Type 1 Rev 3675 Type ii Revenue 240 Revenue 3915 Average per room 34.04348 As the average cost is already 34.04, which means upgrade to Deluxe will only cost $5 more with no revenue plus d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? TI or TII Rooms Avg income Rev from Type I 3675 105 35 Rev from Type II 4170 115 36.26087 So convert type ii as the revenue per room is higher. e. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? Yes, it can be but the demand for next night and the costs will be required Problem 8-25 (Algorithmic) Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here: For example, Cabinetmaker 1 estimates that it will take 45 hours to complete all the oak cabinets and 63 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 35 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 35/45 = 0.78, or 78%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 35/63 = 0.56, or 56%, of the cherry cabinets if it worked only on cherry cabinets. a. Formulate a linear programming model that can be used to determine the proportion of the oak cabinets and the proportion of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects. Let O1 = proportion of Oak cabinets assigned to cabinetmaker 1 O2 = proportion of Oak cabinets assigned to cabinetmaker 2 O3 = proportion of Oak cabinets assigned to cabinetmaker 3 C1 = proportion of Cherry cabinets assigned to cabinetmaker 1 C2 = proportion of Cherry cabinets assigned to cabinetmaker 2 C3 = proportion of Cherry cabinets assigned to cabinetmaker 3 a. b. Solve the model formulated in part (a). What proportion of the oak cabinets and what proportion of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places. Total Cost = $ 3368 c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Y Shadow price of cabinetmaker 1 is -5.52, Allowable increase in RHS of constraint 1 is 10 , Therefore, optimal solution would changed. Yes, If cabinetmaker 2 has 2 hours, then solution changes, because it is within the allowable range. d. Suppose Cabinetmaker 2 reduced its cost to $39 per hour. What effect would this change have on the optimal solution? If required, round your answers to three decimal places. No, if cabinetmaker 2 reduces it cost, solution would not change, because the change is within the allowable decrease. Explain. Problem 8-29 (Algorithmic) La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rose wine, and fruit juice. To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 25% and no more than 35% rose, and exactly 25% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 9500 gallons of white wine and 8000 gallons of rose wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1 3547.71 per gallon for the white and $1.5 per gallon for the rose; the fruit juice can be purchased for $0.5 per gallon. La Jolla Beverage Products can sell all of the wine cooler it can produce for $2.5 per gallon. A. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Sunk The objective of La Jolla is to maximize the profit contribution of wine cooler production by blending the ingredients. But in this situation, by maximizing the amount of ingredients required the profit contribution maximizes proportionately, thus the cost of wine and fruit juices are irrelevant here. The objective can be obtained even these cost values are not available. Cost of wine and fruit juice considered as relevant cost for this situation because these costs are needed to make a wine cooler. Management can recover these cost in futurb. Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution it will realize from this blend. If required, round your answers to one decimal place. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. Profit = (Revenue/gallon)(amount blended) – Cost wine and juice Profit Contribution c. If La Jolla could obtain additional amounts of the white wine, should it do so? Yes, the white wine availability constraint is binding constraint, means if more gallons are made available more amount of cooler will be blended. If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase? 3.05 d. If La Jolla Beverage Products could obtain additional amounts of the rosé wine, should it do so? N If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase? gallons to be purchased for $1.5 per gallon. No additional wine is needed at this point. e. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this shadow price? Requiring 50% plus one gallon of white wine would reduce profits by $3.05. They should reduce 50% minimum to increase profits. f. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this shadow price? Allowing the amount of fruit juice to exceed 20% by just one gallon will increase profits, so they should increase the amount in eac wine cooler.