Chapter 4 study test 2 Dec anal

47. Super City Discount Department Store is open 24 hours a day. The number of cashiers need in each four-hour period of a day is listed below. Period Cashiers Needed 10 p.m. to 2 a.m. 8 2 a.m. to 6 a.m. 4 6 a.m. to 10 a.m. 7 10 a.m. to 2 p.m. 12 2 p.m. to 6 p.m. 10 6 p.m. to 10 p.m. 15 ​ If cashiers work for eight consecutive hours, how many should be scheduled to begin working in each period in order to minimize the number of cashiers needed?

Let TNP = number of cashiers who begin working at 10 p.m. TWA = number of cashiers who begin working at 2 a.m. SXA = number of cashiers who begin working at 6 a.m. TNA = number of cashiers who begin working at 10 a.m. TWP = number of cashiers who begin working at 2 p.m. SXP = number of cashiers who begin working at 6 p.m. Min TNP + TWA + SXA + TNA + TWP + SXP s.t. TNP + TWA ≥ 4 TWA + SXA ≥ 7 SXA + TNA ≥ 12 TNA + TWP ≥ 10 TWP + SXP ≥ 15 SXP + TNP ≥ 8 all variables ≥ 0

21. Media selection problems usually determine a. how many times to use each media source. b. the coverage provided by each media source. c. the cost of each advertising exposure. d. the relative value of each medium.

a

34. Which operations management application model is most likely to have an objective function that minimizes the sum of manufacturing costs, purchasing costs, and overtime costs? a. blending b. make-or-buy decision c. workforce assignment d. production scheduling

b

14. The primary limitation of linear programmings applicability is the requirement that all decision variables be nonnegative. a. True b. False

false

24. The dual price for a constraint that compares funds used with funds available is 0.058. This means that a. the cost of additional funds is 5.8%. b. if more funds can be obtained at a rate of 5.5%, some should be. c. no more funds are needed. d. the objective was to minimize.

B

26. If Pij = the production of product i in period j, then to indicate that the limit on production of the companys three products in period 2 is 400, we would express it as ​ a. P 21 + P 22 + P 23 ≤ 400. b. P 12 + P 22 + P 32 ≤ 400. c. P 32 ≤ 400. d. P 23 ≤ 400.

B

23. An LP model for a marketing research application uses the variable HD to represent the number of homeowners interviewed during the day. The objective function minimizes the cost of interviewing this and other categories. There is a constraint that HD ≥ 100. The solution indicates that interviewing another homeowner during the day will increase costs by 10.00. What does this tell you about the HD variable? ​ a. The objective function coefficient of HD is 10. b. The dual price for the HD constraint is 10. c. The objective function coefficient of HD is −10. d. The dual price for the HD constraint is −10.

D

25. Let M be the number of units to make and let B be the number of units to buy. If it costs $2 to make a unit and $3 to buy a unit, and 4000 units are needed, the objective function is a. Max 2M + 3B. b. Min 4000 (M + B). c. Max 8000M + 12,000B. d. Min 2M + 3B.

D

51. Wes Wheeler is the production manager of Wheeler Wheels, Inc. Wes has just received orders for 1,000 standard wheels and 1,250 deluxe wheels next month and for 800 standard and 1,500 deluxe wheels the following month. All orders are to be filled. ​ The cost of producing standard wheels is $10 and the cost for deluxe wheels is $16. Overtime rates are 50% higher. There are 1,000 hours of regular time and 500 hours of overtime available each month. It takes 0.5 hour to make a standard wheel and 0.6 hour to make a deluxe wheel. The cost of storing one wheel from one month to the next is $2. ​ Wes wants to develop a two-month production schedule for standard and deluxe wheels. Formulate this production planning problem as a linear program.

Define the decision variables We want to determine the production levels, x j , as follows: ​ Month 1 Month 2 ​ Reg. Time Overtime Reg. Time Overtime Standard x1 x2 x5 x6 Deluxe x3 x4 x7 x8 ​ Let y 1 = number of standard wheels stored from month 1 to month 2 y 2 = number of deluxe wheels stored from month 1 to month 2 ​ Define the objective function Minimize total production and storage costs: Min (cost per wheel)(number of wheels produced) + 2y 1 + 2y 2 Min 10x 1 + 15x 2 + 16x 3 + 24x 4 + 10x 5 + 15x 6 + 16x 7 + 24x 8 + 2y 1 + 2y 2 ​ Define the constraints Standard Wheel Production Month 1 = (Requirements) + (Amount Stored) (1) x 1 + x 2 = 1000 + y 1 or x 1 + x 2 - y 1 = 1000 Deluxe Wheel Production Month 1 = (Requirements) + (Amount Stored) (2) x 3 + x 4 = 1250 + y 2 or x 3 + x 4 - y 2 = 1250 Standard Wheel Production Month 2 = (Requirements) - (Amount Stored) (3) x 5 + x 6 = 800 - y 1 or x 5 + x 6 + y 1 = 800 Deluxe Wheel Production Month 2 = (Requirements) - (Amount Stored) (4) x 7 + x 8 = 1500 - y 2 or x 7 + x 8 + y 2 = 1500 Regular Hours Used Month 1 ≤ Regular Hours Available Month 1: (5) 0.5x 1 + 0.6x 3 ≤ 1000 Overtime Hours Used Month 1 ≤ Overtime Hours Available Month 1: (6) 0.5x 2 + 0.6x 4 ≤ 500 Regular Hours Used Month 2 ≤ Regular Hours Available Month 2: (7) 0.5x 5 + 0.6x 7 ≤1000 Overtime Hours Used Month 2 ≤ Overtime Hours Available Month 2: (8) 0.5x 6 + 0.6x 8 ≤ 500

5. Production constraints frequently take the form: ​ Beginning inventory + Sales − Production = Ending inventory a. True b. False

False

6. If a real-world problem is correctly formulated, it is NOT possible to have alternative optimal solutions. a. True b. False

False

8. A company produces two different products from steel. One requires 3 tons of steel and the other requires 5 tons. There are 100 tons of steel available daily. A constraint on daily production could be written as: 3x 1 + 5x 2 ≥ 100. True or false

False

9. Double-subscript notation for decision variables should be avoided unless the number of decision variables exceeds nine. a. True b. False

False

Target Shirt Company makes three varieties of shirts: Collegiate, Traditional, and European. These shirts are made from different combinations of cotton and polyester. ​ The cost per yard of unblended cotton is $5 and for unblended polyester is $4. Target can receive up to 4000 yards of raw cotton and 3000 yards of raw polyester fabric weekly. The table below gives pertinent data concerning the manufacture of the shirts. ​ ​ Shirt Total Yards Fabric Requirement Weekly Contracts Weekly Demand Selling Price Collegiate 1.00 At least 50% cotton 500 600 $14.00 Traditional 1.20 No more than 20% polyester 650 850 $15.00 European 0.90 As much as 80% polyester 280 675 $18.00 ​ Formulate and solve this blending problem as a linear program.

Formulation Decision variables Not only must we decide how many shirts to make and how much fabric to purchase, we also need to decide how much of each fabric is blended into each shirt. ​ Let s j = total number of shirt style j produced fi = number of yards of material i purchased xij = yards of fabric i blended into shirt style j where i = 1 (cotton) or 2 (polyester) and j = 1 (Collegiate), 2 (Traditional), or 3 (European) ​ Objective function Maximize the overall profit. To determine the profit function, subtract the cost of purchasing the fabric from the shirt sales revenue. Thus, the objective function is: ​ Max 14s 1 + 15s 2 + 18s 3 5f 1 4f 2 ​ Constraints Definition of total number of shirts of each style Total number of each style = Total yardage used in making the style/Yardage per shirt) (1) Collegiate: s 1 = (x 11 + x 21 )/1 >>> s 1 x 11 x 21 = 0 (2) Traditional: s 2 = (x 12 + x 22 )/1.2 >>> 1.2s 2 x 12 x 22 = 0 (3) European: s 3 = (x 13 + x 23 )/0.9 >>> 0.95s 3 x 13 x 23 = 0 ​ Definition of total yardage of materials (4) Cotton: f 1 = x 11 + x 12 + x 13 >>> f 1 x 11 x 12 x 13 = 0 (5) Polyester: f 2 = x 21 + x 22 + x 23 >>> f 2 x 21 x 22 x 23 = 0 ​ Weekly availability of the resources (6) Cotton: f 1 < 4000 (7) Polyester: f 2 < 3000 ​ Meet weekly contracts (8) Collegiate: s 1 > 500 (9) Traditional: s 2 > 650 (10) European: s 3 > 280 ​ Do not exceed weekly demand (11) Collegiate: s 1 < 600 (12) Traditional: s 2 < 850 (13) European: s 3 < 675 ​ Fabric requirements Collegiate at least 50% cotton: (Total yds. of cotton in Colleg. shirts) > [(0.5(1.00) yds./shirt)(number of Colleg. shirts)] (14) x 11 > 0.5s 1 >>> x 11 0.5s 1 > 0 ​ Traditional at most 20% polyester: (Total yds. polyester in Tradit. shirts) < [0.2(1.20) yds./shirt)(number of Tradit. shirts)] (15) x 22 < 0.24s2 >>> x 22 0.24s 2 < 0 ​ European at most 80% polyester: (Total yds. polyester in Europ. shirts) < [0.8(0.90) yds./shirt)(number of Europ. shirts)] (16) x 23 < 0.72s 3 >>> x 23 0.72s 3 < 0 ​ Nonnegativity of variables s j , f i , x ij > 0 for i = 1 or 2 and j = 1, 2, or 3 ​ Total profit = $23,152.50 Cotton Collegiate yds. = 300.0, cotton Traditional yds. = 816.0, cotton European yds. = 121.5, polyester Collegiate yds. = 300.0, polyester Traditional yds. = 204.0, polyester European yds. = 486.0, Collegiate shirts = 600, Traditional shirts = 850, European shirts = 675, cotton yards = 1237.5, polyester yards = 990.0

46. Evans Enterprises bought a prime parcel of beachfront property and plans to build a luxury hotel on it. After meeting with the architectural team, the Evans family has drawn up some information to make preliminary plans for construction. Excluding the suites, which are not part of this decision, the hotel will have four kinds of rooms: beachfront non-smoking, beachfront smoking, lagoon view non-smoking, and lagoon view smoking. To decide how many of the four kinds of rooms to plan for, the Evans family will consider the following information: ​ a. ​ ​ After adjusting for expected occupancy, the average nightly revenue for a beachfront non- smoking room is $175. The average nightly revenue for a lagoon view non-smoking room is $130. Smokers will be charged an extra $15. b. ​ ​ ​ Construction costs vary. The cost estimate is $12,000 for a lagoon view room and $15,000 for a beachfront room. Air purifying systems and additional smoke detectors and sprinklers ad $3000 to the cost of any smoking room. Evans Enterprises has raised $6.3 million in construction guarantees for this portion of the building. c. There will be at least 100 but no more than 180 beachfront rooms. d. ​ Design considerations require that the number of lagoon view rooms be at least 1.5 times the number of beachfront rooms, and no more than 2.5 times that number. e. ​ Industry trends recommend that the number of smoking rooms be no more than 50% of the number of non-smoking rooms. ​ Develop the linear programming model to maximize revenue.

Let BN = number of beachfront non-smoking rooms BS = number of beachfront smoking rooms LN = number of lagoon view non-smoking rooms LS = number of lagoon view smoking rooms Max 175BN + 190BS + 130LN + 145LS s.t. 15,000BN + 18,000BS + 12,000LN + 15,000LS ≤ 6,300,000 BN + BS ≥ 100 BN + BS ≤ 180 −1.5BN − 1.5BS + LN + LS ≥ 0 −2.5BN − 2.5BS + LN + LS ≤ 0 −0.5BN + BS − 0.5LN + LS ≤ 0 BN, BS, LN, LS ≥ 0

43. FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below. Ingredient Cost Volume Fat Grams Calories Dried fruit 0.35 1/4 cup 0 150 Nut mix 0.50 3/8 cup 10 400 Cereal mix 0.20 1 cup 1 50 ​ The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The blend should contain no more than 1000 calories and no more than 25 grams of fat. Dried fruit must be at least 20% of the volume of the mixture, and nuts must be no more than 15% of the weight of the mixture. Develop a model that meets these restrictions and minimizes the cost of the blend.

Let D = number of ounces of dried fruit mix in the blend N = number of ounces of nut mix in the blend C = number of ounces of cereal mix in the blend Min 0.35D + 0.50N + 0.20C s.t. 0.25D + 0.375N + C ≥ 3 0.25D + 0.375N + C ≤ 4 150D + 400N + 50C ≤ 1000 10N + C ≤ 25 0.2D − 0.075N − 0.2C ≥ 0 −0.15D + 0.85N − 0.15C ≤ 0 D, N, C ≥ 0

39. Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles. Current schedules yield the following information: Requirements Cost to Cost to Component Plastic Time Space Manufacture Purchase Front 3 10 2 8 12 Seat/frame 4 6 2 6 9 Rear wheel (each) 0.5 2 0.1 1 3 Available 50,000 160,000 30,000 The company obviously does not have the resources available to manufacture everything needed for the completion of 12,000 tricycles so it has gathered purchase information for each component. Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased in order to provide 12,000 fully completed tricycles at the minimum cost.

Let FM = number of fronts made SM = number of seats made WM = number of wheels made FP = number of fronts purchased SP = number of seats purchased WP = number of wheels purchased Min 8FM + 6SM + 1WM + 12FP + 9SP + 3WP s.t. 3FM + 4SM + 0.5WM ≤ 50,000 10FM + 6SM + 2WM ≤ 160,000 2FM + 2SM + 0.1WM ≤ 30,000 FM + FP ≥ 12,000 SM + SP ≥ 12,000 WM + WP ≥ 24,000 FM, SM, WM, FP, SP, WP ≥ 0

42. G&P Manufacturing would like to minimize the labor cost of producing dishwasher motors for a major appliance manufacturer. Although two models of motors exist, the finished models are indistinguishable from one another; their cost difference is due to a different production sequence. The time in hours required for each model in each production area, along with the labor cost, is shown in the table below. Model 1 Model 2 Area A 15 3 Area B 4 10 Area C 4 8 Cost 80 65 ​ Currently, labor assignments provide for 10,000 hours in both areas A and B and 18,000 hours in area C. If 2000 hours are available to be transferred from area B to area A and 3000 hours are available to be transferred from area C to either area A or area B, develop the linear programming model whose solution would tell G&P how many of each model to produce and how to allocate the workforce.

Let P 1 = number of model 1 motors to produce P 2 = number of model 2 motors to produce A A = number of hours allocated to area A A B = number of hours allocated to area B A C = number of hours allocated to area C T BA = number of hours transferred from B to A T CA = number of hours transferred from C to A T CB = number of hours transferred from C to B Min 80P 1 + 65P 2 s.t. 15P 1 + 3P 2 − A A ≤ 0 4P 1 + 10P 2 − A B ≤ 0 4P 1 + 8P 2 − A C ≤ 0 A A − T BA − T CA = 10,000 A B − T CB + T BA = 10,000 A C + T CA + T CB = 18,000 T BA ≤ 2000 T CA + T CB ≤ 3000 all variables ≥ 0

40. Tots Toys Company is trying to schedule production of two very popular toys for the next three months: a rocking horse and a scooter. Information about both toys is given below. Beg. Inv. Required Required Production Production Toy June 1 Plastic Time Cost Cost Rocking horse 25 5 2 12 1 Scooter 55 4 3 14 1.2 ​ Plastic Time Monthly Demand Monthly Demand Summer Schedule Available Available Horse Scooter June 3500 2100 220 450 July 5000 3000 350 700 August 4800 2500 600 520 ​ Develop a model that would tell the company how many of each toy to produce during each month. You are to minimize total cost. Inventory cost will be levied on any items in inventory on June 30, July 31, or August 31 after demand for the month has been satisfied. Your model should make use of the relationship: Beginning inventory + Production − Demand = Ending inventory for each month. The company wants to end the summer with 150 rocking horses and 60 scooters as beginning inventory for September 1. Dont forget to define your decision variables.

Let P ij = number of toy i to produce in month j S ij = surplus (inventory) of toy i at end of month j Min 12P 11 + 12P 12 + 12P 13 + 14P 21 + 14P 22 + 14P 23 + 1S 11 + 1S 12 + 1S 13 + 1.2S 21 + 1.2S 22 + 1.2S 23 s.t. P 11 − S 11 = 195 S 11 + P 12 − S 12 = 350 S 12 + P 13 − S 13 = 600 S 13 ≥ 150 P 21 − S 21 = 395 S 21 + P 22 − S 22 = 700 S 22 + P 23 − S 23 = 520 S 23 ≥ 60 5P 11 + 4P 21 ≤ 3500 5P 12 + 4P 22 ≤ 5000 5P 13 + 4P 23 ≤ 4800 2P 11 + 3P 21 ≤ 2100 2P 12 + 3P 22 ≤ 3000 2P 13 + 3P 23 ≤ 2500 P ij , S ij ≥ 0

41. Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order was for 1200 door lock units for an apartment complex developer. The sales and production departments must work together to determine delivery schedules. Each lock unit consists of three components: the knob and face plate, the actual lock itself, and a set of two keys. Although the processes used in the manufacture of the three components vary, there are three areas where the production manager is concerned about the availability of resources. These three areas, their usage by the three components, and their availability are detailed in the table below. Resource Knob and Plate Lock Key (each) Available Brass alloy 12 5 1 15,000 units Machining 18 20 10 36,000 minutes Finishing 15 5 1 12,000 minutes ​ A quick look at the amounts available confirms that Larkin does not have the resources to fill this contract. A subcontractor, who can make an unlimited number of each of the three components, quotes the prices below. Component Subcontractor Cost Larkin Cost Knob and plate 10.00 6.00 Lock 9.00 4.00 Keys (set of 2) 1.00 0.50 ​ Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.

Let PM = number of knob and plate units to make PB = number of knob and plate units to buy LM = number of lock units to make LB = number of lock units to buy KM = number of key sets to make KB = number of key sets to buy Min 6PM + 10PB + 4LM + 9LB + 0.5KM + 1KB s.t. 12PM + 5LM + 2KM ≤ 15,000 18PM + 20LM + 20KM ≤ 36,000 15PM + 5LM + 2KM ≤ 12,000 PM + PB ≥ 1200 LM + LB ≥ 1200 KM + KB ≥ 1200 PM, PB, LM, LB, KM, KB ≥ 0

45. Island Water Sports provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made on how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person and can accommodate six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. The company would like to maximize the number of customers served while generating at least $1800 in revenue and honoring all reservations.

Let R = number of Wildlife Raft Trips to schedule S = number of Group Sailing Lessons to schedule Max 6R + 4S s.t. R + S ≤ 12 R + 2S ≤ 18 6R ≥ 30 4S ≥ 20 120R + 160S ≥ 1800 R ≤ 10 2S ≤ 12 R,S>/=0

37. An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below. Medium Cost per Ad # Reached Exposure Quality TV 500 10,000 30 Radio 200 3000 40 Newspaper 400 5000 25 If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10,000, develop the model that will maximize the number reached and achieve an exposure quality of at least 1000.

Let T = number of TV ads R = number of radio ads N = number of newspaper ads Max 10,000T + 3000R + 5000N s.t. 500T + 200R + 400N ≤ 10,000 30T + 40R + 25N ≥ 1000 T − R ≤ 4 T, R, N ≥ 0

48. National Wing Company (NWC) is gearing up for the new B-48 contract. Currently, NWC has 100 equally qualified workers. Over the next three months, NWC has made the following commitments for wing production: Month Wing Production May 20 June 24 July 30 ​ Each worker can either be placed in production or train new recruits. A new recruit can be trained to be an apprentice in one month. The next month, he, himself, becomes a qualified worker (after two months from the start of training). Each trainer can train two recruits. The production rate and salary per employee are estimated below. Employee Production Rate (Wings/Month) Salary per Month Production 0.6 $3,000 Trainer 0.3 3,300 Apprentice 0.4 2,600 Recruit 0.05 2,200 ​ At the end of July, NWC wishes to have no recruits or apprentices but at least 140 full-time workers. Formulate and solve a linear program for NWC to accomplish this at minimum total cost.

P i = number of producers in month i (where i = 1, 2, or 3) T i = number of trainers in month i (where i = 1 or 2) A i = number of apprentices in month i (where i = 2 or 3) R i = number of recruits in month i (where i = 1 or 2) Min 3000P 1 + 3300T 1 + 2200R 1 + 3000P 2 + 3300T 2 + 2600A 2 + 2200R 2 + 3000P 3 + 2600A 3 s.t. 0.6P 1 + 0.3T 1 + 0.05R 1 ≥ 20 0.6P 1 + 0.3T 1 + 0.05R 1 + 0.6P 2 + 0.3T 2 + 0.4A 2 + 0.05R 2 ≥ 44 0.6P 1 + 0.3T 1 + 0.05R 1 + 0.6P 2 + 0.3T 2 + 0.4A 2 + 0.05R 2 + 0.6P 3 + 0.4A 3 ≥ 74 P 1 − P 2 + T 1 − T 2 = 0 P 2 − P 3 + T 2 + A 2 = 0 A 2 − R 1 = 0 A 3 − R 2 = 0 2T 1 − R 1 ≥ 0 2T 2 − R 2 ≥ 0 P 1 + T 1 = 100 P 3 + A 3 ≥ 140 P j , T j , A j , R j ≥ 0 for all j P 1 = 100, T 1 = 0, R 1 = 0, P 2 = 80, T 2 = 20, A 2 = 0, R 2 = 40, P 3 = 100, A 3 = 40 Total cost = $1,098,000

1. Media selection problems can maximize exposure quality and use number of customers reached as a constraint, or maximize the number of customers reached and use exposure quality as a constraint. a. True b. False

True

11. Compared to the problems in the textbook, real-world problems generally require more variables and constraints. a. True b. False

True

16. It can take longer to collect the data for a large-scale linear programming model than it does for either the formulation of the model or the development of the computer solution. a. True b. False

True

2. Linear programming is appropriate for financial problem situations involving capital budgeting, asset allocation, financial planning, and portfolio selection. a. True b. False

True

3. Portfolio selection problems should acknowledge both risk and return. a. True b. False

True

7. To properly interpret dual prices, one must know how costs were allocated in the objective function. a. True b. False

True

50. BP Cola must decide how much money to allocate for new soda and traditional soda advertising over the coming year. The advertising budget is $10,000,000. Because BP wants to push its new sodas, at least one-half of the advertising budget is to be devoted to new soda advertising. However, at least $2,000,000 is to be spent on its traditional sodas. BP estimates that each dollar spent on traditional sodas will translate into 100 cans sold, whereas, because of the harder sell needed for new products, each dollar spent on new sodas will translate into 50 cans sold. ​ To attract new customers, BP has lowered its profit margin on new sodas to 2 cents per can as compared to 4 cents per can for traditional sodas. How should BP allocate its advertising budget if it wants to maximize its profits while selling at least 750 million cans?

X 1 = amount invested in new soda advertising X 2 = amount invested in traditional soda advertising Max X 1 + 4X 2 s.t. X 1 + X 2 ≤ 10,000,000 X 1 ≥ 5,000,000 X 2 ≥ 2,000,000 50X 1 + 100X 2 ≥ 750,000,000 X 1 , X 2 ≥ 0 Spend $5,000,000 on new soda ad, spend $5,000,000 on traditional ad, profit = $25 million

49. John Sweeney is an investment advisor who is attempting to construct an "optimal portfolio" for a client who has $400,000 cash to invest. There are ten different investments, falling into four broad categories that John and his client have identified as potential candidates for this portfolio. The following table lists the investments and their important characteristics. Note that Unidyne equities (stocks) and Unidyne debt (bonds) are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment. ​ Expected Annual Liquidity Risk Category Investment After-Tax Return Factor Factor Equities Unidyne Corp. 15.0% 100 60 Col. Mustard Restaurant 17.0% 100 70 First General REIT 17.5% 100 75 Debt Metropolitan Electric 11.8% 95 20 Unidyne Corp. 12.2% 92 30 Lemonville Transit 12.0% 79 22 Real Estate Fairview Apartment Partnership 22.0% 0 50 First General REIT (see above) (see above) (see above) Money T-Bill Account 9.6% 80 0 Money Market Fund 10.5% 100 10 All Savers Certificates 12.6% 0 0 ​ Formulate and solve a linear program to accomplish John's objective as an investment advisor, which is to construct a portfolio that maximizes his clients total expected after-tax return over the next year, subject to the following constraints placed upon him by the client for the portfolio: 1. Its (weighted) average liquidity factor must be at least 65. 2. The (weighted) average risk factor must be no greater than 55. 3. At most, $60,000 is to be invested in Unidyne stocks or bonds. 4. No more than 40% of the investment can be in any one category except the money category. 5. No more than 20% of the investment can be in any one investment except the money market fund. 6. At least $1,000 must be invested in the money market fund. 7. The maximum investment in All Savers Certificates is $15,000. 8. The minimum investment desired for debt is $90,000. 9. At least $10,000 must be placed in a T-bill account.

X j = $ invested in investment j; where j = 1(Uni Eq.), 2(Col. Must.), 3(1st Gen REIT), 4(Met. Elec.), 5(Uni Debt), 6(Lem. Trans.), 7(Fair. Apt.), 8(T-Bill), 9(Money Market), 10(All Saver's) Max ​ 0.15X 1 + 0.17X 2 + 0.175X 3 + 0.118X 4 + 0.122X 5 + 0.12X 6 + 0.22X 7 + 0.096X 8 + 0.105X 9 + 0.126X 10 s.t. X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 + X 9 + X 10 = 400,000 100X 1 + 100X 2 + 100X 3 + 95X 4 + 92X 5 + 79X 6 + 80X 8 + 100X 9 ≥ 65(X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 + X 9 + X 10 ) 60X 1 + 70X 2 + 75X 3 + 20X 4 + 30X 5 + 22X 6 + 50X 7 + 10X 9 ≤ 55(X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 + X 9 + X 10 ) X 1 + X 5 ≤ 60,000 X 1 + X 2 + X 3 ≤ 160,000 X 4 + X 5 + X 6 ≤ 160,000 X 3 + X 7 ≤ 160,000 X 1 ≤ 80,000 X 2 ≤ 80,000 X 3 ≤ 80,000 X 4 ≤ 80,000 X 5 ≤ 80,000 X 6 ≤ 80,000 X 7 ≤ 80,000 X 8 ≤ 80,000 X 9 ≥ 1,000 X 10 ≤ 15,000 X 4 + X 5 + X 6 ≥ 90,000 X 8 ≥ 10,000 X j ≥ 0 j = 1, ... , 10 Solution: ​ X 1 = 0; X 2 = 80,000; X 3 = 80,000; X 4 = 0; X 5 = 60,000; X 6 = 74,000; X 7

27. Let Pij = the production of product i in period j. To specify that production of product 1 in period 3 and in period 4 differs by no more than 100 units, we would write it as ​ a. P 13 − P 14 ≤ 100; P 14 − P 13 ≤ 100. b. P 13 − P 14 ≤ 100; P 13 − P 14 ≥ 100. c. P 13 − P 14 ≤ 100; P 14 − P 13 ≥ 100. d. P 13 − P 14 ≥ 100; P 14 − P 13 ≥ 100.

a

33. The production scheduling problem modeled in the textbook involves capacity constraints on all of the following types of resources EXCEPT a. material. b. labor. c. machine. d. storage.

a

44. Meredith Ribbon Company produces paper and fabric decorative ribbon that it sells to paper products companies and craft stores. The demand for ribbon is seasonal. Information about projected demand and production for a particular type of ribbon is given. Demand (yards) Production Cost per Yard Production Capacity (yards) Quarter 1 10,000 0.03 30,000 Quarter 2 18,000 0.04 20,000 Quarter 3 16,000 0.06 20,000 Quarter 4 30,000 0.08 15,000 ​ An inventory holding cost of $0.005 is levied on every yard of ribbon carried over from one quarter to the next. a. Define the decision variables needed to model this problem. b. The objective is to minimize total cost, the sum of production and inventory holding cost. Give the objective function. c. Write the production capacity constraints. d. Write the constraints that balance inventory, production, and demand for each quarter. Assume there is no beginning inventory in quarter 1. e. To attempt to balance the production and avoid large changes in the workforce, production in period 1 must be within 5000 yards of production in period 2. Write this constraint.

a. Let P i = production in yards in quarter i S i = ending surplus (inventory) in quarter i b. Min 0.03P 1 + 0.04P 2 + 0.06P 3 + 0.08P 4 + 0.005(S 1 + S 2 + S 3 + S 4 ) c. P 1 ≤ 30,000 P 2 ≤ 20,000 P 3 ≤ 20,000 P 4 ≤ 15,000 d. P 1 − S 1 = 10,000 S 1 + P 2 − S 2 = 18,000 S 2 + P 3 − S 3 = 16,000 S 3 + P 4 − S 4 = 30,000 P 1 ≤ 30,000 P 2 ≤ 20,000 P 3 ≤ 20,000 P 4 ≤ 15,000

22. To study consumer characteristics, attitudes, and preferences, a company would engage in a. client satisfaction processing. b. marketing research. c. capital budgeting.

b

30. Blending problems arise whenever a manager must decide how to a. mix several different asset types in one investment strategy. b. mix two or more resources to produce one or more products. c. combine the results of two or more research studies into one. d. allocate workers with different skill levels to various work shifts.

b

31. In a production scheduling LP, the demand requirement constraint for a time period takes the form a. Beginning inventory + Production + Ending inventory >/=Demand. b. Beginning inventory - Production + Ending inventory = Demand. c. Beginning inventory + Production - Ending inventory = Demand. d. Beginning inventory - Production - Ending >/= Demand.

c

28. Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, then a. B ≤ 5. b. A − 0.5B + C ≤ 0. c. 0.5A − B − 0.5C ≤ 0. d. −0.5A + 0.5B − 0.5C ≤ 0.

d

29. Department 3 has 2500 hours. Transfers are allowed to departments 2 and 4, and from departments 1 and 2. If A i measures the labor hours allocated to department i and Tij the hours transferred from department i to department j, then ​ a. T 13 + T 23 − T 32 − T 34 − A 3 = 2500. b. T 31 + T 32 − T 23 − T 43 + A 3 = 2500. c. A 3 + T 13 + T 23 − T 32 − T 34 = 2500. d. A 3 − T 13 − T 23 + T 32 + T 34 = 2500.

d

32. A mutual fund manager must decide how much money to invest in Atlantic Oil (A) and how much to invest in Pacific Oil (P). At least 60% of the money invested in the two oil companies must be in Pacific Oil. A correct modeling of this constraint is a. 0.4A + 0.6P >/= 0. b. -0.4A + 0.6P >/= 0. c. 0.6A + 0.4P >/= 0. d. -0.6A + 0.4P >/=0

d

35. Blending problems occur frequently in which of the following industries? a. chemical b. petroleum c. food services d. All of these are correct.

d

10. Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the right-hand side is acceptable since both are a measure of time. a. True b. False

false

13. A company makes two products, A and B. A sells for $100 and B sells for $90. The variable production costs are $30 per unit for A and $25 for B. The companys objective could be written as: Max 190x1 − 55x2 . a. True b. False

false

15. A decision maker would be wise to not deviate from the optimal solution found by an LP model because it is the best solution. a. True b. False

false

4. It is improper to combine manufacturing costs and overtime costs in the same objective function. a. True b. False

false

12. For the multi period production scheduling problem in the textbook, period n − 1's ending inventory variable was also used as period n's beginning inventory variable. a. True b. False

true

17. The purpose of media selection applications of linear programming is to help marketing managers allocate a fixed advertising budget to various advertising media. a. True b. False

true

18. The marketing research model presented in the textbook involves minimizing total interview cost subject to interview quota guidelines. a. True b. False

true

19. A marketing research firm must determine how many daytime interviews (D) and evening interviews (E) to conduct. At least 40% of the interviews must be in the evening. A correct modeling of this constraint is: -0.4D + 0.6E >/= 0. a. True b. False

true

20. An important part of the LP process is to take the time to ensure the linear programming model accurately reflects the real problem. a. True b. False

true

52. Comfort Plus Inc. (CPI) manufactures a standard dining chair used in restaurants. The demand forecasts for quarter 1 (January-March) and quarter 2 (April-June) are 3700 chairs and 4200 chairs, respectively. CPI has a policy of satisfying all demand in the quarter in which it occurs. The chair contains an upholstered seat that can be produced by CPI or purchased from DAP, a subcontractor. DAP currently charges $12.50 per seat, but has announced a new price of $13.75, effective April 1. CPI can produce the seat at a cost of $10.25. CPI can produce up to 3800 seats per quarter. ​ Seats that are produced or purchased in quarter 1 and used to satisfy demand in quarter 2 cost CPI $1.50 each to hold in inventory, but maximum inventory cannot exceed 300 seats. ​ Formulate this problem as a linear programming problem.

x1 = number of seats produced by CPI in quarter 1 x2 = number of seats purchased from DAP in quarter 1 x3 = number of seats carried in inventory from quarters 1 to 2 x4 = number of seats produced by CPI in quarter 2 x5 = number of seats purchased from DAP in quarter 2 ​ Min 10.25x 1 + 12.5x 2 + 1.5x 3 + 10.25x 4 + 13.75x 5 (costs) ​ s.t. x 1 + x 2 - x 3 ≥ 3700 (quarter 1 demand) x 3 + x 4 + x 5 ≥> 4200 (quarter 2 demand) x 1 ≤ 3800 (CPIs production capacity in quarter 1) x 4 ≤ 3800 (CPI's production capacity in quarter 2) x 3 ≤< 300 (inventory capacity)

38. Information on a prospective investment for Wells Financial Services is given below. Period 1 2 3 4 Loan funds available 3000 7000 4000 5000 Investment income (% of previous periods investment) 110% 112% 113% Maximum investment 4500 8000 6000 7500 Payroll payment 100 120 150 100 ​ In each period, funds available for investment come from two sources: loan funds and income from the previous periods investment. Expenses, or cash outflows, in each period must include repayment of the previous periods loan plus 8.5% interest, and the current payroll payment. In addition, to end the planning horizon, investment income from period 4 (at 110% of the investment) must be sufficient to cover the loan plus interest from period 4. The difference in these two quantities represents net income and is to be maximized. How much should be borrowed, and how much should be invested each period?

​ Let L t = loan in period t, t = 1, ... ,4 I t = investment in period t, t = 1, ... ,4 Max 1.1I 4 − 1.085L 4 s.t. L 1 ≤ 3000 I 1 ≤ 4500 L 1 − I 1 = 100 L 2 ≤ 7000 I 2 ≤ 8000 L 2 + 1.1I 1 − 1.085L 1 − I 2 = 120 L 3 ≤ 4000 I 3 ≤ 6000 L 3 + 1.12I 2 − 1.085L 2 − I 3 = 150 L 4 ≤ 5000 I 4 ≤ 7500 L 4 + 1.13I 3 −1.085L 3 −I 4 = 100 1.10I 4 − 1.085L 4 ≥ 0 L t , I t ≥ 0

36. A&C Distributors represents many outdoor products companies and schedules deliveries to discount stores, garden centers, and hardware stores. Currently, scheduling needs to be done for two lawn sprinklers, the Water Wave and Spring Shower models. Requirements for shipment to a warehouse for a national chain of garden centers are shown below. Month Shipping Capacity Product Minimum Requirement Unit Cost to Ship Per-Unit Inventory Cost March 8000 Water Wave 3000 0.30 0.06 Spring Shower 1800 0.25 0.05 April 7000 Water Wave 4000 0.40 0.09 Spring Shower 4000 0.30 0.06 May 6000 Water Wave 5000 0.50 0.12 Spring Shower 2000 0.35 0.07 Let S ij be the number of units of sprinkler i shipped in month j, where i = 1 or 2, and j = 1, 2, or 3. Let W ij be the number of sprinklers that are at the warehouse at the end of a month, in excess of the minimum requirement. a Write the portion of the objective function that minimizes shipping costs. b. An inventory cost is assessed against this ending inventory. Give the portion of the objective function that represents inventory cost. c. There will be three constraints that guarantee, for each month, that the total number of sprinklers shipped will not exceed the shipping capacity. Write these three constraints. d. There are six constraints that work with inventory and the number of units shipped, making sure that enough sprinklers are shipped to meet the minimum requirements. Write these six constraints.

​ a. Min 0.3S 11 + 0.25S 21 + 0.40S 12 + 0.30S 22 + 0.50S 13 + 0.35S 23 b. Min 0.06W 11 + 0.05W 21 + 0.09W 12 + 0.06W 22 + 0.12W 13 + 0.07W 23 c. S 11 + S 21 ≤ 8000 S 12 + S 22 ≤ 7000 S 13 + S 23 ≤ 6000 d. S 11 − W 11 = 3000 S 21 − W 21 = 1800 W 11 + S 12 − W 12 = 4000 W 21 + S 22 − W 22 = 4000 W 12 + S 13 − W 13 = 5000 W 22 + S 23 − W 23 = 2000