Chapter 9 Homework Problems (Linear Programming Business Application)

1. 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 10000 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 \$10000, develop the model that will maximize the number reached and achieve an exposure quality if at least 1000.

2. 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) .5 2 .1 1 3 Available 50000 160000 30000

The company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles so 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 12000 fully completed tricycles at the minimum cost.

3. The 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.

 Begin. Invty. 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 Sept. 1. Don’t forget to define your decision variables.

4. Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order has been 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.

 Resource Knob and Plate Lock Key (each) Available Brass Alloy 12 5 1 15000 units Machining 18 20 10 36000 minutes Finishing 15 5 1 12000 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 .50

Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.

5. G and 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 is tabled here, along with the labor cost.

 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 each of Areas A and B and 18000 hours in Area C. If 2000 hours are available to be transferred from area B to Area A, 3000 hours are available to be transferred from area C to either Areas A or 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.

6. 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 .35 1/4 cup 0 150 Nut Mix .50 3/8 cup 10 400 Cereal Mix .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.

 7. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of 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.

8. Evans Enterprises has bought a prime parcel of beachfront property and plans to build a luxury hotel. 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. In order to decide how many of each 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 for a lagoon view room is \$12,000 and for a beachfront room is \$15,000. 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.

9. 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?

10. 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 can 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 is estimated below.

 Employee Production Rate (Wings/Month) Salary Per Month Production .6 \$3,000 Trainer .3 3,300 Apprentice .4 2,600 Recruit .05 2,200

At the end of July, NWC wishes to have no recruits or apprentices but have at least 140 full-time workers. Formulate and solve a linear program for NWC to accomplish this at minimum total cost.

 11. 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?

 12. 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.

 13. A food processing company makes meatloaf to be sold in the frozen food section of supermarkets. Each week the recipe used changes based on the current cost of ingredients. Ingredients and current costs are as shown below: ​ Ingredient                                Cost/pound pork                                         \$2.87 hamburger                               \$1.63 wheat filler                              \$0.22 corn filler                                 \$0.16 ​ For each batch made, at least 300 pounds of pork and 100 pounds of hamburger are required. No more than 200 pounds of wheat filler can be used per batch and the amount of corn filler has to range between 50 and 150 pounds. Moreover, each batch must contain at least 500 pounds of meat and no more than 200 pounds of filler.  Develop the LP formulate for this problem. ​ ​