Exam 4: Linear Programming Applications in Marketing, Finance and Operations Management

Portfolio selection problems should acknowledge both risk and return.

Let M be the number of units to make and 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

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.. to 2.. 8 2. to 6. 4 6.. to 10. 7 10. to 2.. 12 2.. to 6.. 10 6. to 10.. 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?

If a real-world problem is correctly formulated, it is not possible to have alternative optimal solutions.

For the marketing research problem presented in the textbook, the research firm's objective is to conduct the market survey so as to meet the client's needs at a minimum cost.

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. Toy Begin. Invty. June 1 Required Plastic Required Time Production Cost Production 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.

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 0 150 Nut Mix .50 3/8 10 400 Cereal Mix .20 1 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.

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.

Why should decision makers who are primarily concerned with marketing or finance or production know about linear programming?

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.

Discuss the need for the use of judgment or other subjective methods in mathematical modeling.

Let P_ij = 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,

The objective function for portfolio selection problems usually is maximization of expected return or

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.

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.

If an LP problem is not correctly formulated, the computer software will indicate it is infeasible when trying to solve it.

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 T_ij the hours transferred from department i to department j, then

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 period's 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 period's investment. Expenses, or cash outflows, in each period must include repayment of the previous period's 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?

Give examples of how variations in the workforce assignment model presented in the textbook could be applied to other types of allocation problems.

The dual price for a constraint that compares funds used with funds available is .058. This means that