Exam 9: Linear Programming Applications in Marketing, Finance, and Operations Management

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 and there is a constraint that HD  100.The solution indicates that interviewing another homeowner during the day will increase costs by 10.00.What do you know?

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.

The marketing research model presented in the textbook involves minimizing total interview cost subject to interview quota guidelines.

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. Component Requirements Plastic Time Space Cost to Manufacture Cost to 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.

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.

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 Summer Schedule Plastic Available Time Available Monthly Demand Horse Monthly Demand 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.

To study consumer characteristics,attitudes,and preferences,a company would engage in

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

A&C Distributors is a company that 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 .30 .06 Spring Shower 1800 .25 .05 April 7000 Water Wave 4000 .40 .09 Spring Shower 4000 .30 .06 May 6000 Water Wave 5000 .50 .12 Spring Shower 2000 .35 .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.

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.

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

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?

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.

Compared to the problems in the textbook,real-world problems generally require more variables and constraints.

The Meredith Ribbon Company produces paper and fabric decorative ribbon which 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 .03 30,000 Quarter 2 18,000 .04 20,000 Quarter 3 16,000 .06 20,000 Quarter 4 30,000 .08 15,000 An inventory holding cost of $.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 objection 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 company makes two products from steel; one requires 2 tons of steel and the other requires 3 tons.There are 100 tons of steel available daily.A constraint on daily production could be written as: 2x₁ + 3x₂  100.

For the multiperiod production scheduling problem in the textbook,period n  1's ending inventory variable was also used as period n's beginning inventory variable.

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.

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.

It is improper to combine manufacturing costs and overtime costs in the same objective function.