Exam 4: Linear Programming Applications in Marketing, finance, and OM

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: 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.

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. ​ 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.

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

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

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.

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?

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

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 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

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

Information on a prospective investment for Wells Financial Services is given below. ​ 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? ​

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. ​ 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?

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. 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.​ ​

In a production scheduling LP,the demand requirement constraint for a time period takes the form

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.

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

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

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,we would write it as ​

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. ​ 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.​

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 ​