Exam 8: Linear Programming Applications

Table 8-6 The following is a linear programming formulation of a labor planning problem.There are four overlapping shifts,and management must decide how many employees to schedule to start work on each shift.The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day.The variables X₁ - X₄ represent the number of employees starting work on each shift (shift 1 through shift 4). -According to Table 8-6,which describes a labor planning problem and its solution,how many workers would actually be on duty during shift 1?

Table 8-1 A small furniture manufacturer produces tables and chairs.Each product must go through three stages of the manufacturing process: assembly,finishing,and inspection.Each table requires 3 hours of assembly,2 hours of finishing,and 1 hour of inspection.Each chair requires 2 hours of assembly,2 hours of finishing,and 1 hour of inspection.The profit per table is $120 while the profit per chair is $80.Currently,each week there are 200 hours of assembly time available,180 hours of finishing time,and 40 hours of inspection time.Linear programming is to be used to develop a production schedule.Define the variables as follows: T = number of tables produced each week C = number of chairs produced each week -According to Table 8-1,which describes a production problem,suppose it is decided that the number of hours used in the assembly process must be at least 80 percent of the time available.How would this constraint be written?

Table 8-5 Each coffee table produced by Timothy Kent Designers nets the firm a profit of $9.Each bookcase yields a $12 profit.Kent's firm is small and its resources limited.During any given production period (of 1 week),10 gallons of varnish and 12 lengths of high quality redwood are available.Each coffee table requires approximately 1 gallon of varnish and 1 length of redwood.Each bookcase takes 1 gallon of varnish and 2 lengths of wood. T = number of tables produced each week B = number of bookcases produced each week -Referring to Table 8-5,suppose that this problem requires that you use all the varnish for the week.How would the linear programming representation change?

The linear programming model of the production mix problem only includes constraints of the less than or equal form.

Table 8-2 Diamond Jeweler's is trying to determine how to advertise in order to maximize their exposure.Their weekly advertising budget is $10,000.They are considering three possible media: tv,newspaper,and radio.Information regarding cost and exposure is given in the table below: Let T = the # of tv ads,N = the # of newspaper ads,and R = the # of radio ads -According to Table 8-2,what is the optimal solution?

Table 8-5 Each coffee table produced by Timothy Kent Designers nets the firm a profit of $9.Each bookcase yields a $12 profit.Kent's firm is small and its resources limited.During any given production period (of 1 week),10 gallons of varnish and 12 lengths of high quality redwood are available.Each coffee table requires approximately 1 gallon of varnish and 1 length of redwood.Each bookcase takes 1 gallon of varnish and 2 lengths of wood. T = number of tables produced each week B = number of bookcases produced each week -Referring to Table 8-5,the solution to the problem is

Cedar Point amusement park management is preparing the park's annual promotional plan for the coming season.Several advertising alternatives exist: newspaper,television,radio,and displays at recreational shows.The information below shows the characteristics associated with each of the advertising alternatives,as well as the maximum number of placements available in each medium.Given an advertising budget of $250,000,how many placements should be made in each medium to maximize total audience exposure? Formulate this as a linear programming problem.

The linear programming model of the production scheduling process can include the impact of hiring and layoffs,regular and overtime pay rates,and the desire to have a constant and stable production schedule over a several-month period.

Table 8-4 A small furniture manufacturer produces tables and chairs.Each product must go through three stages of the manufacturing process: assembly,finishing,and inspection.Each table requires 4 hours of assembly,3 hours of finishing,and 1 hour of inspection.Each chair requires 3 hours of assembly,2 hours of finishing,and 2 hours of inspection.The selling price per table is $140 while the selling price per chair is $90.Currently,each week there are 220 hours of assembly time available,160 hours of finishing time,and 45 hours of inspection time.Assume that one hour of assembly time costs $5.00;one hour of finishing time costs $6.00;one hour of inspection time costs $4.50;and that whatever labor hours are not required for the table and chairs can be applied to another product.Linear programming is to be used to develop a production schedule.Define the variables as follows: T = number of tables produced each week C = number of chairs produced each week -According to Table 8-4,which describes a production problem,suppose you realize that you can trade off assembly hours for finishing hours,but that the total number of finishing hours,including the trade-off hours,cannot exceed 240 hours.How would this constraint be written?

A small furniture manufacturer produces tables and chairs.Each product must go through three stages of the manufacturing process: assembly,finishing,and inspection.Each table requires 3 hours of assembly,2 hours of finishing,and 1 hour of inspection.Each chair requires 2 hours of assembly,2 hours of finishing,and 1 hour of inspection.The profit per table is $120,while the profit per chair is $80.Currently,each week there are 200 hours of assembly time available,180 hours of finishing time,and 40 hours of inspection time.To keep a balance,the number of chairs produced should be at least twice the number of tables.Also,the number of chairs cannot exceed six times the number of tables.Formulate this as a linear programming problem.Carefully define all decision variables.

Which of the following is considered a decision variable in the media selection problem of minimizing interview costs in surveying?

A fast food restaurant uses full-time and part-time help to meet fluctuating demand during the day.The following table presents projected need for workers at different times of the day: There is a maximum of four full-time workers and the other workers are part-time workers.Each full-time worker is there from 9:00 until 5:00,while the part-time workers will work for 4 consecutive hours at a cost of $4.00 per hour.The cost of the full-time worker is $50 per day.The company wishes to minimize total cost while meeting the demands.Formulate this as a linear programming problem.Carefully define all decision variables.

Table 8-2 Diamond Jeweler's is trying to determine how to advertise in order to maximize their exposure.Their weekly advertising budget is $10,000.They are considering three possible media: tv,newspaper,and radio.Information regarding cost and exposure is given in the table below: Let T = the # of tv ads,N = the # of newspaper ads,and R = the # of radio ads -According to Table 8-2,what is the advertising budget constraint?

A manufacturer of microcomputers produces four models: Portable,Student,Office,and Network.The profit per unit on each of these four models is $500,$350,$700,and $1000,respectively.The models require the labor and materials per unit shown below. Formulate this product mix problem using linear programming.

Which of the following does not represent a factor a manager might typically consider when employing linear programming for a production scheduling?

The linear programming transportation model allows us to solve problems where supply does not equal demand.

In the linear programming transportation model,the coefficients of the objective function can represent either the cost or the profit from shipping goods along a particular route.

In general,linear programming is unable to solve complex labor planning as the objective function is usually not definable.

Table 8-7 Ivana Myrocle wishes to invest her inheritance of $200,000 so that her return on investment is maximized,but she also wishes to keep her risk level relatively low.She has decided to invest her money in any of three possible ways: CDs,which pay a guaranteed 6 percent;stocks,which have an expected return of 13 percent;and a money market mutual fund,which is expected to return 8 percent.She has decided that any or all of the $200,000 may be invested,but any part (or all)of it may be put in any of the 3 alternatives.Thus,she may have some money invested in all three alternatives.In formulating this as a linear programming problem,define the variables as follows: C = dollars invested in CDs S = dollars invested in stocks M = dollars invested in the money market mutual fund -According to Table 8-7,which describes an investment problem,suppose that Ivana has decided that the amount invested in stocks cannot exceed one-fourth of the total amount invested.Which is the best way to write this constraint?

Dr.Malcomb Heizer wishes to invest his retirement fund of $2,000,000 so that his return on investment is maximized,but he also wishes to keep the risk level relatively low.He has decided to invest his money in any of three possible ways: CDs that pay a guaranteed 4 percent;stocks that have an expected return of 14 percent;and a money market mutual fund that is expected to return 18 percent.He has decided that the total $2,000,000 will be invested,but any part (or all)of it may be put in any of the three alternatives.Thus,he may have some money invested in all three alternatives.He has also decided to invest,at most,30 percent of this in stocks and at least 20 percent of this in money market funds.Formulate this as a linear programming problem and carefully define all the decision variables.