Exam 8: Linear Programming Applications

In production scheduling LP problems,inventory at the end of this month is set equal to

First Securities,Inc. ,an investment firm,has $380,000 on account.The chief investment officer would like to reinvest the $380,000 in a portfolio that would maximize return on investment while at the same time maintaining a relatively conservative mix of stocks and bonds.The following table shows the investment opportunities and rates of return. The Board of Directors has mandated that at least 60 percent of the investment consist of a combination of municipal and federal bonds,25 percent Blue Chip Stock,and no more than 15 percent High Tech Stock.Formulate this portfolio selection problem using linear programming.

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 be assigned to shift 1?

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,which of the following constraints would be used?

In a production scheduling problem,the inventory at the end of this month is set equal to the inventory at the end of last month + last month's production − sales this month.

Swearingen and McDonald,a small furniture manufacturer,produces fine hardwood tables and chairs.Each product must go through three stages of the manufacturing process: assembly,finishing,and inspection.Each table requires 12 hours of assembly,20 hours of finishing,and 2 hours of inspection.Each chair requires 4 hours of assembly,16 hours of finishing,and 3 hours of inspection.The profit per table is $150 while the profit per chair is $100.Currently,each week there are 300 hours of assembly time available,220 hours of finishing time,and 30 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 6 times the number of tables.Formulate this as a linear programming problem.Carefully define all decision variables.Find the solution.

The linear programming model of the production scheduling process is usually used when we have to schedule the production of a single product,requiring a mix of resources,over time.

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. How many of each model should be produced to maximize profit.What is the maximum profit?

Describe the shipping problem linear programming application.

What is the objective in the diet problem?

Table 8-3 A marketing research firm would like to survey undergraduate and graduate college students about whether or not they take out student loans for their education.There are different cost implications for the region of the country where the college is located and the type of degree.The survey cost table is provided below: The requirements for the survey are as follows: 1.The survey must have at least 1500 students 2.At least 400 graduate students 3.At least 100 graduate students should be from the West 4.No more than 500 undergraduate students should be from the East 5.At least 75 graduate students should be from the Central region 6.At least 300 students should be from the West The marketing research firm would like to minimize the cost of the survey while meeting the requirements.Let X₁ = # of undergraduate students from the East region,X₂ = # of graduate students from the East region,X₃ = # of undergraduate students from the Central region,X₄ = # of graduate students from the Central region,X₅ = # of undergraduate students from the West region,and X₆ = # of graduate students from the West region. -According to Table 8-3,the constraint that there must be at least 400 graduate students is expressed as

Table 8-3 A marketing research firm would like to survey undergraduate and graduate college students about whether or not they take out student loans for their education.There are different cost implications for the region of the country where the college is located and the type of degree.The survey cost table is provided below: The requirements for the survey are as follows: 1.The survey must have at least 1500 students 2.At least 400 graduate students 3.At least 100 graduate students should be from the West 4.No more than 500 undergraduate students should be from the East 5.At least 75 graduate students should be from the Central region 6.At least 300 students should be from the West The marketing research firm would like to minimize the cost of the survey while meeting the requirements.Let X₁ = # of undergraduate students from the East region,X₂ = # of graduate students from the East region,X₃ = # of undergraduate students from the Central region,X₄ = # of graduate students from the Central region,X₅ = # of undergraduate students from the West region,and X₆ = # of graduate students from the West region. -According to Table 8-3,the constraint that the survey must have at least a total of 1500 students is expressed as

Which of the following statements is false regarding the portfolio selection problem?

Determining the mixture of ingredients for a most economical feed or diet combination would be described as a production mix type of linear program.

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 assigned the following risk factors to each investment instrument - CDs (C): 1.2;stocks (S): 4.8;money market mutual fund (M): 3.2.If Ivana decides that she wants the risk factor for the whole investment to be less than 3.3,how should the necessary constraint be written?

Ivana Miracle wishes to invest up to her full inheritance of $300,000,and her goal is to minimize her risk subject to an expected annual return of at least $30,000.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 15 percent;and a money market mutual fund,which is expected to return 8 percent.Risk factors are 1.0 for the CDs,3.6 for the stocks,and 1.8 for the money market fund.Formulate this as a linear program.

Describe a diet problem linear programming application.

Ivana Miracle wishes to invest up to her full inheritance of $300,000,and her goal is to minimize her risk subject to an expected annual return of at least $30,000.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 15 percent;and a money market mutual fund,which is expected to return 8 percent.Risk factors are 1.0 for the CDs,3.6 for the stocks,and 1.8 for the money market fund.What is the optimal solution and minimum risk value?

Linear programming variable names such as X₁₁,X₁₂,X₁₃,could possibly be used to represent production of a product (X_1j)over several months.

Table 8-3 A marketing research firm would like to survey undergraduate and graduate college students about whether or not they take out student loans for their education.There are different cost implications for the region of the country where the college is located and the type of degree.The survey cost table is provided below: The requirements for the survey are as follows: 1.The survey must have at least 1500 students 2.At least 400 graduate students 3.At least 100 graduate students should be from the West 4.No more than 500 undergraduate students should be from the East 5.At least 75 graduate students should be from the Central region 6.At least 300 students should be from the West The marketing research firm would like to minimize the cost of the survey while meeting the requirements.Let X₁ = # of undergraduate students from the East region,X₂ = # of graduate students from the East region,X₃ = # of undergraduate students from the Central region,X₄ = # of graduate students from the Central region,X₅ = # of undergraduate students from the West region,and X₆ = # of graduate students from the West region. -According to Table 8-3,the minimum cost is