Exam 13: Linear Optimization

Use the scenario given below to answer the following question(s). The Ranger group of industries has a factory that produces three types of action figures,types 1,2,and 3,on a make-to-order basis.The factory operates five days a week for 9 hours each day.The key decision facing the manager is what type of machine should be used to process each action figure during the coming quarter (13 weeks).Two types of machines can be used: captain and regular.Captain machines can be used to make all action figures and are the only machines that can produce certain action figures,such as "The move action figures." Demands,variable costs for each type of machine,and production rates on the machines are given in the table below. The factory has 30 regular machines and 6 captain machines.After manufacturing,the action figures are sent to the finishing department and then sold.Any action figure that cannot be manufactured in the factory because of limited capacity will be purchased from an external supplier,finished at the factory,and sold at the market selling price. -If C_i,R_i,and P_i (where i = 1,2,3…)are used to denote the number of action figures to be produced on captain machines,the number of action figures to be produced on regular machines,and the number of action figures purchased from an outside supplier respectively,which of the following are the constraints that meet production requirements?

An optimal solution is ________.

Use the case below to answer the following question(s). Stoudemire Inc.produces televisions at two plants: Arizona and Arkansas.They ship them to major distribution centers in San Antonio,Washington,Miami,and Dallas.The table shown below provides the unit cost of shipping between any plant and distribution center,plant capacities over the next planning period,and distribution center demands.Stoudemire's supply chain manager faces the problem of determining how much to ship between each plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Assume X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Arizona,i = 2 represents Arkansas,j = 1 represents San Antonio,j = 2 represents Washington,and so on. -Calculate the total number of television sets shipped from the plant in Arkansas.(Hint: Use Solver)

Use the case below to answer the following question(s). Stoudemire Inc.produces televisions at two plants: Arizona and Arkansas.They ship them to major distribution centers in San Antonio,Washington,Miami,and Dallas.The table shown below provides the unit cost of shipping between any plant and distribution center,plant capacities over the next planning period,and distribution center demands.Stoudemire's supply chain manager faces the problem of determining how much to ship between each plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Assume X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Arizona,i = 2 represents Arkansas,j = 1 represents San Antonio,j = 2 represents Washington,and so on. -For the data given above,the objective function is ________.

What are the four basic steps involved in building linear optimization models?

The shadow price ________.

Use the case below to answer the following question(s). S&N Manufacturing makes four different steel sheets,A,B,C,and D,with different percentages of iron.Each sheet must flow through five departments,forming,rolling,sintering,painting,and quality check.The table below shows the relevant data.Production rates are shown in units/hour.(Sheet A uses imported steel and does not require rolling.) S&N wants to determine the number of steel sheets to produce in order to maximize monthly profit. Let A,B,C,and D denote the number of units of model A,B,C,and D to be produced,respectively. -For the data given above,the objective function is ________.

Consider the scenario given below.Use Excel solver to answer the following question(s). Peca Inc.is a small manufacturer of two types of office chairs,the swivel and no-swivel models.The manufacturing process consists of two principal departments: fabrication and finishing.The fabrication department has 24 skilled workers,each of whom works 7 hours per day.The finishing department has 6 workers,who also work a 7-hour shift.A swivel type requires 7 labor hours in the fabricating department and 2 labor hours in finishing.The no-swivel model requires 8 labor hours in fabricating and 3 labor hours in finishing.Peca Inc.makes a net profit of $100 on the swivel model,and $130 on the no-swivel model.The company anticipates selling at least twice as many no-swivel models as swivel models.The company wants to determine how many of each model should be produced on a daily basis to maximize net profit. -In which of the following cases is the Cell Value not equal to the right-hand side of the value of the constraint?

Use the scenario given below to answer the following question(s). Ray Inc.manufactures two types of gaming consoles,the sp² hybrid and the sp³ hybrid models.The manufacturing process consists of two principal departments: design and production.The design department has 58 skilled workers,each of whom works 7 hours per day.The production department has 25 workers,who also work a 7-hour shift.On an average,to produce an sp² hybrid,Ray Inc.requires 3.5 labor hours for design and 2 labor hours for production.The sp³ model requires 4 labor hours for design and 1.5 labor hours in production.The company anticipates selling at least 1.5 times as many sp³ models as sp² models.The company operates five days per week and makes a net profit of $130 on the sp² model,and $150 on the sp³ hybrid model.Ray Inc.wants to determine how many of each model should be produced on a weekly basis to maximize net profit. -Which of the following statements is true of the number of sp³ and sp² hybrids to be manufactured in a week?

Consider the scenario given below.Use Excel solver to answer the following question(s). Peca Inc.is a small manufacturer of two types of office chairs,the swivel and no-swivel models.The manufacturing process consists of two principal departments: fabrication and finishing.The fabrication department has 24 skilled workers,each of whom works 7 hours per day.The finishing department has 6 workers,who also work a 7-hour shift.A swivel type requires 7 labor hours in the fabricating department and 2 labor hours in finishing.The no-swivel model requires 8 labor hours in fabricating and 3 labor hours in finishing.Peca Inc.makes a net profit of $100 on the swivel model,and $130 on the no-swivel model.The company anticipates selling at least twice as many no-swivel models as swivel models.The company wants to determine how many of each model should be produced on a daily basis to maximize net profit. -Determine the objective function.

Use the scenario given below to answer the following question(s). Ray Inc.manufactures two types of gaming consoles,the sp² hybrid and the sp³ hybrid models.The manufacturing process consists of two principal departments: design and production.The design department has 58 skilled workers,each of whom works 7 hours per day.The production department has 25 workers,who also work a 7-hour shift.On an average,to produce an sp² hybrid,Ray Inc.requires 3.5 labor hours for design and 2 labor hours for production.The sp³ model requires 4 labor hours for design and 1.5 labor hours in production.The company anticipates selling at least 1.5 times as many sp³ models as sp² models.The company operates five days per week and makes a net profit of $130 on the sp² model,and $150 on the sp³ hybrid model.Ray Inc.wants to determine how many of each model should be produced on a weekly basis to maximize net profit. -If the numbers of sp² and sp³ hybrid gaming consoles produced each week are denoted as X₁ and X₂ respectively,the function that describes the design department's labor constraint for a week is ________.

Consider the scenario given below.Use Excel solver to answer the following question(s). Peca Inc.is a small manufacturer of two types of office chairs,the swivel and no-swivel models.The manufacturing process consists of two principal departments: fabrication and finishing.The fabrication department has 24 skilled workers,each of whom works 7 hours per day.The finishing department has 6 workers,who also work a 7-hour shift.A swivel type requires 7 labor hours in the fabricating department and 2 labor hours in finishing.The no-swivel model requires 8 labor hours in fabricating and 3 labor hours in finishing.Peca Inc.makes a net profit of $100 on the swivel model,and $130 on the no-swivel model.The company anticipates selling at least twice as many no-swivel models as swivel models.The company wants to determine how many of each model should be produced on a daily basis to maximize net profit. -What is the allowable decrease per unit of swivel chair produced?

Use the case below to answer the following question(s). S&N Manufacturing makes four different steel sheets,A,B,C,and D,with different percentages of iron.Each sheet must flow through five departments,forming,rolling,sintering,painting,and quality check.The table below shows the relevant data.Production rates are shown in units/hour.(Sheet A uses imported steel and does not require rolling.) S&N wants to determine the number of steel sheets to produce in order to maximize monthly profit. Let A,B,C,and D denote the number of units of model A,B,C,and D to be produced,respectively. -Which of the following is the inequality/equality that describes the time constraint of the sintering department?

Use the data given below to answer the following question(s). Ball Inc.specializes in producing disinfectants.In developing a new disinfectant,company scientists have specified that the mixture must contain at least 26% hydrogen,at least 30% oxygen,and no more than 28% water.The percentages of each of these constituents in two types of ingredients that can be used to produce the disinfectant are given in the table,along with the wholesale cost per pound. Let X₁ and X₂ denote DDT and solid carbon dioxide,respectively. -Identify the inequality that describes the hydrogen constraint.

Use the case below to answer the following question(s). S&N Manufacturing makes four different steel sheets,A,B,C,and D,with different percentages of iron.Each sheet must flow through five departments,forming,rolling,sintering,painting,and quality check.The table below shows the relevant data.Production rates are shown in units/hour.(Sheet A uses imported steel and does not require rolling.) S&N wants to determine the number of steel sheets to produce in order to maximize monthly profit. Let A,B,C,and D denote the number of units of model A,B,C,and D to be produced,respectively. -Determine the optimal number of D-type steel sheets produced.(Hint: Use Solver)

In the expression 3.0A + 3.5B + 2.3C ≥ 850,where A,B,and C are decision variables of a problem,the constraint function is the ________.

A binding constraint is one ________.

Use the case below to answer the following question(s). Stoudemire Inc.produces televisions at two plants: Arizona and Arkansas.They ship them to major distribution centers in San Antonio,Washington,Miami,and Dallas.The table shown below provides the unit cost of shipping between any plant and distribution center,plant capacities over the next planning period,and distribution center demands.Stoudemire's supply chain manager faces the problem of determining how much to ship between each plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Assume X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Arizona,i = 2 represents Arkansas,j = 1 represents San Antonio,j = 2 represents Washington,and so on. -In which of the following cases is the Cell Value not equal to the right-hand side value of the constraint?

Use the case below to answer the following question(s). S&N Manufacturing makes four different steel sheets,A,B,C,and D,with different percentages of iron.Each sheet must flow through five departments,forming,rolling,sintering,painting,and quality check.The table below shows the relevant data.Production rates are shown in units/hour.(Sheet A uses imported steel and does not require rolling.) S&N wants to determine the number of steel sheets to produce in order to maximize monthly profit. Let A,B,C,and D denote the number of units of model A,B,C,and D to be produced,respectively. -Which of the following is the inequality/equality that describes the time constraint of the forming department?

Use the case below to answer the following question(s). Stoudemire Inc.produces televisions at two plants: Arizona and Arkansas.They ship them to major distribution centers in San Antonio,Washington,Miami,and Dallas.The table shown below provides the unit cost of shipping between any plant and distribution center,plant capacities over the next planning period,and distribution center demands.Stoudemire's supply chain manager faces the problem of determining how much to ship between each plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Assume X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Arizona,i = 2 represents Arkansas,j = 1 represents San Antonio,j = 2 represents Washington,and so on. -The number of television sets shipped from the plant in Arizona to the distribution center in Dallas is ________.(Hint: Use Solver)