Exam 13: Linear Optimization

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 total cost incurred by Stoudemire Inc.is ________.(Hint: Use Solver)

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. -The difference between the right- and left-hand sides of the fabrication labor constraint is ________.

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 day are denoted as X₁ and X₂ respectively,the function that describes the production 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. -If the limitation in the finishing department is changed to 43 labor hours,the total profit ________.

Whenever a constraint has negative slack,the shadow price is zero.

The statement,"The amount invested in online advertising cannot be more than thrice the amount invested in print advertising," is an example of a(n)________.

Briefly describe the four possible outcomes of a linear optimization model.

What are the three options provided by the basic solver in Excel for the solving method?

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. -Determine the equality/inequality that explains the demand constraint for the distribution center in Washington.

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 inequality/equality that explains the capacity constraint for Arizona 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. -The number of television sets shipped from the plant in Arizona to the distribution center in Washington is ________.(Hint: Use Solver)

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 is a decision variable in the given problem?

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. -What is the objective function?

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,determine the objective function for the data given above.

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 number of sp² and sp³ models manufactured by the company are 100 and 150 respectively,the number hours used in the design department is ________.

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,determine the constraint on available production time on captain machines.

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. -The inequality/equality that describes the water constraint is ________.

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. -Using Solver,determine the number of hours used in the forming department.

In the constraints section obtained in solver reports,what is the Cell Value?

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 rolling department?