Exam 14: Integer, nonlinear, and Advanced Optimization Methods

Excel functions such as IF,ABS,MIN,and MAX lead to nonsmooth models.

Use the scenario shown below to answer the following question(s). An automobile rental is considering a major remodeling effort and needs to determine the best combination of rates and products to maximize revenues.Currently,the rental has 375 automobiles with the following history: Each segment has its own price/demand elasticity.Estimates are: The rental owners want to keep the price of a scooter rental between $35 and $45; a chopper rental between $45 and $60; and a car rental between $65 and $74.50. Define S = price of a scooter rental,Ch = price of a chopper rental,and Ca = price of a car rental. -Determine the objective function for nonsmooth optimization of the below fixed-cost problem for Hudson Creators.

Use the scenario shown below to answer the following question(s). An automobile rental is considering a major remodeling effort and needs to determine the best combination of rates and products to maximize revenues.Currently,the rental has 375 automobiles with the following history: Each segment has its own price/demand elasticity.Estimates are: The rental owners want to keep the price of a scooter rental between $35 and $45; a chopper rental between $45 and $60; and a car rental between $65 and $74.50. Define S = price of a scooter rental,Ch = price of a chopper rental,and Ca = price of a car rental. -Given below is the spreadsheet model for an automobile rental.To better understand and mitigate the inherent risk existing in the model,identify the uncertain variable to which assumptions can be defined using Crystal Ball.

OptQuest enhances the analysis capabilities of Crystal Ball by automatically searching for optimal solutions within Crystal Ball simulation model spreadsheets.

Use the scenario below to answer the following question(s). A company makes standard 115-inch-wide rolls of thin sheet metal,and slits them into smaller rolls to meet customer orders for widths of 13,16,and 25 inches.The demands for these widths vary from week to week.From a 115-inch roll,there are many different ways to slit 13-,16-,and 25-inch pieces. A cutting pattern is a configuration of the number of smaller rolls of each type that are cut from the raw stock.Of course,one would want to use as much of the roll as possible to avoid costly scrap.For example,one could cut seven 16-inch rolls,leaving a 3-inch piece of scrap.Finding good cutting patterns for a large set of end products is in itself a challenging problem.Suppose that the company has proposed the following cutting patterns: Demands this week are 1,000 13-inch rolls,1,430 16-inch rolls,and 1,260 25-inch rolls.The problem is to develop a model that will determine how many 115-inch rolls to cut into each of the six patterns in order to meet demand and scrap. Define X_i to be the number of 115-inch rolls to cut using cutting pattern i,for i = 1,…,6. Note that X_i needs to be a whole number because each roll that is cut generates a different number of end items. -What is the number of 16-inch rolls produced?

Why can't an individual running Solver generate sensitivity information for integer models in the same manner as for linear models?

A nonlinear optimization model differs from a mixed integer linear optimization model in that the nonlinear optimization model ________.

Use the case below to answer the following question(s). Rolling Inc.produces cars at two plants: Baltimore,Maryland,and Orlando,Florida.They ship them to major distribution centers in Minnesota,Dallas,Los Angeles,and Memphis.The Accounting,Production,and Marketing departments have provided the information in the table below,which shows the unit cost of shipping between any plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Suppose demand forecasts exceed the existing capacity and the company is considering adding a new plant from among two choices: Ohio or Kansas.Both plants would have a capacity of 3,000 units but only one can be built.The table below shows the revised data. Use double-subscripted variables to simplify the formulation.Define X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Baltimore,and j = 1 represents Minnesota. Define a binary variable for the decision of which plant to build: Y₁ = 1 if the Ohio plant is built,and Y₂ = 1 if the Kansas plant is built. -To guarantee that only one new plant is built,we must have the constraint ________.

Use the case below to answer the following question(s). Rolling Inc.produces cars at two plants: Baltimore,Maryland,and Orlando,Florida.They ship them to major distribution centers in Minnesota,Dallas,Los Angeles,and Memphis.The Accounting,Production,and Marketing departments have provided the information in the table below,which shows the unit cost of shipping between any plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Suppose demand forecasts exceed the existing capacity and the company is considering adding a new plant from among two choices: Ohio or Kansas.Both plants would have a capacity of 3,000 units but only one can be built.The table below shows the revised data. Use double-subscripted variables to simplify the formulation.Define X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Baltimore,and j = 1 represents Minnesota. Define a binary variable for the decision of which plant to build: Y₁ = 1 if the Ohio plant is built,and Y₂ = 1 if the Kansas plant is built. -Which of the following is the capacity constraint for the Orlando plant?

Sensitivity information for integer models can be generated in the same manner as for linear models.

Use the scenario shown below to answer the following question(s). An automobile rental is considering a major remodeling effort and needs to determine the best combination of rates and products to maximize revenues.Currently,the rental has 375 automobiles with the following history: Each segment has its own price/demand elasticity.Estimates are: The rental owners want to keep the price of a scooter rental between $35 and $45; a chopper rental between $45 and $60; and a car rental between $65 and $74.50. Define S = price of a scooter rental,Ch = price of a chopper rental,and Ca = price of a car rental. -Determine the constraint that models the rental capacity.

Use the scenario below to answer the following question(s). A company makes standard 115-inch-wide rolls of thin sheet metal,and slits them into smaller rolls to meet customer orders for widths of 13,16,and 25 inches.The demands for these widths vary from week to week.From a 115-inch roll,there are many different ways to slit 13-,16-,and 25-inch pieces. A cutting pattern is a configuration of the number of smaller rolls of each type that are cut from the raw stock.Of course,one would want to use as much of the roll as possible to avoid costly scrap.For example,one could cut seven 16-inch rolls,leaving a 3-inch piece of scrap.Finding good cutting patterns for a large set of end products is in itself a challenging problem.Suppose that the company has proposed the following cutting patterns: Demands this week are 1,000 13-inch rolls,1,430 16-inch rolls,and 1,260 25-inch rolls.The problem is to develop a model that will determine how many 115-inch rolls to cut into each of the six patterns in order to meet demand and scrap. Define X_i to be the number of 115-inch rolls to cut using cutting pattern i,for i = 1,…,6. Note that X_i needs to be a whole number because each roll that is cut generates a different number of end items. -Identify the constraint specified for 13-inch rolls.

Use the case below to answer the following question(s). Rolling Inc.produces cars at two plants: Baltimore,Maryland,and Orlando,Florida.They ship them to major distribution centers in Minnesota,Dallas,Los Angeles,and Memphis.The Accounting,Production,and Marketing departments have provided the information in the table below,which shows the unit cost of shipping between any plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Suppose demand forecasts exceed the existing capacity and the company is considering adding a new plant from among two choices: Ohio or Kansas.Both plants would have a capacity of 3,000 units but only one can be built.The table below shows the revised data. Use double-subscripted variables to simplify the formulation.Define X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Baltimore,and j = 1 represents Minnesota. Define a binary variable for the decision of which plant to build: Y₁ = 1 if the Ohio plant is built,and Y₂ = 1 if the Kansas plant is built. -The inequality/equality that explains the demand constraint for Minnesota is ________.

Use the case below to answer the following question(s). The Tipton Hotel is considering a major remodeling effort and needs to determine the best combination of rates and suite sizes to maximize revenues.Currently,the hotel has 755 suites with the following history: Each market segment has its own price/demand elasticity.Estimates are: This means,for example,that a 1% decrease in the price of a standard suite will increase the number of suites sold by 1.5%.Similarly,a 1% increase in the price will decrease the number of suites sold by 1.5%.For any pricing structure (in $),the projected number of suites of a given type sold (we will allow continuous values for this problem)can be found using the formula: (Historical average number of suites sold)+ (Elasticity)(New price - Current price)(Historical average number of suites sold)/(Current price) The hotel owners want to keep the price of a standard suite between $70 and $90; a gold suite between $90 and $110; and a platinum suite between $120 and $149. Define S = price of a standard suite,G = price of a gold suite,and P = price of a platinum suite. -Which of the following is the expression that can be used to determine the projected number of platinum suites sold?

Use the case below to answer the following question(s). The Tipton Hotel is considering a major remodeling effort and needs to determine the best combination of rates and suite sizes to maximize revenues.Currently,the hotel has 755 suites with the following history: Each market segment has its own price/demand elasticity.Estimates are: This means,for example,that a 1% decrease in the price of a standard suite will increase the number of suites sold by 1.5%.Similarly,a 1% increase in the price will decrease the number of suites sold by 1.5%.For any pricing structure (in $),the projected number of suites of a given type sold (we will allow continuous values for this problem)can be found using the formula: (Historical average number of suites sold)+ (Elasticity)(New price - Current price)(Historical average number of suites sold)/(Current price) The hotel owners want to keep the price of a standard suite between $70 and $90; a gold suite between $90 and $110; and a platinum suite between $120 and $149. Define S = price of a standard suite,G = price of a gold suite,and P = price of a platinum suite. -If the number of available suites increases by 1 to 756,using the Lagrange multiplier,the total revenue would ________.

Use the scenario below to answer the following question(s). A company makes standard 115-inch-wide rolls of thin sheet metal,and slits them into smaller rolls to meet customer orders for widths of 13,16,and 25 inches.The demands for these widths vary from week to week.From a 115-inch roll,there are many different ways to slit 13-,16-,and 25-inch pieces. A cutting pattern is a configuration of the number of smaller rolls of each type that are cut from the raw stock.Of course,one would want to use as much of the roll as possible to avoid costly scrap.For example,one could cut seven 16-inch rolls,leaving a 3-inch piece of scrap.Finding good cutting patterns for a large set of end products is in itself a challenging problem.Suppose that the company has proposed the following cutting patterns: Demands this week are 1,000 13-inch rolls,1,430 16-inch rolls,and 1,260 25-inch rolls.The problem is to develop a model that will determine how many 115-inch rolls to cut into each of the six patterns in order to meet demand and scrap. Define X_i to be the number of 115-inch rolls to cut using cutting pattern i,for i = 1,…,6. Note that X_i needs to be a whole number because each roll that is cut generates a different number of end items. -Integer models are discontinuous by their very nature.Hence,________.

Any optimization model is completely deterministic,that is,no inherent risk exists in using the optimal solution obtained from the model.

Use the case below to answer the following question(s). Rolling Inc.produces cars at two plants: Baltimore,Maryland,and Orlando,Florida.They ship them to major distribution centers in Minnesota,Dallas,Los Angeles,and Memphis.The Accounting,Production,and Marketing departments have provided the information in the table below,which shows the unit cost of shipping between any plant and distribution center to minimize the total transportation cost,not exceed available capacity,and meet customer demand. Suppose demand forecasts exceed the existing capacity and the company is considering adding a new plant from among two choices: Ohio or Kansas.Both plants would have a capacity of 3,000 units but only one can be built.The table below shows the revised data. Use double-subscripted variables to simplify the formulation.Define X_ij = amount shipped from plant i to distribution center j,where i = 1 represents Baltimore,and j = 1 represents Minnesota. Define a binary variable for the decision of which plant to build: Y₁ = 1 if the Ohio plant is built,and Y₂ = 1 if the Kansas plant is built. -If we are to use Solver to determine where the new plant has to be built,what is the amount shipped from Kansas to Memphis?

Use the case below to answer the following question(s). The Tipton Hotel is considering a major remodeling effort and needs to determine the best combination of rates and suite sizes to maximize revenues.Currently,the hotel has 755 suites with the following history: Each market segment has its own price/demand elasticity.Estimates are: This means,for example,that a 1% decrease in the price of a standard suite will increase the number of suites sold by 1.5%.Similarly,a 1% increase in the price will decrease the number of suites sold by 1.5%.For any pricing structure (in $),the projected number of suites of a given type sold (we will allow continuous values for this problem)can be found using the formula: (Historical average number of suites sold)+ (Elasticity)(New price - Current price)(Historical average number of suites sold)/(Current price) The hotel owners want to keep the price of a standard suite between $70 and $90; a gold suite between $90 and $110; and a platinum suite between $120 and $149. Define S = price of a standard suite,G = price of a gold suite,and P = price of a platinum suite. -The projected number of total suites sold or the total suite capacity is modeled by the constraint ________.

Use the case below to answer the following question(s). The Tipton Hotel is considering a major remodeling effort and needs to determine the best combination of rates and suite sizes to maximize revenues.Currently,the hotel has 755 suites with the following history: Each market segment has its own price/demand elasticity.Estimates are: This means,for example,that a 1% decrease in the price of a standard suite will increase the number of suites sold by 1.5%.Similarly,a 1% increase in the price will decrease the number of suites sold by 1.5%.For any pricing structure (in $),the projected number of suites of a given type sold (we will allow continuous values for this problem)can be found using the formula: (Historical average number of suites sold)+ (Elasticity)(New price - Current price)(Historical average number of suites sold)/(Current price) The hotel owners want to keep the price of a standard suite between $70 and $90; a gold suite between $90 and $110; and a platinum suite between $120 and $149. Define S = price of a standard suite,G = price of a gold suite,and P = price of a platinum suite. -Using Solver,determine the new price of a single standard suite.