Exam 14: Integer, nonlinear, and Advanced Optimization Methods

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. -Determine the projected revenue for selling the projected number of total suites.

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. -Briefly describe the basic process for using OptQuest in Crystal Ball.

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 projected number of standard suites sold.

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.

Write a brief note on integer optimization models with binary variables.

In a nonlinear optimization model,the terms cannot be written as a constant times a variable.

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 Memphis is ________.

Use the data shown below to answer the following question(s). Malone Inc.has identified five potential new processes; however,the firm is constrained by its available budget and human resources.Each process is expected to generate a return (given by the net present value)but requires a fixed amount of cash and personnel.Because the resources are limited,all processes cannot be selected.Processes cannot be partially completed; thus,either the process must be undertaken completely or not at all.The data are given in the table below.If a process is selected,it generates the full value of the expected return and requires the full amount of cash and personnel shown in the table.Define X_i = 1 if process i is selected,and 0 otherwise. -Using Solver,the total return for the optimal solution is ________.

Use the data shown below to answer the following question(s). Suppose that an unincorporated township wishes to find the best locations for emergency rooms.Assume that the township is divided into smaller layouts and that transportation studies have estimated the response time for emergency vehicles to travel between each pair of layouts.The township wants to locate the emergency rooms so that all layouts can be reached within a 9-minute response time. The table below shows the estimated response time in minutes between each pair of layouts: Define X_j = 1 if an emergency room is located in layout j,and 0 if not. -Determine the objective function.

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. -Using Solver,determine the amount to be shipped from the plant in Orlando to the distribution center in Minnesota.

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. -What is the new price of a single gold suite,using Solver?

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. -Using Solver,determine the number of 115-inch rolls produced with the cutting pattern 6.

Any sequence of integers in the decision variable range is called a permutation.

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. -The constraint for 16-inch rolls is formulated as ________.

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. -Determine the constraint modeled for 25-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. -Using Solver,determine the optimal cost involved in shipping cars from Baltimore to Memphis.

Use the data shown below to answer the following question(s). Hudson's Creators is a company that produces embroidered clothes.Forecasts of sales for the next year are 200 in the autumn,450 in the winter,and 100 in the spring.Plain clothes are purchased from a supplier for $25.The cost of capital is estimated to be 28% per year (or 7% per quarter); thus,the holding cost per item is 0.07($25)= $1.75 per quarter.Hudson hires art students part-time to produce his designs during the autumn,and they earn $6.50 per hour.Because of the high demand for part-time help during the winter holiday season,labor rates are higher in the winter,and workers earn $8.00 per hour.In the spring,labor is more difficult to keep,and the owner must pay $7.25 per hour to retain qualified help.Each embroidered cloth takes 2 hours to complete.Define the model based on how the production should be planned over the three quarters to minimize the combined production and inventory holding costs. Use the following terms to define the decision variables: P_A = amount to produce in autumn P_W = amount to produce in winter P_S = amount to produce in spring I_A = inventory held at the end of autumn I_W = inventory held at the end of winter I_S = inventory held at the end of spring Suppose that Hudson must rent some equipment to produce his products,which costs $75 for three months.The equipment can be rented or returned each quarter,so if nothing is produced in a quarter,it makes no sense to incur the rental cost. The fixed costs can be incorporated into the model by defining an additional set of variables: Y_A = 1 if production occurs during the autumn,and 0 if not Y_W = 1 if production occurs during the winter,and 0 if not Y_S = 1 if production occurs during the spring,and 0 if not -Which of the following is the basic material balance equation for the autumn season?

Use the data shown below to answer the following question(s). Hudson's Creators is a company that produces embroidered clothes.Forecasts of sales for the next year are 200 in the autumn,450 in the winter,and 100 in the spring.Plain clothes are purchased from a supplier for $25.The cost of capital is estimated to be 28% per year (or 7% per quarter); thus,the holding cost per item is 0.07($25)= $1.75 per quarter.Hudson hires art students part-time to produce his designs during the autumn,and they earn $6.50 per hour.Because of the high demand for part-time help during the winter holiday season,labor rates are higher in the winter,and workers earn $8.00 per hour.In the spring,labor is more difficult to keep,and the owner must pay $7.25 per hour to retain qualified help.Each embroidered cloth takes 2 hours to complete.Define the model based on how the production should be planned over the three quarters to minimize the combined production and inventory holding costs. Use the following terms to define the decision variables: P_A = amount to produce in autumn P_W = amount to produce in winter P_S = amount to produce in spring I_A = inventory held at the end of autumn I_W = inventory held at the end of winter I_S = inventory held at the end of spring Suppose that Hudson must rent some equipment to produce his products,which costs $75 for three months.The equipment can be rented or returned each quarter,so if nothing is produced in a quarter,it makes no sense to incur the rental cost. The fixed costs can be incorporated into the model by defining an additional set of variables: Y_A = 1 if production occurs during the autumn,and 0 if not Y_W = 1 if production occurs during the winter,and 0 if not Y_S = 1 if production occurs during the spring,and 0 if not -Which of the following is the basic material balance equation for the spring season?

Integer models are continuous by their very nature.

Integer linear optimization models differ from nonlinear optimization models in that integer linear optimization models ________.