Exam 13: Linear Optimization

Use the following information to answer the questions) below. Cerebro Manufacturing produces four types of structural support fittings-plugs, rails, rivets, and clips-which are machined on two CNC machining centers. The machining centers have a capacity of 250,000 minutes per year. The gross margin per unit and machining requirements are shown in the spreadsheet below. A B C D E F 1 Cerebro Manufacturing Model 2 3 Product Plugs Rails Rivets Clips Machine Capacity mins./year) 4 Gross margin/unit \ 0.40 \ 1.20 \ 0.80 \ 1.10 5 Minutes/unit 1 2 3 1.5 250,000 6 Gross margin/minute 7 Maximum production 8 Profit -What is the maximum possible production of rails based on the machine capacity?

Use the table below to answer the following questions). The Riviera Transport Company RTC) produces car accessories at two plants: Dallas and Atlanta. They ship them to major distribution centers in Houston, San Jose, Jacksonville, 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, plant capacities over the next planning period, and distribution center demands. RTC'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 Xij = amount shipped from plant i to distribution center j, where i = 1 represents Dallas, i = 2 represents Atlanta, j = 1 represents Houston, and so on. Transpartation Model Data Distribution Center Plant Houston Sarn Jose Jacksorville Mernphis Capacity Dallas 13.00 15.25 10.99 18.48 1250 Atlarita 10.75 15.16 9.65 18.50 750 Dernand 175 325 480 950 Answer the following questions) using a linear optimization model. -Which of the following is the constraint for total amount shipped from Atlanta?

Use the table below to answer the following questions). The Riviera Transport Company RTC) produces car accessories at two plants: Dallas and Atlanta. They ship them to major distribution centers in Houston, San Jose, Jacksonville, 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, plant capacities over the next planning period, and distribution center demands. RTC'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 Xij = amount shipped from plant i to distribution center j, where i = 1 represents Dallas, i = 2 represents Atlanta, j = 1 represents Houston, and so on. Transpartation Model Data Distribution Center Plant Houston Sarn Jose Jacksorville Mernphis Capacity Dallas 13.00 15.25 10.99 18.48 1250 Atlarita 10.75 15.16 9.65 18.50 750 Dernand 175 325 480 950 Answer the following questions) using a linear optimization model. -Which of the following is the constraint of nonnegativity for all values of i and j?

Use the table below to answer the following questions). Letherin Hides is a company that makes boots specifically targeting college students. Forecasts of sales for the next year are 200 in the summer, 450 in the autumn, and 500 in the winter. Accessories that are used on the boots are purchased from a supplier for $31.66. The cost of capital is estimated to be 24% per year or 6% per quarter); thus, the holding cost per item is 0.06$31.66) = $1.9 per quarter rounded figure). Letherin Hides hires freelance art designers at part-time to craft designs during the summer, and they earn $6 per hour. In the autumn, labor is more difficult to keep, and the owner must pay $6.5 per hour to retain qualified help. Because of the high demand for part-time help during the winter holiday season, labor rates are higher in the winter, and workers earn $7.75 per hour. Each boot design takes 2 hours to complete. How should production be planned over the three quarters to minimize the combined production and inventory holding costs? The table below provides information on Letherin Hides boot design cost and production. Datherin Hides Data Sumuner Autumn Winter Unit Production Cost 12 13 15.5 Unit Inventory Holding Cost 1.9 1.9 1.9 demand 200 450 500 Use a linear optimization model based on the data to answer the following questions. -According to the linear optimization model, what is the total amount to be produced in summer?

Which of the following best defines decision variables in an optimization model?

Infeasible problems can occur when the demand requirement is higher than the available capacity.

What are the two basic properties of a linear optimization model?

Use the following information to answer the questions) below. Cerebro Manufacturing produces four types of structural support fittings-plugs, rails, rivets, and clips-which are machined on two CNC machining centers. The machining centers have a capacity of 250,000 minutes per year. The gross margin per unit and machining requirements are shown in the spreadsheet below. A B C D E F 1 Cerebro Manufacturing Model 2 3 Product Plugs Rails Rivets Clips Machine Capacity mins./year) 4 Gross margin/unit \ 0.40 \ 1.20 \ 0.80 \ 1.10 5 Minutes/unit 1 2 3 1.5 250,000 6 Gross margin/minute 7 Maximum production 8 Profit -What is the maximum production of clips based on the machine capacity?

Consider the spreadsheet for Stone Age Surfboards provided below. Use this data to answer the following questions), using the Standard Solver. Stone Age Surfboards is a small manufacturer of two types of popular low-tide surfboards, the Graystone and the Lava models. The manufacturing process consists of two departments: fabrication and finishing. The fabrication department has 8 skilled workers, each of whom works 9.25 hours per day. The finishing department has 5 workers, each of whom works a 6-hour shift per day. Each pair of Graystone surfboards requires 2.5 labor hours in the fabrication department and 2 labor hours in finishing. The Lava model requires 4.2 labor-hours in fabrication and 3.6 labor-hours in finishing. The company operates 6 days a week. It makes a per unit profit of $40 on the Graystone model and $60 on the Lava model. The company anticipates selling at least twice as many Lava models as Graystone models. A B C D 1 Stone Age Surfboards 2 3 Data 4 Product 5 Department Graystone Lava Limitation hours) 6 Fabrication 2.5 4.2 74 7 Finishing 2 3.6 30 8 9 Profit/Unit \ 40.00 \ 60.00 10 11 12 Model 13 Graystone Lava 14 Quantity Produced Hours Used 15 Fabrication 16 Finishing 17 18 Excess Lava 19 Market Mixture 20 21 Total Profit 22 Profit Contribution -What is the Allowable Decrease for Graystone surfboards produced?

Problem statements such as production must be sufficient to meet promised customer orders would be categorized as a constraint.

Use the table below to answer the following questions). Letherin Hides is a company that makes boots specifically targeting college students. Forecasts of sales for the next year are 200 in the summer, 450 in the autumn, and 500 in the winter. Accessories that are used on the boots are purchased from a supplier for $31.66. The cost of capital is estimated to be 24% per year or 6% per quarter); thus, the holding cost per item is 0.06$31.66) = $1.9 per quarter rounded figure). Letherin Hides hires freelance art designers at part-time to craft designs during the summer, and they earn $6 per hour. In the autumn, labor is more difficult to keep, and the owner must pay $6.5 per hour to retain qualified help. Because of the high demand for part-time help during the winter holiday season, labor rates are higher in the winter, and workers earn $7.75 per hour. Each boot design takes 2 hours to complete. How should production be planned over the three quarters to minimize the combined production and inventory holding costs? The table below provides information on Letherin Hides boot design cost and production. Datherin Hides Data Sumuner Autumn Winter Unit Production Cost 12 13 15.5 Unit Inventory Holding Cost 1.9 1.9 1.9 demand 200 450 500 Use a linear optimization model based on the data to answer the following questions. -According to the linear optimization model, what is the total cost incurred by Letherin Hides for the combined seasons?

Use the following information to answer the questions) below. Cerebro Manufacturing produces four types of structural support fittings-plugs, rails, rivets, and clips-which are machined on two CNC machining centers. The machining centers have a capacity of 250,000 minutes per year. The gross margin per unit and machining requirements are shown in the spreadsheet below. A B C D E F 1 Cerebro Manufacturing Model 2 3 Product Plugs Rails Rivets Clips Machine Capacity mins./year) 4 Gross margin/unit \ 0.40 \ 1.20 \ 0.80 \ 1.10 5 Minutes/unit 1 2 3 1.5 250,000 6 Gross margin/minute 7 Maximum production 8 Profit -What is the total profit generated from the manufacture of only plugs?

Use the table below to answer the following questions). Letherin Hides is a company that makes boots specifically targeting college students. Forecasts of sales for the next year are 200 in the summer, 450 in the autumn, and 500 in the winter. Accessories that are used on the boots are purchased from a supplier for $31.66. The cost of capital is estimated to be 24% per year or 6% per quarter); thus, the holding cost per item is 0.06$31.66) = $1.9 per quarter rounded figure). Letherin Hides hires freelance art designers at part-time to craft designs during the summer, and they earn $6 per hour. In the autumn, labor is more difficult to keep, and the owner must pay $6.5 per hour to retain qualified help. Because of the high demand for part-time help during the winter holiday season, labor rates are higher in the winter, and workers earn $7.75 per hour. Each boot design takes 2 hours to complete. How should production be planned over the three quarters to minimize the combined production and inventory holding costs? The table below provides information on Letherin Hides boot design cost and production. Datherin Hides Data Sumuner Autumn Winter Unit Production Cost 12 13 15.5 Unit Inventory Holding Cost 1.9 1.9 1.9 demand 200 450 500 Use a linear optimization model based on the data to answer the following questions. -According to the linear optimization model, what is the net production for autumn?

Which of the following is an example of a simple bounds constraint?

constraints essentially state that input = output and ensure that the flow of material or money is accounted for at locations or between time periods.

Use the table below to answer the following questions). The Riviera Transport Company RTC) produces car accessories at two plants: Dallas and Atlanta. They ship them to major distribution centers in Houston, San Jose, Jacksonville, 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, plant capacities over the next planning period, and distribution center demands. RTC'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 Xij = amount shipped from plant i to distribution center j, where i = 1 represents Dallas, i = 2 represents Atlanta, j = 1 represents Houston, and so on. Transpartation Model Data Distribution Center Plant Houston Sarn Jose Jacksorville Mernphis Capacity Dallas 13.00 15.25 10.99 18.48 1250 Atlarita 10.75 15.16 9.65 18.50 750 Dernand 175 325 480 950 Answer the following questions) using a linear optimization model. -The mathematical form Y $\le$ 450 would be considered as representing a simple bounds constraint.

Consider the spreadsheet for Stone Age Surfboards provided below. Use this data to answer the following questions), using the Standard Solver. Stone Age Surfboards is a small manufacturer of two types of popular low-tide surfboards, the Graystone and the Lava models. The manufacturing process consists of two departments: fabrication and finishing. The fabrication department has 8 skilled workers, each of whom works 9.25 hours per day. The finishing department has 5 workers, each of whom works a 6-hour shift per day. Each pair of Graystone surfboards requires 2.5 labor hours in the fabrication department and 2 labor hours in finishing. The Lava model requires 4.2 labor-hours in fabrication and 3.6 labor-hours in finishing. The company operates 6 days a week. It makes a per unit profit of $40 on the Graystone model and $60 on the Lava model. The company anticipates selling at least twice as many Lava models as Graystone models. A B C D 1 Stone Age Surfboards 2 3 Data 4 Product 5 Department Graystone Lava Limitation hours) 6 Fabrication 2.5 4.2 74 7 Finishing 2 3.6 30 8 9 Profit/Unit \ 40.00 \ 60.00 10 11 12 Model 13 Graystone Lava 14 Quantity Produced Hours Used 15 Fabrication 16 Finishing 17 18 Excess Lava 19 Market Mixture 20 21 Total Profit 22 Profit Contribution -What will be the total profit contribution from Graystone surfboards?

Use the table below to answer the following questions). Qualis Investments is a private limited business that manages personal financial portfolios. The company manages six mutual funds and has a client that has acquired $750,000 from an inheritance. Characteristics of the funds are given in the table below. Qualis Mutual Fund Data Expected Annual Return Risk Measure 1 Qualis Low-priced Stock Fund 7.23\% 11.87 2 Qualis Multinational Fund 8.42\% 12.62 3 Qualis Mid-cap Stock Fund 6.66\% 13.72 4 Qualis Mortgage Fund 2.72\% 3.37 5 Qualis Income Equity Fund 8.89\% 8.4 6 Qualis Balanced Fund 3.50\% 6.91 The company recommends that no more than $300,000 be invested in any individual fund, and that at least $80,000 be invested in each of the mid-cap stock and mortgage fund. The total amount invested in the low-priced and multinational funds are at least 35% of the total investment. The client would like an average return of at least 6%. Assume that X1- X6 represent the dollar amounts invested in funds 1 through 6, respectively. -Which of the following is the objective function minimize total risk?

Consider the spreadsheet for Stone Age Surfboards provided below. Use this data to answer the following questions), using the Standard Solver. Stone Age Surfboards is a small manufacturer of two types of popular low-tide surfboards, the Graystone and the Lava models. The manufacturing process consists of two departments: fabrication and finishing. The fabrication department has 8 skilled workers, each of whom works 9.25 hours per day. The finishing department has 5 workers, each of whom works a 6-hour shift per day. Each pair of Graystone surfboards requires 2.5 labor hours in the fabrication department and 2 labor hours in finishing. The Lava model requires 4.2 labor-hours in fabrication and 3.6 labor-hours in finishing. The company operates 6 days a week. It makes a per unit profit of $40 on the Graystone model and $60 on the Lava model. The company anticipates selling at least twice as many Lava models as Graystone models. A B C D 1 Stone Age Surfboards 2 3 Data 4 Product 5 Department Graystone Lava Limitation hours) 6 Fabrication 2.5 4.2 74 7 Finishing 2 3.6 30 8 9 Profit/Unit \ 40.00 \ 60.00 10 11 12 Model 13 Graystone Lava 14 Quantity Produced Hours Used 15 Fabrication 16 Finishing 17 18 Excess Lava 19 Market Mixture 20 21 Total Profit 22 Profit Contribution -What is the total number of hours used for the finishing operation?

Use the table below to answer the following questions). Qualis Investments is a private limited business that manages personal financial portfolios. The company manages six mutual funds and has a client that has acquired $750,000 from an inheritance. Characteristics of the funds are given in the table below. Qualis Mutual Fund Data Expected Annual Return Risk Measure 1 Qualis Low-priced Stock Fund 7.23\% 11.87 2 Qualis Multinational Fund 8.42\% 12.62 3 Qualis Mid-cap Stock Fund 6.66\% 13.72 4 Qualis Mortgage Fund 2.72\% 3.37 5 Qualis Income Equity Fund 8.89\% 8.4 6 Qualis Balanced Fund 3.50\% 6.91 The company recommends that no more than $300,000 be invested in any individual fund, and that at least $80,000 be invested in each of the mid-cap stock and mortgage fund. The total amount invested in the low-priced and multinational funds are at least 35% of the total investment. The client would like an average return of at least 6%. Assume that X1- X6 represent the dollar amounts invested in funds 1 through 6, respectively. -Which of the following describes the constraint on the mid-cap fund?