1. [20 points] A mattress manufacturer has three production facilities located in Pittsburg (PA), Houston (TX),...

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1. [20 points] A mattress manufacturer has three production facilities located in Pittsburg (PA), Houston (TX), and Los Angeles (CA). They supply their products to three distribution centers in Boston (MA), Atlanta (GA), and San Jose (CA). Each distribution center has specific demands to meet. The capacity, demand, and transportation cost per unit between facilities are given in the following map: Demand = 1100 $4 Fresho Las vegas Salt Lake City Montone Wyoming Denver $24 Phoenix Capacity = 1000 Tuana Mexical Tucson Ensenada Cudad Juárez $14 0 Sonara Saint Paul South Dakoto Milwaukee Québec Demand 1200 Nebraska Lincoln $22 Capacity 1100 $5 Kansas $12 irginia Richmond Raleigh Memphis charlotte North Caroling New Mexico $20 Dallas $15 Demand = 800 $9 Austin Jacksonville San Antonio Capacity = 1300 Philadelphia For example, the capacity of Los Angeles facility is 1000 units. Demand of San Jose distribution center is 1100 units. Transportation cost per unit from Los Angeles to San Jose is $4. Develop a linear programming model defining the decision variables, objective function, and constraints. Using an appropriate linear programming solver, determine how many units should be shipped from each production facility to each distribution center to meet demand while minimizing transportation costs. Calculate the total transportation cost for this distribution plan. 2. [20 points] Javier Mendez is the head coach of the renowned AKA mixed martial arts gym. AKA has three full-time coaches already, each with years of experience in various martial arts. Their experience in martial arts in a scale of 1-5, 1 being beginner and 5 being expert, is given in the following table: Martial Art Coach Boxing Wrestling Jui Jitsu Muay Thai Taekwondo MMA Derek Yuen 5 3 2 2 2 4 Wayne Phillips 2 4 5 2 2 4 Ron Kelsar 2 2 2 4 5 4 With the help of an MBA student, who is also a student at AKA, coach Mendez has developed a linear program to assign coaches to each martial arts class to maximize the overall teaching rating. The LPSolve model is given in 'Q2 Model.mod' file and Excel Solver Model is given in 'Q2 Model.xlsx'. In this model, the class rating is maximized (higher experience coach in a class is related to higher rating), while each coach will be teaching 2 classes or less and each class will have exactly one coach. After developing the LP, coach Mendez hired three former champions as coaches to help with the increasing presence of athletes. The new coaches are also very experienced. Their rating is given in the table below: Martial Art Coach Boxing Wrestling Jui Jitsu Muay Thai Taekwondo MMA Daniel Cormier 4 5 3 2 2 4 Cain Velasquez 4 5 3 2 2 5 Luke Rockhold 3 3 2 5 4 4 Coach Mendez wants to reduce the workload of coaches by assigning one coach to only one class. a. Develop a linear programming model stating the decision variables, objective function, and constraints to represent the problem. b. Solve the model for the previous situation (2 classes to 1 coach). Which coach should be assigned to what class? What is the total rating? (Hint: *.mod files can be opened with Notepad or similar text editor software in addition to LP Solve). C. Solve the model with the current situation (1 class to 1 coach). Modify the Linear Programming model and find the best assignment of 6 coaches to 6 classes to maximize the overall rating. Which coach should be assigned to what class? What is the total rating? d. If a coach can teach a maximum of 2 classes, would the solution change? Explain. 3. [20 points] Elizabeth Bailey is the owner and general manager of Princess Brides, which provides a wedding planning service in southwestern Georgia. She uses social media advertising to market her business. She plans to run advertisements on two social media platforms, Instagram and Facebook, for the next year to maximize the customer reach. Each Instagram ad costs $62 and reaches 150 people, while the Facebook ad costs $41 and reaches 100 customers. Bailey has budgeted $2,000 for advertising. Based on comments from her customers, Bailey wants to have at least 20 Instagram ads and at least 10 Facebook ads. (Note: ads cannot be fractions). a. Formulate this as a linear program. Define decision variables, objective function, and constraints. b. Find the optimal solution for the linear program by using an appropriate linear program solver. Using the optimal solution for the linear program, find an integer solution (because ads cannot be fractions) by rounding the linear optimal solution or making an educated guess. What is the integer solution? How many Instagram and Facebook ads should be run? What is the total customer reach? C. Solve this as an integer programming problem. (See class examples to make a linear model to an integer model in LPSolve and Excel Solver). What is the optimal integer solution? How many Instagram and Facebook ads should be run? What is the total customer reach? d. Discuss the two solutions in (b) and (c). What is the reason for the difference? 4. [20 points] Innis Construction Company specializes in building moderately priced homes in Cincinnati, Ohio. Tom Innis has identified eight potential locations to construct new single-family dwellings, but he cannot put up homes on all of the sites because he has only $300,000 to invest in all projects. The following table shows the cost of constructing homes in each area and the expected profit to be made from the sale of each home. Note that the home-building costs differ considerably due to low costs, site preparation, and differences in the models to be built. Note also that a fraction of a home cannot be built. LOCATION COST OF BUILDING AT THIS SITE ($) EXPECTED PROFIT ($) Clifton 60,000 5,000 Mt. Auburn 50,000 6,000 Mt. Adams 82,000 10,000 Amberly 103,000 12,000 Norwood 50,000 8,000 Covington 41,000 3,000 Roselawn 80,000 9,000 Eden Park 69,000 10,000 a. Formulate Innis's problem using 0-1 integer programming. Define decision variables, objective function, and constraints. b. Solve with LPSolve or Excel Solver. What is the solution? What is the profit? 5. [20 points] Geraldine Shawhan is president of Shawhan File Works, a firm that manufactures two types of metal file cabinets. The demand for her two-drawer model is up to 600 cabinets per week; the demand for a three-drawer cabinet is limited to 400 per week. Shawhan File Works has a weekly operating capacity of 1,300 hours, with the two-drawer cabinet taking 1 hour to produce and the three-drawer cabinet requiring 2 hours. Each two-drawer model sold yields a $10 profit, and the profit for the large model is $15. Shawhan has listed the following goals: • Attain a profit as close to $11,000 as possible each week. • Avoid underutilization of the firm's production capacity. • Sell as many two- and three-drawer cabinets as the demand indicates. Set this up as a goal programming problem and solve it using computer software. What is the solution? 1. [20 points] A mattress manufacturer has three production facilities located in Pittsburg (PA), Houston (TX), and Los Angeles (CA). They supply their products to three distribution centers in Boston (MA), Atlanta (GA), and San Jose (CA). Each distribution center has specific demands to meet. The capacity, demand, and transportation cost per unit between facilities are given in the following map: Demand = 1100 $4 Fresho Las vegas Salt Lake City Montone Wyoming Denver $24 Phoenix Capacity = 1000 Tuana Mexical Tucson Ensenada Cudad Juárez $14 0 Sonara Saint Paul South Dakoto Milwaukee Québec Demand 1200 Nebraska Lincoln $22 Capacity 1100 $5 Kansas $12 irginia Richmond Raleigh Memphis charlotte North Caroling New Mexico $20 Dallas $15 Demand = 800 $9 Austin Jacksonville San Antonio Capacity = 1300 Philadelphia For example, the capacity of Los Angeles facility is 1000 units. Demand of San Jose distribution center is 1100 units. Transportation cost per unit from Los Angeles to San Jose is $4. Develop a linear programming model defining the decision variables, objective function, and constraints. Using an appropriate linear programming solver, determine how many units should be shipped from each production facility to each distribution center to meet demand while minimizing transportation costs. Calculate the total transportation cost for this distribution plan. 2. [20 points] Javier Mendez is the head coach of the renowned AKA mixed martial arts gym. AKA has three full-time coaches already, each with years of experience in various martial arts. Their experience in martial arts in a scale of 1-5, 1 being beginner and 5 being expert, is given in the following table: Martial Art Coach Boxing Wrestling Jui Jitsu Muay Thai Taekwondo MMA Derek Yuen 5 3 2 2 2 4 Wayne Phillips 2 4 5 2 2 4 Ron Kelsar 2 2 2 4 5 4 With the help of an MBA student, who is also a student at AKA, coach Mendez has developed a linear program to assign coaches to each martial arts class to maximize the overall teaching rating. The LPSolve model is given in 'Q2 Model.mod' file and Excel Solver Model is given in 'Q2 Model.xlsx'. In this model, the class rating is maximized (higher experience coach in a class is related to higher rating), while each coach will be teaching 2 classes or less and each class will have exactly one coach. After developing the LP, coach Mendez hired three former champions as coaches to help with the increasing presence of athletes. The new coaches are also very experienced. Their rating is given in the table below: Martial Art Coach Boxing Wrestling Jui Jitsu Muay Thai Taekwondo MMA Daniel Cormier 4 5 3 2 2 4 Cain Velasquez 4 5 3 2 2 5 Luke Rockhold 3 3 2 5 4 4 Coach Mendez wants to reduce the workload of coaches by assigning one coach to only one class. a. Develop a linear programming model stating the decision variables, objective function, and constraints to represent the problem. b. Solve the model for the previous situation (2 classes to 1 coach). Which coach should be assigned to what class? What is the total rating? (Hint: *.mod files can be opened with Notepad or similar text editor software in addition to LP Solve). C. Solve the model with the current situation (1 class to 1 coach). Modify the Linear Programming model and find the best assignment of 6 coaches to 6 classes to maximize the overall rating. Which coach should be assigned to what class? What is the total rating? d. If a coach can teach a maximum of 2 classes, would the solution change? Explain. 3. [20 points] Elizabeth Bailey is the owner and general manager of Princess Brides, which provides a wedding planning service in southwestern Georgia. She uses social media advertising to market her business. She plans to run advertisements on two social media platforms, Instagram and Facebook, for the next year to maximize the customer reach. Each Instagram ad costs $62 and reaches 150 people, while the Facebook ad costs $41 and reaches 100 customers. Bailey has budgeted $2,000 for advertising. Based on comments from her customers, Bailey wants to have at least 20 Instagram ads and at least 10 Facebook ads. (Note: ads cannot be fractions). a. Formulate this as a linear program. Define decision variables, objective function, and constraints. b. Find the optimal solution for the linear program by using an appropriate linear program solver. Using the optimal solution for the linear program, find an integer solution (because ads cannot be fractions) by rounding the linear optimal solution or making an educated guess. What is the integer solution? How many Instagram and Facebook ads should be run? What is the total customer reach? C. Solve this as an integer programming problem. (See class examples to make a linear model to an integer model in LPSolve and Excel Solver). What is the optimal integer solution? How many Instagram and Facebook ads should be run? What is the total customer reach? d. Discuss the two solutions in (b) and (c). What is the reason for the difference? 4. [20 points] Innis Construction Company specializes in building moderately priced homes in Cincinnati, Ohio. Tom Innis has identified eight potential locations to construct new single-family dwellings, but he cannot put up homes on all of the sites because he has only $300,000 to invest in all projects. The following table shows the cost of constructing homes in each area and the expected profit to be made from the sale of each home. Note that the home-building costs differ considerably due to low costs, site preparation, and differences in the models to be built. Note also that a fraction of a home cannot be built. LOCATION COST OF BUILDING AT THIS SITE ($) EXPECTED PROFIT ($) Clifton 60,000 5,000 Mt. Auburn 50,000 6,000 Mt. Adams 82,000 10,000 Amberly 103,000 12,000 Norwood 50,000 8,000 Covington 41,000 3,000 Roselawn 80,000 9,000 Eden Park 69,000 10,000 a. Formulate Innis's problem using 0-1 integer programming. Define decision variables, objective function, and constraints. b. Solve with LPSolve or Excel Solver. What is the solution? What is the profit? 5. [20 points] Geraldine Shawhan is president of Shawhan File Works, a firm that manufactures two types of metal file cabinets. The demand for her two-drawer model is up to 600 cabinets per week; the demand for a three-drawer cabinet is limited to 400 per week. Shawhan File Works has a weekly operating capacity of 1,300 hours, with the two-drawer cabinet taking 1 hour to produce and the three-drawer cabinet requiring 2 hours. Each two-drawer model sold yields a $10 profit, and the profit for the large model is $15. Shawhan has listed the following goals: • Attain a profit as close to $11,000 as possible each week. • Avoid underutilization of the firm's production capacity. • Sell as many two- and three-drawer cabinets as the demand indicates. Set this up as a goal programming problem and solve it using computer software. What is the solution?