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Problem 3: Magic Star [25 pts] A Magic Star in a star polygon with numbers in each of its vertices and intersections (see Figure 1). o Figure 1: Magic star with 5 vertices The goal of the Magic Star is to place numbers such that: . The star only contains integer numbers from 1 to 10 The four numbers on each line sum to 24 No number is repeated Mathematically formulate a linear integer optimization model to find the solution to any Magic Star of this size. Proceed as follows: 1. [7 pts) Formulate this problem identifying parameters, decision variables, constraints, and objective function. 2. 15 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report your solution using a star diagram. Include a printout of your mod and .dat files. 3. (6 pts) Report the solution of the instances shown in Figure 2. Describe any modification to your mathematical model to accommodate the partially filled instances. Include a printout of your mod and .dat files. 4. [7 pts) Somebody claims that there is no Magic Star in which the numbers located in the vertices are all larger than those located at the intersections, i.e, {A,B,C,G,H) > {D, E, F, I, J} in Figure 1. Is this true? Explain. Describe any modification to your mathematical model and code that is needed to answer this question. Include a printout of your .mod and .dat files. Problem 2: Volunteer scheduling (30 pts) 1. (15 pts) The AZ Department of Health needs volunteers to cover multiple logistical duties at vaccination centers during the next week. Because of the variability in the demand, some days require more volunteers than others, as shown in Table 1. Volunteers must work 3 consecutive days at the same site. What is the minimum number of volunteers required to cover the given demands? Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Cardinals Stadium 17 13 15 19 14 16 11 Phoenix Municipal Stadium 23 15 8 28 18 22 8 Table 1: Volunteer requirements (a) [8 pts) Model this situation using a linear integer formulation. Identify parameters, decision variables, constraints, and objective function. (b) 17 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report the optimal number of volunteers and their corresponding schedules for each location. Include a printout of your .mod and .dat files. 2. (15 pts) The AZ Department of Health is considering adding part-time volunteers that work 2 consecutive days only. As an incentive, full- and part-time volunteers will receive a gift card worth Sc and Sd, respectively. The goal is to decide if allowing part-time drivers is beneficial for the operation (a) (5 pts) Model this situation using a linear integer formulation. Identify parameters, decision variables, constraints, and objective function, (b) (5 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report the optimal number of volunteers and their corresponding schedules at each location. Include a printout of your .mod and .dat files. (c) (5 pts) Compare the results with Part 1 assuming that gift cards are used as incentive. Which strategy one is better for AZDH when c = 50 and d=30? Is this result expected? Explain. Problem 3: Magic Star [25 pts] A Magic Star in a star polygon with numbers in each of its vertices and intersections (see Figure 1). o Figure 1: Magic star with 5 vertices The goal of the Magic Star is to place numbers such that: . The star only contains integer numbers from 1 to 10 The four numbers on each line sum to 24 No number is repeated Mathematically formulate a linear integer optimization model to find the solution to any Magic Star of this size. Proceed as follows: 1. [7 pts) Formulate this problem identifying parameters, decision variables, constraints, and objective function. 2. 15 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report your solution using a star diagram. Include a printout of your mod and .dat files. 3. (6 pts) Report the solution of the instances shown in Figure 2. Describe any modification to your mathematical model to accommodate the partially filled instances. Include a printout of your mod and .dat files. 4. [7 pts) Somebody claims that there is no Magic Star in which the numbers located in the vertices are all larger than those located at the intersections, i.e, {A,B,C,G,H) > {D, E, F, I, J} in Figure 1. Is this true? Explain. Describe any modification to your mathematical model and code that is needed to answer this question. Include a printout of your .mod and .dat files. Problem 2: Volunteer scheduling (30 pts) 1. (15 pts) The AZ Department of Health needs volunteers to cover multiple logistical duties at vaccination centers during the next week. Because of the variability in the demand, some days require more volunteers than others, as shown in Table 1. Volunteers must work 3 consecutive days at the same site. What is the minimum number of volunteers required to cover the given demands? Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Cardinals Stadium 17 13 15 19 14 16 11 Phoenix Municipal Stadium 23 15 8 28 18 22 8 Table 1: Volunteer requirements (a) [8 pts) Model this situation using a linear integer formulation. Identify parameters, decision variables, constraints, and objective function. (b) 17 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report the optimal number of volunteers and their corresponding schedules for each location. Include a printout of your .mod and .dat files. 2. (15 pts) The AZ Department of Health is considering adding part-time volunteers that work 2 consecutive days only. As an incentive, full- and part-time volunteers will receive a gift card worth Sc and Sd, respectively. The goal is to decide if allowing part-time drivers is beneficial for the operation (a) (5 pts) Model this situation using a linear integer formulation. Identify parameters, decision variables, constraints, and objective function, (b) (5 pts) Implement and solve this problem using AMPL (using Gurobi as a solver). Report the optimal number of volunteers and their corresponding schedules at each location. Include a printout of your .mod and .dat files. (c) (5 pts) Compare the results with Part 1 assuming that gift cards are used as incentive. Which strategy one is better for AZDH when c = 50 and d=30? Is this result expected? Explain