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1. A grocery store Molokai would like to start a promotional campaign. The advertising team must decide on the types of advertising that the store should use. Three effective advertising media are currently under consideration. A TV commercial costs $300,000 and would reach 10000 customers. A poster in local newspapers would attract 6000 customers at a cost of $200,000. A post on social media from a celebrity can receive 5000 customers at a cost of $180,000.

In addition to the money paid to media, each TV commercial costs $150,000 to produce. The cost of a poster in a local newspaper amounts to $50,000 while the preparation of a social media post costs $40,000. Each type of advertisement requires a separate design cost.

The store has a design budget of $1 million and can spend at most $3 million on all types of media. There are only 3 spots available for running commercials on TV and only 5 newspapers are available during the time of the promotional campaign. The number of social media post is limited to six. The store wants to determine the number of advertisements in each media to maximize the customers’ exposure, i.e., reach as many customers as possible.

1) Provide a complete algebraic formulation to determine the advertisement mix that maximizes the customers’ exposure to the grocery store. Define the variables and write down the objective function and all constraints mathematically.

  1. 2)  Create a spreadsheet model for this problem and solve it with Excel Solver.
  2. 3)  What is the optimal solution? What is the optimal value?

2. More than half of the Washington, D.C., Metro system’s train cars have been taken offline after a derailment in October 2021 and a possible safety issue with wheel assemblies. The Washington Metrorail Safety Commission ordered that all 748 of the system’s 7000-series rail cars, about 60 percent of its rail car fleet, be taken out of service. Kawasaki Rail Car builder has four service centers. The cost of replacing a wheel assembly at each service center, along with the spare parts and labor needed per 7000-series rail car, is shown in the table below.

Service center Cost ($) Labor (hours) Spare parts (units)

1 12000 4 5 2 8000 5 6 3 7200 6 7 4 5600 7 8

The union requires that at least 250 rail cars be serviced at Center 3. In the next two months, 4600 hours of labor and 5,700 units of spare parts are available for allocation to the four service centers.

  1. Formulate the problem as a linear programming problem (i.e., define the variables, and write down the objective function and all constraints in algebraic form).
  2. Create a spreadsheet model for this problem and solve it with Solver. Generate and provide a copy of the Sensitivity Analysis report with the solution.
  3. What is the optimal solution?
  4. What is the optimal value?
  5. What are the binding constraints?
  6. How much will it cost to service one more rail car? How much will we save by servicing one fewer?
  7. How would the optimal solution change if it costs only $6,000 to service one rail car at center 2? For what ranges of service costs at center 2 does the current optimal solution remain optimal?
  8. How much are we willing to pay for one additional labor hour?
  9. How much is the labor contract costing us? What would be the value of reducing the 250 rail cars limit down to 100 cars? To 0 rail cars?
  10. How much is one additional unit of spare parts worth? How many units are we willing to buy at that price?

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