LGT3102 – Management Science

Homework #4

1. The Spencer Shoe Company manufactures a line of inexpensive shoes and distributes to five main distribution centers: Milwaukee, Dayton, Cincinnati, Buffalo, and Atlanta, from which the shoes are shipped to retail shoe stores. The company currently has four production plants located at Pontiac, Cincinnati, Dayton, and Atlanta, and they are considering shutting down some of them. Distribution costs and other data are given as follows:

Normal weekly plant capacity (pairs of shoes) 27,000 40,000 40,000 40,000 Unit production cost $10.80 $10.56 $10.76 $10.48

Weekly fixed cost $28,000 $16,000 $24,000 $28,000

If a production plant is shut down, the weekly fixed cost of that plant can be avoided. All demand must be met. The company wants to determine which plant(s) they should keep, so that total weekly cost, including production, distribution, and fixed costs, is minimized.

• Write down the algebraic mixed integer linear programming (MILP) formulation of this problem. Note: Please treat the number of pairs of shoes shipped from one location to another as a continuous variable, and use the following decision variables:

𝑋𝑋_{𝑖𝑖𝑖𝑖} = number of pairs of shoes shipped from Plant 𝑖𝑖 to DC 𝑗𝑗 (a continuous variable); 𝑌𝑌_𝑖𝑖 = 1 if Plant 𝑖𝑖 is open, and 𝑌𝑌_𝑖𝑖 = 0 if Plant 𝑖𝑖 is closed.

• Use Excel Solver to solve your MILP (see page 5 of the Course Outline for spreadsheet homework instructions). Note: In the Solver Options menu, set Integer Optimality to 0%. Otherwise, the solution that Solver generates may not be optimal.

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2. The following table illustrates a number of possible duties for the drivers of a bus company. We wish to ensure, at the lowest possible cost, that at least one driver is on duty for each hour of the planning period (8 am to 6 pm). Formulate this scheduling problem as a shortest path problem (i.e., provide a network diagram, state clearly what each node represents, and define the length of every arc). Note: There is no need to determine the optimal solution.

3. Schips Department Store operates a fleet of 10 trucks. The trucks arrive at random times throughout the day at the store’s truck dock to be loaded with new deliveries or to have incoming shipments from the regional warehouse unloaded. The arrival rate per truck is 0.25 trucks per hour. The service rate is 4 trucks per hour. Assume that the arrivals of each truck follow a Poisson probability distribution and the service times follow an exponential distribution. Determine the following operating characteristics:
   • The probability no trucks are at the truck dock
   • The average number of trucks waiting for loading/unloading
   • The average number of trucks in the truck dock area
   • The average waiting time before loading/unloading begins
   • The average waiting time in the system
   • What is the hourly cost of operation if the cost is $50 per hour for each truck and $30 per hour for the truck dock?
   • Consider a two-channel truck dock operation where the second channel could be operated for an additional $30 per hour. How much would the average number of trucks in the truck dock area have to be reduced to make the two-channel truck dock economically feasible?