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# (30 points) A cross docking facility is currently operating at full capacity in terms of unloading.

3. (30 points)

A cross docking facility is currently operating at full capacity in terms of unloading. The business has grown so large that they are currently in the process of building a new, larger cross docking facility. They would like to avoid the long waits that trucks endure before being unloaded due to the fact that there are not enough unloading doors. If the company adds a new unloading door, this will result in a fixed cost. However, this fixed cost is negligible compared to the hourly costs incurred throughout the year. So, we will be considering a comparison of hourly costs.

The company would like to know how many doors to plan to have in the new building. For every new door added, a new worker must be hired at a rate of $20/hour. Given the unloading speed of an average worker, you know that with 2 unloading doors (i.e., 2 workers), trucks can be processed at a rate of 10 per hour. With 3 unloading doors, you can process trucks at 15 per hour. With 4 unloading doors, you can process trucks at a rate of 20 per hour. With 5 unloading doors, you can process trucks at a rate of 30 per hour and finally, with 6 unloading doors, you can process 40 per hour. Trucks arrive at a rate of 14 per hour (plus or minus 2). Because, the current facility only has two doors and this is not sufficient, we will disregard this option. Based on the arrival rate of the trucks and the unloading rates, this creates an average line size PER HOUR of:

3.2 (if 3 doors), 0.9 (if 4 doors), 0.2667 (if 5 doors), 0.1286 (if 6 doors). For every hour that a truck spends waiting, it costs the company $50 in truck driver labor costs and loss of profit due to idle time.

a. How many doors should the facility plan for if they want to minimize total HOURLY cost?

**b. Holding labor cost constant at $20, how expensive would idle time per hour have to be to justify increasing the number of doors to one MORE than what you found in part a.?**

**c. Holding the cost of idle time constant at $50, how expensive would labor costs per hour have to be to justify opening one LESS door than what you found in part a.?**

**d. Comment on the sensitivity of your answer to changes in cost. For example, would you be confident that your choice of number of unloading doors would remain the same if there were small fluctuations in labor cost or idle costs?**