Stochastic OR

1. A single-server queueing system has a Poisson arrival process with rate lambda customers per hour. The service distribution has a rate of mu customers per hour, with a standard deviation of sigma customers per hour.

   1. Showing your calculations, determine the values of , , , , and . 0

   2. Unlike part a, suppose that the service rate, and hence the service time, is constant (same rate as in part a. Using the diagram in section 17.7, will for this system be greater or less than the value for the system in part a? If it is shorter, specify the value of . If it is longer, use the same graph to determine the smallest number of servers that will make for this system smaller than in part a and specify the value of .

2. Companies such as Raymour & Flanagan and Bobs have regional warehouses where furniture is stored until deliveries are made to customers. A single crew is used for loading and unloading trucks. Trucks arrive to be loaded/unloaded at a rate of lambda per hour, following a Poisson process. The time needed to load/unload a truck follows an exponential distribution with a mean service time of t minutes divided by the crew size. Trucks are loaded/unloaded one at a time (the crew cant be split). Crew members are paid $S/hour. Trucks that are idle (those waiting in line and a truck that is being loaded/unloaded) have a waiting cost of $W/hour. Using the output, determine the optimal crew size for minimizing expected total cost.

3. A company is setting up a new factory, which will include up to work transfer shops. The factory is 600 by 900 yards. Regardless of the number of transfer shops, the factory will be divided into identical rectangular regions, with each shop being at the middle of the region. (e.g. if there are two shops, the factory will be split into two 600 by 450 regions, and the shop would be at (300, 225). The fixed cost of each shop is $F/hour. If there are no more than two shops, each shop will be staffed by a two-member team that works as a single server with an exponential service time distribution with a mean of a minutes. If there are three shops, each shop will have a single server with a mean service time of b minutes (again, exponential). Customers arrivals across the factory follow a Poisson distribution with a rate of lambda per hour. Labor costs are $S/hour/person. Idle time (travel, plus all time in the service system) costs $W/hour. Determine the expected total cost for the three scenarios and determine which scenario is optimal.

4. A contractor is analyzing three contracts on which to make a bid. Due to the size of the company, the contractor can only make a bid on one of the three contracts. If the bid is accepted, the company will have to hire extra workers. Due to fluctuations in the labor market, the labor cost at the time they would begin work could be low, medium, or high. The payoff table below gives the estimated profit (in thousands of dollars, so $500 would represent $500,000) for each of the three contracts given the three possible labor costs. (see Excel q4 for your individual table)

Low labor cost Medium labor cost. High labor cost

Contract 1

Contract 2

Contract 3

Prior

1. Assuming that the company has the same probability of being awarded each contract, determine which contract they should bid on using: i) the maximin criterion, ii) the maximum likelihood criterion, and iii) Bayes decision rule.

2. Determine the crossover values, assuming the probability of the medium labor cost is correct. Based on the results, does it seem necessary to collect more data on the labor cost market?

FOR Question 1 -the values

lambda mu sigma

4.2 6.8 1.5

For the question 2-the values

lambda t S W

1.09 48 20.5 27.5

For the question 3 - the values

F a b lambda S W

44.9 1.62 2.8 23.5 21.9 31.7

for the question 4 -the values

low medium high

c1 334 165 13 c2. 297 192 34 c3. 239 156 61 Prior. 0.24 0.3 0.46