Operations Research Answers

List the basic properties of linear programming models indicating also the assumptions of linear programming problems

Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean of 3 minutes.

i) What is the probability that a person arriving at the booth will have to wait?

ii) What is the average length of the queues that form time to time?

The telephone department will install a second booth when convinced that an arrival would expect to have to wait at least three minutes for the phone. By how much must the flow of arrivals be increased in order to justify a second booth?

A simple queuing system has the mean interval time of 8 minutes and a mean service time of 4 minutes .

i) Determine the mean service rate and the mean arrival rate.

ii) Determine the traffic intensity.

iii) Determine the mean time a customer spends in the queue and in the system .

iv) What is the expected number of customers in the queue and in the system.

v) What is the probability of having at most four customers in the system.

 Consider the transportation problem presented in the following table:

Destination

Origin 1 2 3 Supply

1 2 7 4 50

2 3 3 1 80

3 5 4 7 70

4 1 6 2 140

Demand 70 90 180 340

Use North West Corner Rule to determine the minimum cost of transportation hence use MODI approach to determine the transportation cost. Use Least Cost Method to determine the minimum cost of transportation. Use Vogel Approximation Method to determine the minimum cost of transportation. 

A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic processing and craftsman time for hand finishing. The table below gives the number of minutes required for each item:

Machine time Craftsman time Item

X 13 20

Y 19 29

The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time is costed at £10 per hour worked and craftsman time is costed at £2 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is £20 for X and £30 for Y. The company has a specific contract to produce 10 items of X per week for a particular customer. Formulate the problem of deciding how much to produce per week as a linear program. Solve this linear program graphically.

Q4). A gold processor has two sources of gold ore, source A and source B. In order to keep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints?

A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye how many acres of each should be planted to maximize profits?

The Copper Mining company owns 2 mines, which produce 3 grades of ore - high, medium, and low. The company has a contract to supply a smelting company with 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore each hour it is in operation. The company has developed the following LP model to determine the number of hours to operate each mine X and Y so that contractual obligations can be met at the lowest cost.