Operations Research Answers

The optimal solution of an ILLP can be obtained by rounding off the optimal solution of its LP relaxation .

The optimal solution for the following LPP is Z* = 30 :

Max Z = X1 - X2 + 3X3

Subject to X1 + X2 + X3 less than or equal to 10

X1, X2, X3 greater than or equal to zero .

Healthy Snacks Co. produces snack mixes. Recently, the company has decided to introduce a new snack mix that has peanuts, raisins, pretzels, dries cranberries, sunflower seeds and pistachios. Each bag of the new snack is designed in order to hold 250 grams of the snack. The company has decided to market the new product with an emphasis on its health benefits. After consulting nutritionists, Healthy Snacks decides to mix the ingredients so that the snack has the following specifications:

 Three custom officers check the luggage of the passengers of an airport. The 

passengers are found to arrive at an average rate of 30 per 8 hours a day. The amount 

of time a custom officer spends with the passenger is found to have an exponential 

distribution with mean service time 32 minutes. (5) 

 (i) Find the probability that all the custom officers are idle. 

 (ii) Find the expected number of passengers in the queues. 

 (iii) Find the expected waiting time of passenger in the system.

Critically examine Discovery’s use of game theory. Provide two examples of game theory as used by Discovery as a whole or any business units.

Question 1

The following information shows the simulation

conducted by the bank management on the waiting line

of the customers in front of 2 counters:

Processing time of Counter 1 (W1)- 1 second

Processing time of Counter 2 (W2) -2 seconds

Customers are arriving at the rate ofl second.

Walking to the counter takes 1 second.

Customers are waiting at point A. Point A is the end of

the queue. The queue can develop to accommodate many

customers. Many customers a, b, c, d...... are waiting at

point A.

Customer walks to counter 1 first whenever possible.

Point W1 and W2 are the front part of the counter 1 and

counter 2 respectively where the officer serves the

customers.

W2

W1

A

t= 0

t=1

t= 10

Simulate the movement of customers of a, b, c, d.....and

etc from point A to point W1 or point W2 from t = 0 to t

= 10 seconds. Fill in the position of various points in the

diagram. How many customers have been served by the

counters

answer.

seconds?

after

Explain

b) The Farmer decides to buy his bags of food A and food B from another store. This store sells a bag of food A for $9 and a bag of food B for $11. However, the farmer realised the new food bags are perishable. If the farmer does not use an entire bag, the food will spoil. The farmer must now use all of the contents of each bag. Use the linear programming process for this scenario to determine how many bags of food A and B the farmer now needs to meet the animals' minimum daily requirements for a minimum cost. Does the Farmer save any money but purchasing from another store?

The manager of a grocery store in the retirement community of Kapiri is interested in providing good service to the senior citizens who shop in his store. Presently, the store has a separate check-out counter for senior citizens. On average, 30 senior citizens per hour arrive at the counter according to a Poisson distribution and are served at an average rate of 35 customers per hour with exponential service times. Find the following;

(a) Utilization of the checkout clerk

(b) Number of customers in a system

(c) Number of customers in line

(d) Time spent in the system

(e) Waiting time in line

The manager of Kapiri grocery wants to answer the following question:

(1) What service rate would be required to have customers average only eight minutes in

A company has three cement plants from which cement has to be transported to four distribution centres. With identical costs of production at the three plants, the only variable costs involved are transportation costs. The monthly demand at the four distribution centres and the distance from the plants to the distribution centres (in km) are given below: Distribution centres Plants Monthly production (tonnes) 500 1,000 150 800 10,000 200 700 500 100 12,000 600 400 100 900 8,000 Monthly demand (tonnes) 9,000 9,000 10,000 4,000 The transportation charges are ? 10 per tonne per km. Suggest optimal transportation schedule and indicate the total minimum transportation cost. Specially Structured Linear Programmes I: Transportation and Transhipment Problems • 295 If, for certain reasons, route from plant C to distribution centre X is closed down, will the transp- ortation schedule change? If so, suggest the new schedule and effect on total cost.

A tractor operator has only one tractor and he operates it on the job order from farmers. The requestors for the jobs arrive with Poisson distribution having interval time of 0.7 day. The average time to do a job is distributed exponentially with mean 0.5 day. Assuming that the tractor can take up the next job immediately on completion of the previous job, determine the following: i. Will a queue be formed? Explain ii. If queue is formed will it statistically stabilize? Explain your answer iii. What is the utilization factor of the tractor? iv. What is the idle time in daily duty of 7 hours? v. What is the mean number of job orders waiting? vi. What is the mean waiting time for job orders in the system? vii. What is the mean waiting time in the queue?