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1. A farmer has 1000 perches of land in which he can grow corn, millet, or cowpea. Each perch of corn costs Rs 100/- for preparation, requires 7 man-days of work and yields a profit of Rs 30/-. A perch of millet costs Rs 120/- for preparation, requires 10 man-days of work and yields a profit of Rs 40/-. A perch of cowpea costs Rs 70/- for preparation, requires 8 man-days of work and yields a profit of Rs 20/-. The farmer has Rs 100,000/- for preparation and can count on 8,000 man-days work.
a. Formulate a linear programming model for the above problem. (Use X1, X2 and X3 for corn, millet, and cowpea respectively). (5 marks)
b. Prepare the first simplex table. (2 marks)
c. Find the pivot element. (2 marks)
d. The final simplex table for the above problem is given below.
Note: s1, s2and s3 relates to preparation cost, man-days of work and land area respectively.
Basic Variables Quantity 30 40 20 0 0 0
x1 x2 x3 s1 s2 s3
X1 250 1 0 -1.625 0.0625 -0.75 0
X2 625 0 1 1.9375 -0.0438 0.625 0
S3 125 0 0 0.6875 -0.0188 0.125 1
Zj 32500 30 40 28.75 0.125 2.5 0
Cj -Zj 0 0 -875 -0.125 -2.5 0
As per the above table;
i. What is the optimal solution? (2 marks)
ii. Interpret the shadow prices of preparation cost and man-days of work. (2 marks)
iii. Construct the dual for the problem. (5 marks)
iv. Find the optimal values for the dual variables. (2 marks)
v. Find the range within which the production of corn can be changed without affecting the optimal solution. (3 marks)
vi. Find the range within which the production of millet can be changed without affecting the optimal solution. (3 marks)
vii. Find the range within which the production of cowpie can be changed without affecting the optimal solution. (3 marks)
viii. Identify the range within which preparation cost can be varied without altering the optimal solution. (3 marks)
ix. Identify the range within which man-days of work can be varied without altering the optimal solution. (3 marks)
(Total 35 marks)
2. A company has 3 auditors. Each auditor can work upto 160 hours during the next month. During the month, the auditors have to complete 3 projects. Project 1 will take 130 hours, project 2 will take 140 hours and project 3 will take 160 hours. The amount in Rupees per hour that can be billed for assigning each auditor to each project is given below:
Auditor P1 P2 P3
A1 1200 1500 1900
A2 1400 1300 1200
A3 1600 1400 1500
a. Represent the data in a transportation tableau. (5 marks)
b. Find the initial basic feasible solution of the following transportation problem using the following methods and compute the transportation cost of the initial feasible solution.
i. North West Corner Method (5 marks)
ii. Least Cost Method (5 marks)
c. Find the optimal solution for the initial solution obtained by North West Corner Method and compute the new cost.
i. using Stepping Stone Method (10 marks)
ii. using MODI Method (10 marks)
d. Find the optimal solution for the initial solution obtained by Least Cost Method and compute the new cost.
i. using Stepping Stone Method (10 marks)
ii. using MODI Method (10 marks)
(Total 55 marks)
3. A factory works for 8 hours a day. Processing time in minutes for each product by each of 5 operators are given below. Find out which product should be assigned to which operator so that the total processing time is minimized. Processing times of each product are given in the table below:
A B C D E
1 4 2 4 1 2
2 2 3 3 2 1
3 4 3 2 2 1
4 3 2 3 1 2
5 2 3 4 1 3
1. A producert purchases an item at the rate of Rs 10/- per piece from a manufacturer. The annual requirement is 2000 units of the item, while the cost per order is Rs 15/- and the holding cost is Rs 8/- per annum. There is a requirement of 5 square feet to store the item.
a. Calculate the economic order quantity (EOQ) of a product. (3 marks)
b. Calculate the cost of maintaining the inventory. (3 marks)
c. Calculate the reorder level of the products if lead time is 20 days. (3 marks)
d. If the stock out cost is Rs. 10/- per unit per month, calculate the EOQ level. (3 marks)
e. If maximum storage capacity is 300 square feet, calculate the EOQ level. (2 marks)
f. If a discount of 2% is offered for orders greater than 50 units but less than 100 units and 3% discount is offered for orders equal or greater than 100 units, calculate the inventory costs for the following scenarios:
i. Order size <=50 ii. 50 < Order size < 100 iii. 100 <= Order size (9 marks) g. Calculate the EOQ if the total budget available to purchase the material is Rs. 1,000/-. (2 marks) (Total: 25 marks) 2. A person designing digital art works finds that the time spent on an art work follows Exponential distribution with mean 20 minutes. If the art works aredesigned in the order in which they come in and their arrival is approximately follows Poisson distribution with an average rate of 15 per 8-hour day. a. What is the probability that the queuing system is empty? (4 marks) b. What is the probability that the designer is idle? (2 marks) c. How many hours do the designer idles per day? (2 marks) d. What is the average length in the queue? (2 marks) e. On average how many art works are there at the designer in a day? (2 marks) f. What is the average waiting time for art works? (2 marks) g. What is the probability that there are 4 art works to be designed? (2 marks) h. What is the probability that there are more than 8 art works to be designed? (2 marks) i. What is the waiting time of an art work in the queue? (2 marks) j. If the arrival rate of art works has doubled per hour, how many designers are required to provide an equilibrium level of service? (5 marks) (Total: 25 marks) 3. The following table gives data on time and cost of a project. Activity Preceding activity Normal Time (days) Crash time (days) Normal Cost (Rs.) Crash cost (Rs.) A - 2 1 10000 15000 B - 8 5 15000 21000 C A 4 3 20000 24000 D B 1 1 7000 7000 E B 2 1 8000 15000 F C,D 5 3 10000 16000 G E 6 2 12000 36000 a) Construct the network diagram for this project. (6 marks) b) Find the critical path of the project. (1 mark) c) Find the total project cost and duration. (2 marks) d) To shorten the project by 3 weeks at the lowest possible cost, which task(s) should be shortened? (15 marks) e) What would be the new cost after shortening of the project? (1 mark) (Total: 25 marks) 4. A company manufactures 30 items per day. The demand of these items during the past 20 days are listed below. 35 28 24 29 27 21 30 35 35 29 23 32 30 26 17 25 26 40 31 28 a. Randomly identify 10 values out of them to carry out a simulation. (For the purpose of this examination, use the random number table given below and carry out the generation of random numbers starting from its top left hand corner value in vertical direction). (10 marks) b. Describe how you selected the values. (5 marks) c. Estimate the profit/ loss of the company for the next 10 days, if the profit per unit of sold products is Rs. 100/- and the loss per unit from unsold products at the end of the day is Rs. 30/-. (10 marks) (Total: 25 Marks)