Sri Balaji College of Engineering and technology, Jaipur

Department of Mechanical Engineering

Session (2011-12)

B.Tech. IV Year/VII Semester Subject: - Operations Research

Branch: - Mechanical Engineering Date of assignment: - 12/09/2011

Prepared by: - Mr. Deepak Agrawal Date of submission: - 15/09/2011

Assignment No. 02 (Unit-II)

1. General Ford has two plants, two warehouses, and three customers. The locations of these

are as follows: Plants: Detroit and Atlanta Warehouses: Denver and New York Customers: Los Angeles, Chicago, and Philadelphia Cars are produced at plants, then shipped to warehouses, and finally shipped to customers. Detroit can produce 150 cars per week, and Atlanta can produce 100 cars per week. Los Angeles requires 80 cars per week; Chicago, 70; and Philadelphia, 60. It costs Rs.10, 000 to produce a car at each plant, and the cost of shipping a car between two cities is given in Table. Determine how to meet General Ford’s weekly demands at minimum cost.

From To Denver New York

Detroit 1253 637 Atlanta 1398 841

From To Los Angeles Chicago Philadelphia

Denver 1059 996 1691 New York 2786 802 100

2. The Chicago board of education is taking bids on the city’s four school bus routes. Four companies have made the bids in following Table.

a) Suppose each bidder can be assigned only one route. Use the assignment method to minimize Chicago’s cost of running the four bus routes.

b) Suppose that each company can be assigned two routes. Use the assignment method to minimize Chicago’s cost of running the four bus routes.

Company Bids Route 1 Route 2 Route 3 Route 4

1 4000 5000 X X 2 X 4000 X 4000 3 3000 X 2000 X 4 X X 4000 5000

3. Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a

desert island with Jennifer Aniston, Courteney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility measures” in Table indicate how much happiness each couple would experience if they spent all their time together. The happiness earned by a couple is proportional to the fraction of time they spend together. For example, if Freddie and Gwyneth spend half their time together, they earn happiness of 4.5.

a) Let xij be the fraction of time that the ith man spends with the jth woman. The goal of the eight people is to maximize the total happiness of the people on the island. Formulate an LP whose optimal solution will yield the optimal values of the xij’s.

b) Explain why the optimal solution in part (a) will have four xij as 1 and twelve xij as 0. The optimal solution requires that each person spend all his or her time with one person of the opposite sex, so this result is often referred to as the Marriage Theorem.

c) Determine the marriage partner for each person.

JA CC GP JR

TC 7 5 8 2 FP 7 8 9 4 HF 3 5 7 9 ML 5 5 6 7

4. A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per

day, and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary, and personal. The processing cost per check depends on the site (see Table). Each day, 5,000 checks of each type must be processed. Formulate and solve a balanced transportation problem to minimize the daily cost of processing checks.

Checks Site

1 2 Vendor 5 3 Salary 4 4

Personal 2 5

5. Consider a transportation problem where the origins are plants and destinations are depots. The unit transportation costs, capacity at the plants, and the requirements at the depots are indicated below:

Plant Depot Capacity

X Y Z A 1 3 15 150 B 3 5 25 300 Requirements 150 150 150 450 When each plant is also considered a destination and each depot is also considered an origin, there are altogether five origins and five destinations. Some additional cost data are also necessary. These are presented in the following Tables.

Unit Transportation cost from Plant to Plant From Plant TO

Plant A Plant B A 0 65 B 1 0

Unit Transportation Cost from Depot to Depot From Depot To

Depot X Depot Y Depot Z X 0 23 1 Y 1 0 3 Z 65 3 0

Unit Transportation cost from Depot to Plant Depot Plant

A B X 3 15 Y 25 3 Z 45 55

6. At the head office of www.universalteacher.com there are five registration counters. Five

persons are available for service. Counter↓ Person

A B C D E 1 30 37 40 28 40 2 40 24 27 21 36 3 40 32 33 30 35 4 25 38 40 36 36 5 29 62 41 34 39

How should the counters be assigned to persons so as to maximize the profit?

7. Best-ride airlines that operates seven days a week has the following time-table.

Crews must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in the smaller layover. For each pair also mention the city where crew should be based.

8. A salesman wants to visit cities A, B, C, D and E .He does not want to visit any city twice before completing his tour of all the cities and wishes to return to the point of starting journey. Cost of going from one city to another (in rupees) is shown in table below. Find the least cost route.

A B C D E A 0 2 5 7 1 B 6 0 3 8 2 C 8 7 0 4 7 D 12 4 6 0 5 E 1 3 2 8 0

9. Solve the following Assignment problem.

A B C D E 1 11 17 8 16 20 2 9 7 12 6 15 3 13 16 15 12 16 4 21 24 17 28 26 5 14 10 12 11 13

10. Find the optimum solution to the following transportation problem in which the cells

contain the transportation in rupees. Warehouse

Factory↓ A B C D E Available1 7 6 4 5 9 40 2 8 5 6 7 8 30 3 6 8 9 6 5 20 4 5 7 7 8 6 10

Required 30 30 15 20 5

Flight No.↓ Delhi-Mumbai Departure from Delhi

Arrival at Mumbai

1 7.00 AM 8.00 AM 2 8.00 AM 9.00 AM 3 1.00 PM 2.00 PM 4 6.00 PM 7.00 PM

Flight No.↓ Mumbai-Delhi Departure from Delhi

Arrival at Mumbai

101 8.00 AM 9.00 AM 102 9.00 AM 10.00 AM 103 12.00 Noon 1.00 PM 104 5.00 PM 6.00 PM