Instructions on Assignments - Operations Research
1. Three of you form a group & appoint a coordinator to liaise with us
2. Each assignment will be entrusted to only one group
3. Only a single report should be submitted for each assignment
4. Each assignment will carry 3 general questions which have to be answered separately, (1 to 2 pages), indicating the name of the participant who did that.
5. If you like to do your own project we encourage that but you have to submit proposal for our approval & we will put the 3 general questions to be done separately.
6. Coordinator of each group is kindly expected to inform the Assignment Number and Names (with Reg. number) of members of the group. If it is a Proposal for a new project Coordinator is expected to give names & proposal.
7. Assignment solutions should be submitted as a single document for each group as hard copy to the Electrical Engineering office. Last date of submission will be 31 March 2011.
GTFde Silva 0771131625
Assignment 1
1. Write a letter to your CEO importance of using CPM for Project Management in contrast to any other method your organization may be using at present
2. Discuss how OR is important for the development effort of the Country
3. Describe how OR can be used to help farmers
A small Engineering Company has planned to manufacture and market garbage disposal tractor equipment. For this venture the Company has to find a small grant to construct a prototype after which they will try to market the equipment to Municipalities and other customers. The Company had identified activities of the project and for each activity time estimates as (a) optimistic (m) moderate and (b) pessimistic, in number of weeks. These data are tabulated below
|
Item
|
Activity
|
a
|
m
|
b
|
|
1
|
Design & cost prototype
|
1
|
2
|
3
|
|
2
|
Get funds for prototype
|
6
|
8
|
10
|
|
3
|
Construct prototype
|
9
|
10
|
11
|
|
4
|
Seek markets
|
12
|
14
|
16
|
|
5
|
Construct equipment
|
12
|
14
|
16
|
|
6
|
Negotiate with buyers
|
1
|
2
|
3
|
|
7
|
Deliver equipment
|
1
|
2
|
3
|
It was noted that in order to expedite the project, items 3 and 4 can be done in parallel following item 2 and also items 4 and 5 can be done in parallel on the completion of item 4.
Draw the network for the project, identify the critical path and find the duration of the project.
Explain briefly how the above mentioned information could be found using suitable software..
Also construct the slack table for the project.
Find in how many weeks you can complete the construction with probabilities of 60 %, 65 %, ...80%.
Each of the items 1 to 4 have got delayed by 10%. Find how the balance work can be completed in
18 weeks. Find also the extra cost that has to be incurred for this crash program. [three of you can use the
"amb" method to estimate additional costs involved.]
Assignment 2.
- Write a letter to your CEO explaining the importance of using OR in your organization.
- Describe situations where CPM can be used in the Country.
- Describe why CPM should be used to assist managing projects.
A small engineering firm is to set up in a building which needs renovations and obtained for the purpose. Following table shows the activities that need be done before opening the office for business. The table also shows with estimations of time that may be needed in weeks by three persons.
|
Item
|
Activity
|
Estimate
1
|
Estimate
2
|
Estimate
3
|
|
A
|
Call application for staff
|
2
|
3
|
4
|
|
B
|
Call quotations for equipment
|
2
|
3
|
4
|
|
C
|
Purchase furniture
|
0
|
1
|
2
|
|
D
|
Modify office premises
|
1
|
3
|
5
|
|
E
|
Install equipment. Internet, etc.
|
1
|
2
|
3
|
|
F
|
Interview & appoint staff
|
2
|
3
|
4
|
|
G
|
Arrange office
|
1
|
1
|
1
|
|
H
|
Opening function
|
1
|
1
|
1
|
Model the activities in order to complete the project in shortest time taking into consideration logical connection between different activities. Find also the total time T, that will be needed to complete the project and the activities that can not be allowed to delay.
Explain briefly how the above mentioned information could be found using suitable software.
Also construct the slack table for the project.
Find in how many weeks you can complete the construction with probabilities of 60 %, 65 %, ...80%.
Each of the items A to E have got delayed by 10%. Find how the balance work can be completed in
5 weeks. Find also the extra cost that has to be incurred for this crash program. [three of you can use the
"amb" method to estimate additional costs involved.]
Assignment 3
- Describe areas in which computers can help in school management
- Describe some benefits satellite telecommunication had brought to Sri Lanka
- In USA some cars are equipped with a screen giving a map of the area showing location of the car and when the driver indicates destinations gives the shortest route. Explain in some detail how this happens.
Girls High School (GHS) has a middle section with grades 6, 7 and 8, each having three classes A, B, C. Nine teachers are to be assigned to these classes as class teachers. The Principal, Deputy and the Sectional Head had a problem of selecting the most suitable teacher to each class in order to maintain quality of education, discipline and extra curricular activities. Having studied the nature of the students of each class they decided to a give a mark out of 10 for each Teacher for each class, which they averaged and is given in the following table:-
Classes
|
Teacher
|
6 A
|
6B
|
6C
|
7A
|
7B
|
7C
|
8A
|
8B
|
8C
|
|
Nalini
|
7
|
2
|
5
|
8
|
4
|
5
|
7
|
4
|
6
|
|
Sujatha
|
6
|
5
|
7
|
4
|
6
|
3
|
6
|
4
|
0
|
|
Padma
|
4
|
0
|
5
|
4
|
7
|
5
|
8
|
5
|
7
|
|
Shiela
|
6
|
3
|
7
|
0
|
6
|
8
|
6
|
5
|
4
|
|
Hania
|
2
|
3
|
3
|
3
|
0
|
8
|
5
|
3
|
4
|
|
Amara
|
0
|
5
|
6
|
5
|
5
|
0
|
7
|
8
|
8
|
|
Mallika
|
4
|
5
|
3
|
5
|
4
|
7
|
0
|
5
|
6
|
|
Ranjani
|
5
|
6
|
4
|
8
|
5
|
3
|
8
|
0
|
5
|
|
Rohini
|
6
|
7
|
5
|
0
|
7
|
8
|
6
|
5
|
4
|
Note: A mark 0 is given if a Teacher has a child in that class since as a policy
Teachers are discouraged from being a class teacher if her own child is
in that class.
Problem is to allocate one Teacher to one class so as to maximise the total marks.
Background
[a statement of the importance of getting a solution]
Methodology
How the data had been obtained, method used to solve the problem
You must try hand calculations (if possible or explain why it can be done that way), by computer software and by any other method if possible
Results
Conclusions
A brief analysis of solutions by different methods, what benefits solution will bring to the school, other situations this model can be used
Assignment 4
1. Discuss use of Operation Research in development
2. Describe the use of software packages to solve OR problems
3. Explain Linear Programming& its uses.
As the manager of your unit you have to plan the typesetting and printing of a technical manual. You have identified the following activities, their connections to predecessor activities and duration which are tabulated as follows:-
|
Item
|
Activity
|
Predecessors
|
Duration
(days)
|
|
A
|
Proofreading of manuscript
|
-
|
3
|
|
B
|
Sample pages typeset
|
-
|
2
|
|
C
|
Design covers
|
-
|
4
|
|
D
|
Preparation of diagrams
|
-
|
4
|
|
E
|
Author's approval of the sample pages
|
A, B
|
2
|
|
F
|
Typesetting of the manuscript
|
E
|
2
|
|
G
|
Author proofreads the draft
|
F
|
2
|
|
H
|
Author checks diagrams
|
D
|
1
|
|
I
|
Production of printing plates
|
G, H
|
2
|
|
J
|
Printing & binding
|
C, I
|
3
|
Prepare the network of the project and following standard procedure identify the critical path.
Identify the activities that can be delayed and by how many days without delaying the project.
Also construct the slack table for the project.
Find in how many days you can complete the construction with probabilities of 60 %, 65 %, ...80%.
Each of the items A to F (inclusive) have got delayed by 10%. Find how the balance work can be
completed in 18 days.
Find also the extra cost that has to be incurred for this crash program. [three of you can use the
"amb" method to estimate additional costs involved.]
Assignment 5
- Describe different types of activities encountered in a critical path activity and how duration of an activity can be best estimated.
- Describe with an example where a big project (over 100 activities) can be monitored using CPM.
- Discuss some projects in Sri Lanka where CPM or at least networks are used to monitor projects.
The widening of a certain road requires the relocation of wayside overhead power lines for which following activities with estimates of duration (in days) have been identified activity.
|
Activity
|
Description
|
Optimistic
|
Mode rate
|
Pessimistic
|
|
a
|
Planning & preliminaries
|
6
|
7
|
14
|
|
b
|
Secure poles & material
|
4
|
6
|
8
|
|
c
|
Pole location coordination
|
1.5
|
2
|
2.5
|
|
d
|
Trimming trees
|
0.5
|
3
|
3.5
|
|
e
|
Set poles
|
4
|
8
|
12
|
|
f
|
Set conductors(cover old place new)
|
0.5
|
3
|
3.5
|
|
g
|
Set energies (de energize old, Energize new)
|
2
|
3
|
4
|
|
h
|
Remove old poles and connectors
|
4
|
6
|
8
|
|
i
|
Finalize (return material etc.)
|
0.25
|
0.5
|
1.75
|
Draw the network diagram based on your judgment, in order to complete the project in shorted feasible time. For instance, you can start with activity a followed by c and d in parallel. Activity b can follow c which can follow e. You can use your own judgement for the rest of activities.
Identify the critical path.
Also construct the slack table for the project.
Find in how many days you can complete the construction with probabilities of 60 %, 65 %, ...80%.
Each of the items a to e (inclusive) have got delayed by 10%. Find how the balance work can be
completed in at least 10 % more than the total period planned . Find also the extra cost that has to be
incurred for this crash program. [three of you can use the "amb" method to estimate additional costs
involved.]
Assignment 6 .
Write short note on following stressing on their use in the world of work.
(i) Linear Programming
(ii) Game Theory
(iii) Graph theory
An Electricity Generating Board (B) wishes to set up a project against which a group of
protestors (P) are agitating. B and P have 3 and 4 strategies each respectively to achieve
their individual objectives. A think tank called by B scored marks from 0 to 9, as points P
will get for a pair of strategies of each, as shown in the following matrix.
Examine using principles of game theory whether the problem has a unique saddle point solution for pure strategies.
Also solve this problem by using an alternate approach.
Assignment 7
- Describe advantage of using game theory in competition
- Discuss importance of Nash's extension to game theory
- Discuss difficulties of using game theory in competitions.
A shop manager is considering three alternatives to an existing milling machine.
a) Retrofit the existing mill with a power feed (PF)
b) Buy a new mill with a computer-aided design (CAD) feature.
c) Replace the mill with a machining center (MC).
The three alternatives are evaluated based on two criteria: monetary and performance. The
following table provides the pertinent data:
Criterion PF CAD MC
Monetary
Initial Cost ($) 12,000 25,000 120,000
Maintenance cost ($) 2,000 4,000 1,500
Training cost ($ ) 3,000 8,000 2,000
Performance Production rate (units/d) 8 14 40 Setup time (min) 30 20 3 Scrap (lb/d) 440 165 44
The manager surmises that the monetary criterion is 1 ½ times as important as the person formance criterion. Additionally, the product rate is 2 times as important as the setup time, and 3 times as important as the scrap. The setup time is regarded as 4 times as important as the scrap. As for the monetary criteria, the manager estimates that the maintenance and training costs are of equal importance, and the initial cost is 2 times as important as either of these two costs.
Analyze the situation, and make an appropriate recommendation.
[from Taha]
Assignment 8
- Write short note on use of OR
- Give an example of a game against nature and show how it ca be solved
- Are mixed strategies always possible? Give examples where they are possible and also non possible.
A company operates a catalog sales operation encompassing more than 200,000 items stocked in many regional warehouse. In the past, the company considered it essential to keep accurate records of the actual inventory in each warehouse . As a result, full inventory count was ordered every year -an intense and unwelcome activity that is done grudgingly by all warehouses. The company followed each count by an audit that sampled about 100 items per warehouse to check the quality of logistical operation in each region. The result of the audit indicated that, on the average, only 64% of the item in each warehouse matched the actual inventory, which was unacceptable. To remedy the situation, the company ordered more frequent counts of the expensive and fast-moving items. A system analyst was assigned the task of setting up procedures for targeting these items. Instead of responding directly to the company's request for identifying the target items, the system analyst decided to identify the cause of the problem. The analyst ended up changing the goal of the study from "How can be increase the frequency of inventory counts?" to "How can be increase the accuracy of inventory counts?" The study led to the following analysis: Given that the proportion of accurately counted items in a warehouse is p, it is reasonable to assume that there is a 95% chance that an item that was counted correctly in the first place will again be recounted correctly in a subsequent recount. For the proportion 1- p that was not counted correctly in the first round, the chance of a correct recount is 80%. Using this information, the analyst developed a decision tree to graph a break - even chart that compared the count accuracy in the first and second rounds. The end result was that the warehouses that had an accuracy level above the break -even threshold were not required to recount inventory. The surprising result of the proposed solution was a zealous effort on the part of each warehouse to get the count right the first time around, with a resounding across - the - board improvement in count accuracy in all the warehouses.
How did the analyst convince management of the viability of the proposed threshold for recounting?
[From Taha]
Assignment 9.
- Write an article to a news paper on OR
- Write a letter to your CEO describing importance of using OR.
- Describe 2 methods how data required for a project that are not available can be estimated.
In the airline industry, working hours are ruled by agreements with the unions. In particular, the maximum length of tour of duty may be limited to 16 hours for Boeing-747 flight and 14 hours for Boeing-707. If these limits are exceeded because of unexpected delays the crew must be replaced by a fresh one. The airlines maintain reserve crews for such eventualities .The average annual cost of a reserve crew member is estimated at $30,000 .Conversely, an overnight delay resulting from the unavailability of a reserve crew could cost as much as $50,000 for each delay. A crew member is on call 12 consecutive hours a day for 4 days of the week and may not be called on during the remaining 3days of the week. The Boeing-747 crew may also be served by two Boeing-707 crews. The following table summarizes the callout probabilities for reserve crews based on 3-year historical data.
Callout probability
Trip category Trip hr B-747 B-707
1 14.0 .014 .072
2 13.0 .0 .019
3 12.5 .0 .006
4 12.0 .016 .006
5 11.5 .003 .003
6 11.0 .002 .003
As an illustration, the data indicate that for 14-hour trips, the probability of a callout is .014 for Boeing -747 and .072 for Boeing -707.
The following is a typical peak day schedule:
Time of day Aircraft Trip category
8:00 707 3
9:00 707 6 707 2
10:00 707 3
11:00 707 2 707 4
15:00 747 6
16:00 747 4
19:00 747 1
The present policy for reserve crews calls for using two (seven - member) crews between 5:00 and 11:00, four between 11:00 and 17:00, and two between 17:00 and 23:00.
Evaluate the effectiveness of the present reserve crew policy. Specifically, is the present reserve crew size too large, to small, or just right?
[From Taha]
Assignment 10
4. Write how the shortest path method can be used for many possible applications in SL
2. Write how the Traveling Salesman method can be used for many possible applications in SL
3. Write how the Postman method can be used for many possible applications in SL
OUSL has a few Regional Centers and many Education Centers, which can be found from the web. Your team is to find out the optimum route for a single vehicle to start from Colombo and take resource material to every Centre and return to Colombo. [You have to get distances between towns from some source.]
Find a good solution if 2 vehicles are to be used.
Assignment 11
1. Describe how shortest route method can be used in planning new high way between 2 towns.
2. Write a few different types of applications of the minimum spanning tree method.
3. In what ways can network methods be used to help a provincial council.
Your team is to advise the Chief Minister of a province. Select a province and set up a road network connecting Pradeshya Sabas of the province. Find the optimum route for the CM to start from any where and return to the same location and to visit all PS s.
If CM does not wish to visit one or more of PS s (in the opposition) construct shortest route.
As usual all main roads connecting PS s have potholes of equal linear density. Since the PC had spent most of the money to buy motor cars & to hold functions little money is left to repair all these roads. Find the roads with minimum total length to be repaired so that people can travel from any PS to any other PS in the same PC in good roads, at least till the next rainy season..[Use actual figures.]
Assignment 12
1. Discuss use of Operation Research in development
2. Describe 2 methods how data required for a project that are not available can be estimated
3. Describe the use of LP in industrial production management
Three types of cutting tools are produced in a factory using a lathe, a grinder and a polisher. The duration in hours required to produce one batch of tools on each of these machines are given in the table 1 along with costs, selling prices of each batch of tools and the minimum number of hours available on each machine per week.
Table 1
|
Tool
type
|
Processing time (hrs.) per batch
|
Cost
(Rs.) per batch
|
Selling prices (Rs.) per batch
|
|
Lathe
|
Grinder
|
Polisher
|
|
A
|
7
|
2
|
5
|
100
|
145
|
|
B
|
3
|
3
|
8
|
65
|
100
|
|
C
|
4
|
4
|
2
|
80
|
120
|
|
per week
|
50
|
46
|
80
|
|
|
(i) Determine the optimum production schedule and maximum profit per week.
(ii) The factory wants to double the capacity of grinder at an additional cost of Rs. 50 per week. Should it go for it ? Substantiate your result.
Assignment 13
1. Write a letter to your CEO explaining the importance of using OR in your organization.
2. Describe situations where CPM can be used in the Country
3. Describe the use of LP in industrial production management
(a) An air conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping cost vary and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200. Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300. The shipping cost per unit from Houston to Dallas is $8, to Atlanta is $12, and to Denver is $10. The cost per unit from Phoenix to Dallas is $10, to Atlanta is $14, and to Denver is $9. The cost per unit from Memphis to Dallas is $11, to Atlanta is $8, and to Denver is $12. How many units should be shipped from each plant to each regional distribution center? What is the total cost for this?
Assignment 14
1. Discuss use of Operation Research in development
2. Describe 2 methods how data required for a project that are not available can be estimated
3. Describe the use of LP in industrial production management
A company manufacturing perfume produces four types of perfume using three kinds of raw material Amounts of raw material used to produce 1 gm of each perfume and the resulting profit are follows:
Table 1
|
|
R1
|
R2
|
R3
|
Profit($/gm)
|
|
P1
|
3
|
1
|
4
|
19
|
|
P2
|
2
|
1
|
3
|
13
|
|
P3
|
1
|
1
|
3
|
12
|
|
P4
|
2
|
1
|
4
|
17
|
The company has 225 units of R1, 117 units of R2, and 420 units of R3 available. Find how much of each perfume should be produced to maximize profit.
Formulating the problem using the Simplex Algorithm we obtain the following :
When we solve this problem using the simplex Algorithm we obtain the final Optimal Table, dictionary;
Table 2 Optimal Simplex Tableau
|
|
x1
|
x2
|
x3
|
x4
|
s1
|
s2
|
s3
|
rhs
|
|
Z
|
0
|
1
|
0
|
0
|
2
|
1
|
0
|
1827
|
|
x1
|
1
|
1
|
0
|
0
|
1
|
2
|
-1
|
39
|
|
x3
|
0
|
1
|
1
|
0
|
0
|
4
|
-1
|
48
|
|
x4
|
0
|
-1
|
0
|
1
|
-1
|
-5
|
2
|
30
|
Solve the following independent variations of this problem.
(a) The profit from p2 increases from $ 13 to $ 15.
(b) The amount of R2 available increases from 117 units to 125 units.
(c) The company develops a new perfume P5 which requires 3 units of R1, one unit of R2, 2 units of R2, 2 units of R3 and gives $14 profit.
(d) The amount of P3 produced can be at most five times the amount of p2.
(e) The company can buy extra units of R1at a cost of $q / unit. For what values of q should it do this? Assuming q = 1, how many units of R 1 should it buy and what is the new maximum profit?
Assignment 15
1. Discuss use of Operation Research in development
2. Describe 2 methods how data required for a project that are not available can be estimated
3. Discuss how Linear transportation can be used to reduce cost of transport of a comodity
A trip from Colombo to Kataragama takes six hours by bus. A typical table of the bus service in both directions is given below.
Table 3 Time table
|
Departure from Colombo
|
Colombo to Kataragama line or route number
|
A rival at Kataragama
|
|
06.30
08.30
11.30
19.00
00.30
|
a
b
c
d
e
|
12.30
14.30
17.30
01.00
06.30
|
Table 4
|
arrival at Colombo
|
Kataragama to Colombo
line or route number
|
Departure from Kataragama
|
|
11.3
15.00
21.00
00.30
06.00
|
1
2
3
4
5
|
5.30
9.00
15.00
18.30
00.00
|
The cost of providing this service by the transport company depends upon the time spent by the bus crew (driver and conductor) away from their Places in addition to service times. There are five crew. There is a constraint that every crew should be provided with more than 4 hours of rest before the return trip again and should not wait for more than 24 hours for the return trip. The company has residential facilities for the crew at Colombo as well as at Kataragama. Suggest an optimal assignment of the crew.