Mth601 Collection of Old Papers

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MTH601 Operations ResearchFinal Term Examination - February 2005

Time Allowed: 150 Minutes

InstructionsPlease read the following instructions carefully before attempting any of the questions:

1. Attempt all questions. Marks are written adjacent to each question.2. Do not ask any questions about the contents of this examination from anyone.

a. If you think that there is something wrong with any of the questions, attempt it to the best of your understanding.b. If you believe that some essential piece of information is missing, make an appropriate assumption and use it to solve the problem.c. Write all steps, missing steps may lead to deduction of marks.

3. In Q3 you can either draw the network diagram or simply write the asked terms for each activity in the exam software, whichever you think is easy for you. But remember that you have to show your work on exam application correspond to each question.

Total Marks: 50 TotalQuestions: 8

Question No. 1 Marks : 4

Define the followings.

(a) Impossible Assignment(b) Unbounded Solution.


Consider the problem

max imize z = x1 + 5x2 + 3x3

Subject to

x1 + 2x2 + x3 = 3

2x1 − x2 = 4

The variable

x1 , x2 , x3 ≥ 0

x3 plays the role of a slack. Use big M-method to solve the problem.


How the problem of the degeneracy arises in a transportation problem?


Define the followings in Queuing Theory.

a. Customer b. Serverc. Priority


What are the variants of the simplex method? Write few lines about the followings.

(a) Unbounded solution. (b) Unrestricted variables.


Five different machines can process any of the five required jobs with different profits resulting from each assignment.

Find the maximum profit possible through optimum assignments.


How can the linear programming be applied to management problem?


Consider the transportation problem having the following parameter table.

Use the northwest corner rule to obtain an initial basic feasible solution.

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FINALTERM EXAMINATION

FALL 2006

MTH601 - OPERATIONS RESEARCH

Marks: 65

Time: 120min

StudentID/LoginID:

Student Name:

Center Name/Code:

Exam Date:

Please read the following instructions carefully before attempting any of the questions:


a. If you think that there is something wrong with any of the questions, attempt it to the best of your understanding.b. If you believe that some essential piece of information is missing, make an appropriate assumption and use it to solve the problem.

3. This examination is closed books, closed notes.4. Calculator is allowed.5. Symbols by using math type should be pasted on the paper direct from the

math type not from theword document otherwise it would not be

visible.6. In order to get full marks do all necessary steps.

**WARNING: Please note that Virtual University takes serious note of unfair means. Anyone found involved in cheating will get an `F` grade in this course.

For Teacher's use onlyQuestion 1 2 3 4 5 6 7 8 9 10 Total

MarksQuestion 11

Marks

Question No: 1 ( Marks: 10 )

Children Hospital Lahore has monthly demand of 6000 units of a tablet namedFlagyl. Each tablet cost Rs. 6 and the cost of one purchase is Rs. 950.Keeping that stock for a year in the hospitals cost Rs. 3 per unit. The shortage cost for a year is Rs. 10 per unit, Determine the following

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(i). The Optimum Order Quantity

1 2 3 4 Supply

1 10 2 20 11 15

2 12 7 9 20 25

3 4 14 16 18 10

Demand 5 15 15 15

(ii). The time between the orders (iii). The number of orders per year (iv). The Optimum Shortages(v). The time of orders being held


Solve the following problem with Big M – Method.Minimize. Z = 3 X1 + 2 X 2 + X 3

Subject to.

X1 + X 2 = 7

3 X1 + X 2 + X 3 ≥ 10

X1 , X 2 , X 3 ≥ 0


Write the DUAL of the following primal problem.Maximize. Z = − 5 X1 + 2 X 2

Subject to.

− X1 + X 2 ≤ − 2

2 X1 + 3 X 2 ≤ 5

X1 , X 2 ≥ 0


Solve the transportation model starting with the Vogal’s ApproximationMethod.


In a college, every 15 minutes one student arrives for admission information. The staff in the only information counter takes 10 minutes for serving a student on an average. State suitable assumption and find.

I. The average queue length.II. Increase in the arrival rate in order to justify for second counter (when the

waiting time of a student is atleast 15 minutes the management will increase one more counter).


A machine costs Rs. 8000. Annual operating cost are Rs. 1000 for the first year,and then increase by Rs. 500 every year. Resale price are Rs. 4000 for the first year and then decrease by Rs. 500 every year. Determine at which age it is profitable toreplace the machine.

Question No: 7 ( Marks: 1 ) - Please choose one

An unrestricted primal variable will result in an equality dual constraint.

► True

► False


In purchasing model with shortage, the formula of order size is

► Q* = 2C2 D

C3

►Q* =

2C2 D C3 +C4

C3 C4

► Q* = 2C2 D R

C3 R−D

► Q* =2C2 D C3 +C4

C3 (1−D R) C4


In Above Network Diagram the critical path is

► (a) A – B – D – G

► (b) A – C – E – H

► (c) A – C – F – G

► (d) Both (a) and (c)


In assignment problem each job requires

► Exactly one resource

► At least one resource

► At most one resource


The dual of the dual problem yields the original primal

► True

► False

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MTH601 Operations ResearchFinal Term Examination – Spring 2005



1. Attempt all questions. Marks are written adjacent to each question.2. Do not ask any questions about the contents of this examination from

anyone.a. If you think that there is something wrong with any of the questions, attempt it to the best of your understanding.b. If you believe that some essential piece of information is missing, make an appropriate assumption and use it to solve the problem.c. Write all steps, missing steps may lead to deduction of marks.

3. The duration of this examination is 150 minutes.4. This examination is closed book, closed notes and closed neighbors.5. Calculator is allowed.6. Symbols by using math type should be pasted on the paper direct

from the math type not from the word document otherwise it would not be visible.

7. In order to get full marks do all necessary steps.


Total Marks: 50 Total Questions: 17


o

o

oo None of the above


According to strategy, replace an item if it fails before the optimum period 'P'

o IRo IPRo CPR (Common Preventive Replacement)




In purchasing model with shortage, the formula of Ordering Size is

o

o

o

o


Similarity between CPM and PERT is

o Both are used to plan the scheduling of individual activities that make up a project

o Both can be used to determine the earliest/latest stat and finish time for each activity

o All of the above


The arrival rate of customers at a banking counter follows Poisson distribution with a mean of 45 customers per hour. The service rate of the counter clerk also follows Poisson distribution with a mean of 60 customers per hour.Find the following

1. The probability of having 10 customers in the system2. Average number of customers waiting in the system3. Average number of customers waiting in the queue4. Average waiting time of customers in the system5. Average waiting time of customers in the queue


The formula for expected number of customers in a system is

o L = λ /(µ - λ)o L = λ /(λ - µ) o L = µ /(λ - µ) o L = µ /(µ - λ)


For LP problem with 'n' decision variables, each of its corner point solution is at the intersection of constraint boundaries.

o (n – 1) constraint boundarieso 'n' constraint boundarieso (n + 1) constraint boundarieso None of the above


In two dimensional LP problem we need only two lines to identify an extreme point. If in any two dimensional problem there are three or more intersecting lines at the corner point then this indicate that

o Problem has Degenerate Solutiono One of the constraints is redundanto Both First and Secondo None of the above



A transportation problem is degenerate, if while deriving a feasible solution an allocation to any cell

o Satisfies the Column requirementso Satisfies the Row requirementso Satisfies column as well as row requirements simultaneouslyo None of the above


o Problem has Multiple Optimal Solutionso Problem has Unbounded Solutionso Problem has Infeasible Solutionso None of the above




o Dividing the Big-M method's objective function by 'M' and then dropping the negligible terms

o Dividing the Big-M method's objective function by '–M' and then dropping the negligible terms

o Multiply the Big-M method's objective function by 'M' and then dropping the negligible terms

o Multiply the Big-M method's objective function by '–M' and then dropping the negligible terms


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MTH601 Operations ResearchFinal Term Examination - February 2005





3. In Q3 you can either draw the network diagram or simply write the asked terms for each activity in the exam software, whichever you think is easy for you. Butremember that you have to show your work on exam application correspond toeach question.



Define the followings.

(a) Impossible Assignment(b) Unbounded Solution.


Consider the problem

max imize z = x + 5x + 3xSubject to

x1 + 2x + x3 = 3

2x1x−2 = 4

x , x , x ≥ 0

The variable x3 plays the role of a slack. Use big M-method to solve the problem.


How the problem of the degeneracy arises in a transportation problem?


Define the followings in Queuing Theory.

a. Customerb. Serverc. Priority


What are the variants of the simplex method? Write few lines about the followings.

(a) Unbounded solution.(b) Unrestricted variables.


Five different machines can process any of the five required jobs with different profits resulting from each assignment.

Find the maximum profit possible through optimum assignments.


How can the linear programming be applied to management problem?


Consider the transportation problem having the following parameter table.

Use the northwest corner rule to obtain an initial basic feasible solution.

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MTH601 Operations ResearchMid Term Examination - November 2004





3. In Q3 you can either draw the network diagram or simply write the asked terms for each activity in the exam software, whichever you think is easy for you. But remember that you have to show your work on exam application correspond to each question.




A small plant makes two types of automobile parts Part A and Part B. It buys castings that are machined, bored and polished. You are given the following data.

≤ 5 0

≤ 2 5

≤ 2 6

≥ 0

Castings for Part A cost Rs. 20/- each and for Part B they cost Rs. 30/- each. They sell for Rs. 50/- and Rs.60/- respectively. The three machines have running costs of Rs. 200/-, Rs.140/- and Rs. 175/- per hour. Assuming that any combination of Part A and B can be sold. Formulate a linear programming model.

Or

Solve the following problem using simplex method:

Maximize Z = 9x1 + 8x2 + 7x3Subject to

2 x + x + 2 x1 2 3

x + 3 x + x1 2 3

x + 2 x + x1 2 3

x , x , x1 2 3


Consider the following Network, figures in the boxes are the durations. Determine(a) The earliest expected completion time for each period,(b) The latest allowable event completion time for each event(c) The slack time (all floats) for each event(d) The Critical Path through the Network


The demand for a purchased item is 2000 units per month and the shortages are allowed. If the unit cost is Rs 20 per unit, the cost of making one purchase is Rs 1000, the holding cost for 1 unit is Rs 20 per year and the cost of one shortage is Rs 500 per year, determine:

(a) The economic purchase quantity. (b) The time between orders.(c) The number of orders per year. (d) The optimum shortages.(e) The maximum inventory.(f) The time of items being held. (g) The optimum annual cost.

OrA company produces 36000 units per year and the demand of the product is 2400 items per year. The unit cost is Rs 10 per item, the holding cost is one rup e e p e r month p e r item and the set upcost is Rs 100, determine the following if no shortages are allowed

(a) Optimum manufacture quantity(b) The maximum inventory (c) The time between setups (d) The number of setups(e) The manufacturing time(f) The total optimum annual cost

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MIDTERM EXAMINATION

FALL 2006

MTH601 - OPERATIONS RESEARCH

Marks: 40

Time: 60min

StudentID/LoginID:

Student Name:

Center Name/Code:

Exam Date:


1. Attempt all questions. Marks are written adjacent to each question

2. Paste the bitmap image for the tables, diagrams etc while solving your questions

3. Do not ask any questions about the contents of this examination from anyone

a. If you think that there is something wrong with any of the questions, attempt it to the best of your understanding.

b. If you believe that some essential piece of information is missing, make an appropriate assumption and use it to solve the problem.

c. Write all steps, missing steps may lead to deduction of marks.

4. This examination is closed book, closed notes, closed neighbors.

5. Calculator is allowed

**WARNING: Please note that Virtual University takes serious note of unfair means.Anyone found involved in cheating will get an `F` grade in this course.

For Teacher's use onlyQuestion 1 2 3 4 5 6 7 8 Total

Marks


2 F = 2 7 K = 4 8

D = 4A = 6 L = 10

E = 9 5

1 B = 2 3 10I = 7

C = 13G = 10 6 M = 5

J = 3

4 H = 6 9

Calculate the(i). EST, EFT, LST and LFT for each activity.(ii). Total Float, Free Float and Independent Float.(iii). Critical activities and Critical Path.


An Automobile factory manufacturers a particular type of gear within the factory. This gear is used in the final assembly. The particulars of this gear are:

Demand rate =14000 units/year Manufacturing rate =35000 units/year Set up cost =Rs 500 per set up Holding cost = Rs 15/units/year

Determine:(a) The optimum manufacturing quantity. (b) The maximum inventory.(c) The time between orders. (d) The number of orders/year. (e) The time of manufacture.(f) The optimum annual cost if the cost of the item per unit is Rs. 25


Consider the following problem, Solve graphically

Minimize. Z = 3X1 + 2X2Subject to

2X1 + X2 ≥ 10-3X1 + 2X2 ≤ 6

X1 + X2 ≥ 6X1 , X2 ≥ 0


Which one is best describe Sectoral planning

► Inventory Planning in agriculture

► Improving the layout of a workshop in a company

► Simulation Modeling of the Economy of the country

► None of these.


is the most appropriate to situations where we maintain a relative stable employment levels and utilize the resource at a more constant rate

► (a). Resource Leveling Program

► (b). Resource Allocation Program

► (c). Both a & b

► (d). None of these.


If the slack time is zero, it means that the project will be

► Delayed

► Completed on schedule


The amount of an activity can be delayed without affecting the early start time of any other job, is called

► Free Slack

► Independent Slack

► Total Slack

► None of these.


trueLet FS = Free Slack, TS = Total Slack, INDS = Independent Slack, then which relation is

► (a). TS ≤ FS

► (b). INDS ≤ FS

► (c). FS ≤ TS

► (d). Both (b) & (c)

► (e). Both (a) & (c)

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MIDTERM EXAMINATION

SEMESTER SPRING 2005

MTH601 – Operational Research

Total Marks: 50

Duration:90mins

Maximum Time Allowed: (11/2 Hour)


1. Attempt all questions. Marks are written adjacent to each question

2. Paste the bitmap image for the tables, diagrams etc while solving your questions

3. Do not ask any questions about the contents of this examination from anyone

a. If you think that there is something wrong with any of the questions, attempt it to the best of your understanding.

b. If you believe that some essential piece of information is missing, make an appropriate assumption and use it to solve the problem.

c. Write all steps, missing steps may lead to deduction of marks.

4. The duration of this examination is 90 minutes.

5. This examination is closed book, closed notes, closed neighbors.

6. Calculator is allowed


For Teacher’s use only

Descriptive Multiple Choice Questions TotalQuestion Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 4+10=14

Activity EST EFT LST LFTA(1 – 2) 0 5 0 5B(2 – 3) 5 12 5 12C(2 – 4) 5 14 8 17D(3 – 5) 12 23 12 23E(4 – 6) 14 17 18 21F(4 – 5) 14 20 17 23G(5 – 7) 23 29 23 29H(6 – 7) 17 25 21 29

Marks + =

Question No: 1 Marks: 10

For the following project diagram find out Earliest start time (EST), Earliest finish time (EFT), Latest start time(LST), Latest finish time (LFT) for each activity. Also find the Critical Path.

NOTE: There is no need of making diagram corresponds to each step, you have to write down only the required terms correspond to each activity and the justification for the critical path.

Solution:

To find the Critical Path we’ll find the Slack (Float) of each activity.

Activity SlackA(1 – 2) 0B(2 – 3) 0C(2 – 4) 3D(3 – 5) 0E(4 – 6) 4F(4 – 5) 3G(5 – 6) 0H(6 – 7) 4

CriticalActivities

OptimisticTime (t0)

Most Likely

Time (t )

PessimisticTime (tp)

Expected Duration

(t )Variance

A 1 2 3 2 0.11

D 10 13 28 15 9

G 8 12 16 12 1.78

H 2 2 2 2 0

Activity Optimistic Time Most Likely Time Pessimistic TimeA 1 2 3B 3 6 8C 5 8 20D 10 13 28E 4 7 12F 2 4 5G 8 12 16H 2 2 2

Critical Path: A – B – D – GQuestion No: 2 Marks: 10

Suppose the following estimates of activity times of a project,

If the critical path of the project is A, D, G, H, then calculate the Expected Completion Time (Length of the path) and corresponding Variance of the project.

Solution:

m e

Expected Completion time = 2 + 15 + 12 + 2= 31

Variance (of the project) = 0.11 + 9 + 1.78 + 0= 10.89


Demand for the product is 600 units per year, and the items are withdrawn at a constant rate. The setup cost for placing an order to replenish inventory is $25. The unit cost of each item is $3, and the inventory holding cost is$0.05 per item per week.

(a). Assuming shortages are not allowed, determine how often to make a production run and what size it should be.

(b). If shortages are allowed but cost $2 per item per month, determine how often to order and what size the order should be.

Solution:

D=600 unit/yearOrdering Cost (C2) = $25/orderItem Cost (C1) = $3/itemHolding Cost (C3) = 0.05/item/week

= 0.05*52/item/year=2.6/item/year

(a). Shortages are not allowed: Q* =?t* = ?

Q* = 2C2 D / C3

Putting all the values in the above formula we have

Q* = 2 * 25 * 600 / 2.6Q* = 107.42

t* = Q*/D= 107.41/600= 0.1796

(b). Shortages are allowed:Shortage Cost (C4) = $2 /item/month

= 24 item/year

Q* = 2C2 D / C3

*(C3 + C4 ) / C4

= 107.41 * (2.6 + 24) / 24

= 223.606 t* = Q*/D

= 0.3726


Rabbani and Sons of Sialkot manufacture sports goods like hockey sticks, cricket bats etc .Each hockey stick yields a profit of Rs.20 and each cricket bat Rs 25.A hockey stick requires four hours of processing at machine A and two hours at machine B. A bat requires six hours of processing at machine A and five hours at machine Band one hour at machine C. Machine A has a maximum of 135 hours of availability capacity, machine B has 75 hours and machine C has 20 hours.

Read the above problem carefully and formulate it as an LP model to maximize the company’s profit.

Solution:

Decision variablesx1 = Number of hockey stick x2 = number of cricket bats

Objective functionSince each hockey stick yields a profit of Rs.20 so x1 number of hockey stick yields a profit of Rs.20x1

Since each cricket bat yields a profit of Rs 25 x2 number of cricket bats yields a profit of Rs.25x2

As we want to maximize the company profit so objective function isMaximize Z = 20x1 + 25x2

Constraints

Product Machine A Machine B Machine CHockey 4 2 0Bat 6 5 1

Hourly Availability of machine A = 135Hourly Availability of machine B = 75Hourly Availability of machine C = 20

4x1 + 6x2 ≤ 1352x1 + 5x2 ≤ 750x1 + x2 ≤ 20x1 ,x2 ≥0

Now LP problem isMaximize Z = 20x1 + 25x2

Subject to4x1 + 6x2 ≤ 1352x1 + 5x2 ≤ 750x1 + x2 ≤ 20x1, x2 ≥ 0

Question No: 5 Marks: 10Select the best choice (Only one) from the given four choices against each question.

1. employs a different modeling and solution logic than linear programming(a). Transportation Model(b). Inventory Control Model (c). Dynamic Programming (d). None of the above

2. To identify and maintain the proper precedence relationship between activities those are not connected by event, we introduce

(a). Parallel Activity (b). Dummy Activity (c). Sequential Activity (d). None of the above

3. EST and EFT of activities are calculated in(a). Forward pass

(b). Backward pass

4. Critical path is obtained by connecting the jobs having(a). Activities having same EST and LST (b). Activities having same EFT and LFT (c). Activities having zero slack(d). All of the above

5. The Variance Vt of expected time is calculated as

(a). Vt

(b). Vt

(c). Vt

= ( tm − t0

)2

6t − t

= ( 0 p

)2

6t − t

= ( p m

)2

6(d). None of the above

6. In LP problems Additivity means that(a). The effect of two different programs of production is the same as that of a joint program(b). The doubling (or tripling) the product will exactly double (or triple) the profit and the required

resource(c). Both (a)& (b)(d). None of the above

7. Two of the first steps of OR process encompass the actual use of OR techniques. These steps are(a). Model Construction and Model Solution(b). Observation and Implementation(c). Definition of the problem and Model Solution(d). Model Solution and Implementation of results

8. Let FS = Free Slack, TS = Total Slack, INDS = Independent Slack, then which relation is true(a). TS ≤ FS (b). INDS ≤ FS (c). FS ≤ TS(d). Both (b) & (c)

9. Best possible time estimate that a given activity would take under normal conditions which often exist, is called(a). Most Likely time estimate (b). Pessimistic time estimate (c). Smallest time estimate(d). None of the above

10. Standard Deviation S.D is(a). One sixth of the difference between pessimistic time estimates and optimistic time estimates (b). One sixth of the difference between pessimistic time estimates and most likely time estimates (c). One sixth of the difference between optimistic time estimates and most likely time estimates

(d). One sixth of the difference between most likely time estimates and optimistic time estimates

Mth601 Collection of Old Papers

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