ZAGAZIG UNIVERSITY

FACULTY OF COMPUTERS & INFORMATICS

DS200 OPERATIONS RESEARCH

FIRST SEMESTER

SECOND YEAR

FINAL EXAM: JAN. 2013

Time allowed: THREE hours

ANSWER ANY FIVE OF THE FOLLOWING SEVEN QUESTIONS. ** INSTRUCTIONS

* Print your name and student number on this page

* Verify that your copy of the exam has all 4 pages Name:………………………………..

* All questions carry equal marks. Student ID: …………………………..

* A list of useful formulae is given as an appendix.

* Calculators are permitted.

1

Q1: a. Consider the following LPP

max 2x1 + x2

s.t.

x1 + 2x2 ≤ 14

2x1 − x2 ≤ 10

x1 − x2 ≤ 3

x1 , x2 ≥ 0.

i. Write the dual problem.

ii. Given that ) is an optimal solution

to this LPP, use the complementary

slackness theorem, to find optimal Solution

to the dual problem.

b. Consider an M/M/3/4 queue with arrival

rate λ = 2/second and service rate μ =

3/second at each server.

i. Draw the transition diagram for this birth-

and-death process.

ii. Find the steady-state distribution of the

system.

iii. When the system is running in steady

state, find:

a. LS ,the expected number in the

system;

b. the average rate at which new arrivals

enter the system;

c. Ws, the expected time spent in the

system;

d. Wq, the expected time spent in the

queue.

c. Define Infeasibility, Unboundedness,

Alternate optimal, Degenerate basic feasible,

Non-degenerate basic feasible, Basic

infeasible, Non-basic feasible Solutions with

respect to an LP solution.

Q2: a. There are four items (A, B, C, and D)

that are to be shipped by air. The weights of

these are 3, 4, 5, and 3 tons, respectively. The

profits (in thousands of dollars) generated by

these are 5 for A, 6 for B, 7 for C, and 6 for

D. There are 2 units of A, 1 unit of B, 2 units

of C, and 3 units of D available to be shipped.

The maximum weight is 16 tons.

Use dynamic programming to determine the

maximum possible profits that may be

generated.

b. Prove that intersection of two convex sets

is also convex set.

c. Write the procedure to solve:

i. an LPP using Dual simplex method.

ii. Maximization Transportation Problem.

Q3: a. Solve the following LPP using

revised simplex method:

Max Z = 6x1-2x2+3x3

s.t

2x1-x2+2x3 ≤ 2

x1+4x3 ≤ 4

x1,x2,x3 ≥ 0

b. What is sensitivity analysis in LP? Which

type of changes in sensitivity analysis affect

the: i. feasibility ii. Optimality

c. Solve the following nonlinear program:

Min w= x12 + 2x2

2 – 8x1 – 12x2 + 34

Subject to: x1

2 + 2x2

2 = 5

d. Define: quadratic programming, pure

integer programming, Multichannel queuing

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system, Binding Constraint, Combinatorial

Optimization, Evolutionary Algorithms and

Heuristics Techniques.

Q4: a. Solve the following problem using

Gomory’s cutting plane algorithm.

min x1 − 2x2

s.t

−4x1 + 6x2 ≤ 9

x1 + x2 ≤ 4

xi≥ 0 , xi ∈ Z2

b. what are the advantages of the revised

simplex method over the original simplex

method?

c. Given the following data and seasonal

index:

(a) Compute the seasonal index using only

year 1 data.

(b) Determine the deseasonalized demand

values using year 2 data and year 1's

seasonal indices.

(c) Determine the trend line on year 2's

deseasonalized data.

(d) Forecast the sales for the first 3 months

of year 3, adjusting for seasonality.

Q5: a. Hess Oil produces oil at two wells.

Well 1 can produce as many as 150,000

barrels per day, and well 2 can produce as

many as 200,000 barrels per day. It is

possible to ship oil directly from the wells to

Hess’s customers in Los Angeles and New

York. Alternatively, Hess could transport oil

to the ports of Mobile and Galveston and

then ship it by tanker to New York or Los

Angeles. Los Angeles requires 140,000

barrels per day, and New York requires

160,000 barrels per day. The costs of

shipping 1,500 barrels between two points

are shown in the following Table:

Formulate a transshipment model (and

equivalent transportation model) that could

be used to minimize the transport costs in

meeting the oil demands of Los Angeles and

New York.

*Note: Dashes indicate shipments that are

not allowed.

b. Consider the following linear

programming problem:

Max 5x1 + 6x2 + 4x3

s.t.

3x1 + 4x2+ 2x3 ≤ 120

x1 +2x2 + x3≤ 50

x1 + 2x2 + 3x3 ≥ 30

x1, x2, x3 ≥0

The optimal simplex tableau is:

i. Compute the range of optimality for c1

ii. Find the dual price for the second

constraint.

iii. Suppose the right-hand side of the first

constraint is increased from 120 to 125.Find

the new optimal solution and its value.

iv. If c1 changed from $5 to $7 , how will the

optimal solution be affected?

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c. Consider the following nonlinear

programming problem.

Maximize Z = 2x12 2x2 4x3 x3

2,

subject to

2x1 + x2 + x3 ≤ 4

and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

Use the KKT conditions to derive an optimal

solution

Q6: a. the following sales data are available

for 2007-2012.

i. Determine a 4-year weighted moving

average forecast for 2013, where weights are

W1 = 0.1, W2 =0.2, W3 = 0.2 and W4 = 0.5.

ii. Assume that the forecasted demand for

2011 is 15. Use the above data set and

exponential smoothing with α = 0.3 to

forecast for 2013.

iii. Forecast for 2013 using Least squares

trend line.

iv. Determine the forecasted demand for

2013 based on adjusted exponential

smoothing with α = 0.2, β = 0.3.(hint: an

initial trend adjustment of 0 for 2011)

b. Max Z = x1 + 6x2

Subject to:

17x1 + 8x2 ≤ 136

3x1 + 4x2 ≤ 36

x1, x2 ≥ 0 and integer

Find the optimal solution.

c. Solve the Travelling Salesman Problem

given by the following data :

c12=4,c13=7,c14=3,c23=6,c24=3 and

c34=7,where cij=cji, and there is no route

between i and j if the value for cij is not

given above.

Q7: a. i what are the four steps in dynamic

programming?

ii. Identify two types of problems that can be

solved by dynamic programming.

iii. List four components of time-series data

iv. List two measures of historical

forecasting errors.

v. In general terms, describe what qualitative

forecasting models are.

b. A telephone help-desk is staffed by two

operators. If a call comes in when both

operators are already busy, the caller is

placed on hold (i.e., told to wait) if there is

no more than one other caller placed on hold.

If a call comes in when there are two callers

already on hold, the new caller is not

connected (i.e., the new caller does not enter

the system). Calls come in as a Poisson

process with an arrival rate of 0.5 calls per

minute. The time to process a call has an

exponential distribution with a rate

parameter of 0.75 calls per minute.

i. Describe this system using Kendall’s

A/S/m/n notation.

ii. Draw a state transition diagram for this

system.

iii. If there are two callers being processed

and no callers on hold, what is the

probability that there are no new calls in the

next 3 minutes?

iv. Find the steady-state distribution of this

system.

v. When the system is in steady-state:

a. At what average rate do callers enter

the system?

b. What is the average time that a

connected caller spends in the

system?

c.

i. Explain briefly what the traveling

salesman problem is.

ii. Formulate the traveling salesman problem

as an ILP.

iii. Why is the traveling salesman problem

considered “difficult”, although it can be

solved e.g.as a ILP?

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d. Use dynamic programming to solve the

following LPP:

Maz Z = 3x1+7x2

s.t.

2x1+4x2≤ 8

x2≤ 3

x1,x2 ≥ 0

FORMULAE

*Exponential Smoothing

Ft = Ft – 1 + a(At – 1 - Ft – 1)

*Exponential Smoothing with Trend

Adjustment: FIT = Ft + Tt

Ft = a(At - 1) + (1 - a)(Ft - 1 + Tt - 1)

Tt = b(Ft - Ft - 1) + (1 - b)Tt - 1

*Linear Regression Equation

Ft = y = a + bx where,

n

Xb

n

Ya

n

XX

n

YXYXb

iii

i

ii

ii

,

)(/

2

2

HINT: you may need to find b before you can find a

Good Luck

Prof. Ibrahim El-henawy