ZAGAZIG UNIVERSITY

FACULTY OF COMPUTERS & INFORMATICS

DS200 OPERATIONS RESEARCH

FIRST SEMESTER

SECOND YEAR

Final Exam: DEC. 2014

Time Allowed: 180 Minutes

**** Answer all the questions****

** INSTRUCTIONS

Calculators are permitted.

Do not use highlighters or correction fluid.

You are reminded of the need for clear presentation in your answers.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this exam is 90.

Show your working in detailed steps.

If you get stuck on a question, move on and come back later if you have time.

If you find a problem confusing or ambiguous, state your assumptions and solve.

A list of useful formulae is given as an appendix. GOOD LUCK!

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Q1: i. Max Z= 5x1 + 6x2

Subject to: 17x1 + 8x2 ≤ 136

3x1 + 4x2 ≤ 36

x1, x2 ≥ 0 and integer.

Find the optimal solution. [7]

ii. Max Z = 5x1 + 3x2

Subject to: 6x1 + 2x2 ≤ 18

15x1 + 20x2 ≤ 60

x1 , x2 ≥ 0

Determine the sensitivity range for the first objective function coefficient. [4]

iii. TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. [10]

1. A feasible solution violates at least one of the constraints.

2. A minimization model of a linear program contains only surplus variables.

3. Sensitivity analysis is the analysis of the effect of parameter changes on the optimal

solution.

4. Types of forecasting methods are time series, regression, and qualitative.

5. A seasonal pattern is an up-and-down repetitive movement within a trend occurring

periodically.

6. In a 0 - 1 integer model, the solution values of the decision variables are 0 or 1.

7. The sensitivity range for a right-hand-side value is the range of values over which the

quantity values can change without changing the solution variable mix, including slack

variables.

8. In a mixed integer model, some solution values for decision variables are integer and

others can be non-integer.

9. Constraints for nonlinear programs are usually nonlinear.

10. The sensitivity range for an objective coefficient is the range of values over which the

current optimal solution point will remain optimal.

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Q2: i. Solve the following nonlinear program:

Min w= x12 + 2x2

2 – 8x1 – 12x2 + 34

Subject to: x1

2 + 2x2

2 = 4 [6]

ii. Consider the following LPP

max Z= 2x1 + x2

s.t.

x1 + 2x2 ≤ 14

2x1 − x2 ≤ 10

x1 − x2 ≤ 3

x1 , x2 ≥ 0.

Given that ) is an optimal solution to this LPP, Use the complementary slackness

theorem, to find optimal Solution to the dual problem. [7]

iii. Consider the following transportation tableau with the initial solution provided. Use the

stepping stone method to identify a better allocation. [7]

Q3: i. the following sales data are available for 2007-2012.

a. Determine a 3-year weighted moving average forecast for 2013, where weights are W1 = 0.3,

W2 =0.2 and W3 = 0.5 . [3]

b. Assume that the forecasted demand for 2011 is 15. Use the above data set and exponential

smoothing with α = 0.5 to forecast for 2013. [3]

c Forecast for 2013 using linear trend line. [3]

d. Determine the forecasted demand for 2013 based on adjusted exponential smoothing with α =

0.3, β = 0.4.(hint: an initial trend adjustment 0 for 2011) [3]

ii. There are 4 jobs in a 4 person office that must be done as quickly as possible. The supervisor

has used the assignment method to assign people to jobs. She had to rush off to a meeting and has

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asked to assign the jobs to the people before she returns. The original matrix which shows the

expected completion time for each person for each job, shown below.

Job Ruth Steve Valerie Wayne

A 8 26 17 11

B 13 28 4 26

C 38 19 18 15

D 19 26 24 10

Which person should be assigned to which job? How long will the jobs take? [6]

iii. Given the following gasoline data

a. Compute the seasonal index for each quarter. [3]

b. Suppose we expect year 3 to have annual demand of 800, what is the forecast value for

each quarter in year 3? [4]

Q4: The Stratford House Furniture Company makes two kinds of tables — end tables (x1) and coffee

tables (x2). The manufacturer is restricted by material and labor constraints, as shown in the

following linear programming formulation.

The final optimal simplex tableau for this problem is as follows.

a. Formulate the dual for this problem. [3]

b. Define the dual variables and indicate their values. [2]

c. What profit for coffee tables will result in no end tables being produced, and what will

the new optimal solution values be? [3]

d. What will be the effect on the optimal solution if the available wood is increased from

135 to165 board feet? [4]

e. Determine the optimal ranges for c1 . [3]

f. Determine the feasible ranges for q1 (labor hours) . [4]

g. What is the maximum price the Stratford House Furniture Company would be willing

to pay for additional wood, and how many board feet of wood could be purchased at that

price? [3]

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h. If the furniture company wanted to secure additional units of only one of the resources,

labor or wood, which should it be? [2]

Appendix

1. Exponential Smoothing:

Ft = Ft – 1 + (At – 1 - Ft – 1)

2. Exponential Smoothing with Trend Adjustment: FIT = Ft + Tt where,

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

Tt = β(Ft - Ft - 1) + (1 - β)Tt - 1

3. Linear Regression Equation

Ft = 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. Naser H. R. Dr. Mohamed A. M.