ZAGAZIG UNIVERSITY

FACULTY OF COMPUTERS & INFORMATICS

DS200 OPERATIONS RESEARCH

FIRST SEMESTER

SECOND YEAR

Midterm EXAM: DEC. 2014

Time allowed: 60 MINUTES

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

* Verify that your copy of the exam has all 3 pages

* All questions carry equal marks.

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

* Calculators are permitted.

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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. Define: Quadratic Programming ,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. What is sensitivity analysis in LP? Which type of changes in sensitivity analysis affect the:

i. feasibility ii. Optimality

b. Solve the following nonlinear program:

Min w= x12 + 2x2

2 – 8x1 – 12x2 + 34

Subject to: x1

2 + 2x2

2 = 5

Q3: i. 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.

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(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.

ii. 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

Q4: 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?

Q5:

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 = .02 and W4 = 0.5.

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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 linear 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)

Appendix *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. Naser H. R. Dr. Mohamed A. M.