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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.