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(IELM3020)[2011](f)midterm~ychenaa^_55889.pdf
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IELM 3020: Introduction to Operations Research
Sample Mid-Term Exam
You will have 110 minutes to answer all questions. For most questions, only modeling is required. Do not need to solve an LP unless you are explicitly asked to do so.
Question 1. (30%)
The feasible region of an LP is shown as the shadow area in the following figure. Suppose the objective of the LP is to maximize the sum of 4x and y.
1) Write the formulation of the LP. (10%)
2) Identify the corner point that corresponds to the optimal solution. (10%)
3) Write the standard form of the LP. (10%)
Question 2. (20%)
Considering the following linear programming problem LP1:
LP1: Max Z = -2X1+6X2
Subject to X1+X2 2
-X1+X2 1
X1+ 4X3 = 9
X10, X20, X3 is unrestricted
(1)
Write the dual problem of LP1. (10%)
(2)
Given the fact that LP1 is unbounded, use the duality theory to explain that the following linear programming problem LP2 has no feasible solution. (10%)
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LP2: Min w=20Y1+3Y2
Subject to 3Y1-3Y2-6
2Y1+2Y212
Y10, Y20.
Question 3. (20%)
Zales Jewelers uses rubies and sapphires to produce two types of rings. A type 1 ring requires 2 rubies, 3 sapphires, and 1 hour of labor. A type 2 ring uses 3 rubies, 2 sapphires, and 2 hours of labor. Each type 1 ring sells for $300, and types 2 ring sells for $500. At present, Zales has 100 rubies, 120 sapphires, and 70 hours of labor. Extra rubies can be purchased at a cost of $100 per ruby. Market demand requires that the company produce at least 20 type 1 rings and at most 15 type 2 rings. The goal is to maximize profit. To make an LP formulation, we have defined the following variables:
X1 = type 1 rings produced
X2 = type 2 rings produced
R = number of rubies purchased
(1)
Develop an LP to maximize the total profit. (8%)
(2)
Read the sensitivity report generated by Excel Solver, answer the following questions, and state your reasons. (12%)
Name X1X2R Final Value 30 15 5 Reduced Cost 0 0 0 Objective Coefficient 300 500 -100 Allowable Increase 200 1E+30 100 Allowable Decrease 100 133.3333333 50
Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease
(1) 100 100 100 5 1E+30
(2) 120 33.33333333 120 30 7.5
(3) 60 0 70 1E+30 10
(4) 30 0 20 10 1E+30
(5) 15 133.3333333 15 7.5 3
a) If the price of ruby becomes 120, what is the impact to the production plan and
the profit?
b) If Zales currently has 90 rubies, how will the maximum profit be changed?
c) If an agent can help Zales sell 3 extra type 2 rings, and ask for a reward of $500.
Will the company take such an offer?
d) If the maximum available ruby to purchase is 10, how will you re-formulate the LP? Predict the solution change with this additional constraint.
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Question 4. (30%)
In this question, you need to build a transportation model or a minimum cost network