320/520
Thursday ORIE October 5 2006
PRELIM 1
7:309:00pm
Fall 2006
90 minutes
Name: NetID:
Recitation section (underline one): Monday 10:10 (Tim) Tuesday 12:20 (Arijit/Tim) Friday 10:10 (Tim) Monday 4:30 (Arijit) Wednesday 12:20 (Chandra) Friday 12:20 (Chandra)
Instructions
1. Print your name and NetID above. 2. Underline your Recitation section above. 3. You may use a basic scientic calculator. No other aids permitted. 4. Answer all questions, explaining your answers clearly. 5. Write in the space provided. If you run out of space, use the back of the preceding page. 6. This test consists of 4 questions and 7 pages (including this page).
Problem Out of (points) Grade Graded by 1 8 2 6 3 8 4 8 Total 30
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1. (a) [2 marks] Dene the term extreme point.
Now consider a system of constraints Ax = b x 0. Suppose that the vector [2, 3, 0, 4, 0]T is a basic feasible solution, and that the vector [0, 1, 2, 2, 4]T is a feasible solution that may or may not be basic. Answer the following questions, justifying your answers carefully. (b) [2 marks] Exactly one of the following statements is correct. State which one, and explain why it is correct. (i) A must have no more than 3 rows. (ii) A must have at least 3 rows.
(c) [2 marks] Find an extreme point of the feasible region.
(d) [2 marks] region.
Find a feasible solution that is not an extreme point of the feasible
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2. [6 marks] Use the simplex method to solve the linear program maximize subject to x1 x2 x3 2x1 + x2 x3 3x1 + 2x2 x1 + x3 x1 , x2 , x 3 2 4 0 0.
At each iteration, list the basic variables, the corresponding basic feasible solution, its objective value, and the entering and leaving variables.
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3. Consider linear the program maximize subject to 3x1 x2 2x1 x2 + x3 = 1 5x1 + 2x2 x3 4 x1 , x2 , x3 0.
(a) [4 marks] Use Phase 1 of the simplex method to nd an initial feasible tableau.
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Question 3 continued: (b) [3 marks] Suppose Phase 1 shows that the following system of equations is equivalent to the original system of linear constraints: 2x1 + x2 x3 =1 x1 + x3 + s = 2 (where s is the slack variable for the inequality constraint). Starting from the corresponding basis, solve the original linear program.
(c) [1 mark] Use your answer in part (b) to nd a feasible solution with objective value 1000.
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4. A juice stall gets supplies of banana puree and orange juice each morning. The stall sells bottles of three dierent banana/orange blends: Hi-C, Imm-Boost, and Sun-Blend. The following AMPL data le describes the supplies each day (measured in cups), the amount of each ingredient in a bottle of each blend (in cups), and the prot (in dollars) from making a bottle of each blend. set INGREDIENT set BLEND := BANANA ORANGE ; := HIC IMMBOOST SUNBLEND ;
param supply := BANANA 50 ORANGE 220 ; param amount : BANANA 1 1 1 := 6 7 4
HIC IMMBOOST SUNBLEND param profit HIC IMMBOOST SUNBLEND
ORANGE 4 5 3 ;
:=
;
We must decide how many bottles of each blend to make each day in order to maximize total prot. (a) [6 marks] Write an AMPL model le that will solve this problem using the given data le.
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Question 4 continued:
(b) [2 marks] Suppose each day the number of bottles made of each blend should be at least 10% of the total number of bottles made. How could you introduce extra linear constraints to your model to ensure this? (You may introduce extra variables too, if necessary.)
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