The University of QueenslandENGG7302 Advanced Computational Techniques in EngineeringLecturer: Hanna...

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The University of QueenslandENGG7302 Advanced Computational Techniques in EngineeringLecturer: Hanna KurniawatiAssignment 3 – OptimizationDue: Monday, 12 June 2017, 23:59Submission: A .pdf file of your report. Please name your file as A3-studentID.pdf andsubmit through turnitin.Reminder: NO cheating!!!1. [20 points]You are doing grocery shopping for grapes, broccoli, and fish. Suppose youcan buy fractional portions of any item. You have $20 available. The price of a basicunit of each item is $2, $3, and $5 respectively. The satisfaction gained from a unitof each item is 8, 2, and 5 respectively. You cannot have more than 3 units of grapes.Due to the weight you need to carry, the total units of all items combined cannot exceed6 units. Please use the tableau form of simplex method to compute how many unitsshould you buy grapes, broccoli, and fish, so that you can maximize satisfaction withinthe constraints.2. [20 points] You have 10 $2 coins and 9 $1 coins, and no other money. You have to paya given amount C, where no change is given. Of course, you want to minimize overpay.State this problem in mathematical terms. Is this problem linear?3. [30 points] A paper recycling machine can produce toilet paper, writing pads, and papertowels, which sell for 18, 29 and 25 cents and consume 0.5, 0.22 and 0.75 kilograms ofnewspaper and 0.2, 0.4, and 0.22 minutes. Each day 10 hours and 1500 kilograms ofnewspaper are available, and at least 1000 rolls of toilet paper, 200 writing pads and 400rolls of paper towels are required.(a) [10 points] Formulate an appropriate LP to maximize revenue using tableau form ofsimplex method.(b) [20 points] Suppose a company asked to buy your newspapers, instead. What shouldbe the minimum price this? Hint: You need to specify the dual problem.4. [30 points] Suppose you have the following QP problem:min x2 + y2 ?? xy ?? 24x ?? 20ysubject to x + 2y  0x + 2y  9x + y  8x + y  0(a) [15 points] Is this a convex optimization problem?(b) [15 points] Please solve the QP problem using Langrangian method.1

Robert · Accepted Answer

SOLUTION 
Sol 1: -
Let the number of units of grapes bought x
Let the number of units of broccoli bought y
Let the number of units of fish bought z



 
 
In this problem we need to maximize our satisfaction. For this we need to maximize the following 
relation  
8 2 5z x y z  
 
 
Now, from the question we have the following conditions 
Constraint I 
  $20Total amount spent 
  

2 3 5 20x y z  
          (1) 
 
Constraint II 
      6Total unit consisting of all items 
 

6x y z  
          (2) 
 
Constraint III 
   3Total number of grapes 
 

3x 
           (3) 
 
Thus, from above equation we have  
 
,  8 2 5
2 3 5 20
6
3
, , 0
Maximize z x y z
st
x y z
x y z
x
x y z
  
  
  


 
 
  
Now, from above LPP we have 
,  8 2 5
2 3 5 1 20
2 6
3 3
, , 0
Maximize z x y z
st
x y z s
x y z s
x s
x y z
  
   
   
 

 
 
INITIAL SIMPLEX TABLEAU 
Basic 
Variable 
z x y z s1 s2 s3   Ratio 
1 -8 -2 -5 0 0 0 0 
s1 0 2 3 5 1 0 0 20 20/2=10 
s2 0 1 1 1 0 1 0 6 6/1=6 
s3 0 1 0 0 0 0 1 3 3/1=3 
 
Entering element = x 
Departing element = s3 
FIRST ITERATION 
Basic 
Variable 
z x y z s1 s2 s3 
 Ratio 
1 0 -2 -5 0 0 8 24 
s1 0 0 3 5 1 0 -2 14 14/5=2.8 
s2 0 0 1 1 0 1 -1 3 3/1=3 
s3 0 1 0 0 0 0 1 3 3/0 
 
Entering element = z 
Departing element = s1 
 
SECOND ITERATION 
Basic 
Variable 
z x y z s1 s2 s3   
1 0 1 0 1 0 6 38 
s1 
0 0 0.6 1 0.2 0 -0.4 2.8 
s2 
0 0 0.4 0 -0.2 1 -0.6 0.2 
s3 
0 1 0 0 0 0 1 3 
 
Since, there are no negative coefficients in first row. Hence this is the optimal solution  
 
Optimal satisfaction, z = 38 
 
And 
Number of unit of grapes bought = 3 
Number of unit of broccoli bought = 0 
Number of unit of fish bought = 2.8
Sol 2:

The University of Queensland ENGG7302 Advanced Computational Techniques in Engineering Lecturer: Hanna Kurniawati Assignment 3 – Optimization Due: Monday, 12 June 2017, 23:59 Submission: A .pdf file...

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