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MA3264 Mathematical ModellingLecture 9
Chapter 7Discrete Optimization Modelling Continued
Theorem 1 p. 257-258
1x
Question Where is objective function max?
2x
Increasing directionof objective function
Level curves
feasible region is blue
Theorem 1 p. 257-258
1x
Question Where is objective function max?
2x
Increasing directionof objective function
Theorem 1 p. 257-258
1x
Question Where is objective function max?
2x
Increasing directionof objective function
Example 1 page 261
How many tables and how many bookcases should a carpenter make each week to maximize profit? He realizes a profit of $25 per table and $30 per bookcase. He has 690 feet of lumber per week and 120 hours of labor per week. Each table requires 20 feet of lumber and 5 hours of labor. Each bookcase requires 30 feet of lumber and 4 hours of labor.
This Carpenter Problem has different parameters from that on page 238
Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6903020 21 xx12045 21 xx01 x02 x
constraints
explicitly stated
common sense
Figure 7.12 p. 261
6903020 21 xx
1x
)23,0(
)0,5.34(
Question Is the feasible region convex?
)30,0(
)0,24(
)15,12(
2x
12045 21 xx
02 x
01 x
Slack Variables
6903020 121 yxx
1x
)23,0(
)0,5.34(
Question SV satisfy which inequalities?
)30,0(
)0,24(
)15,12(
2x 12045 221 yxx
02 x
01 x
01 y
02 y
03025 21 zxx
Reformulation
6903020 121 yxx
1x
)23,0(
)0,5.34(
Question SV satisfy which inequalities?
)30,0(
)0,24(
)15,12(
2x
12045 221 yxx
02 x
01 x
01 y
02 y
Maximize z
03025 1 zx
Subject to
0,,, 2121 yyxx
Simplex Method
6903020 121 yxx
1x
)23,0(
)0,5.34(
Remark At each extreme point at least 2 of these 4 variables equal 0.
)30,0(
)0,24(
)15,12(
2x
12045 221 yxx
02 x
01 x
01 y
02 y
03025 1 zx
Always assume constraints
0,,, 2121 yyxxBut manipulate these
Simplex Method
1x
)23,0(
)0,5.34(
These pairs of independent variables depend on the extreme point
)30,0(
)0,24(
)15,12(
2x
02 x
01 x
01 y
02 y
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
At each step 1 independent variable and 1 dependent variable change places
Simplex Method
1x
)23,0(
)0,5.34(
)30,0(
)0,24(
)15,12(
2x
02 x
01 x
01 y
02 y
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
In step 1 independent variable 2x becomes dependent since it has the largest negativecoeff. in equation for the objective variable z
03025 21 zxx
Simplex Method
6903020 121 yxx
1x
)23,0(
)0,5.34(
The variable
)30,0(
)0,24(
)15,12(
2x
12045 221 yxx
02 x
01 x
01 y
02 y03025 21 zxx
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
1ythe eqn. it is in restricts 2x
becomes independent since
to be smallest.
Simplex Method
6903020 121 yxx
1x
)23,0(
)0,5.34(
Equations express dependent
)30,0(
)0,24(
)15,12(
2x
12045 221 yxx
02 x
01 x
01 y
02 y03025 21 zxx
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
zyy ,, 21
functions of 21, xxas
independent at
Simplex Method
2321301
132 xyx
1x
)23,0(
)0,5.34(
We pivot to express dependent
)30,0(
)0,24(
)15,12(
2x
2821152
137 yyx
02 x
01 x
01 y
02 y 6905 11 zyx
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
zyx ,, 22
functions of 11, yxas
independent at
Simplex Method
2321301
132 xyx
1x
)23,0(
)0,5.34(
In step 2 independent variable
)30,0(
)0,24(
)15,12(
2x
2821152
137 yyx
02 x
01 x
01 y
02 y 6905 11 zyx
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
1xdependent since it has the largest negative
becomes
coeff. in equation for the objective variable z
Simplex Method
2321301
132 xyx
1x
)23,0(
)0,5.34(
The variable
)30,0(
)0,24(
)15,12(
2x
2821152
137 yyx
02 x
01 x
01 y
02 y 6905 11 zyx
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
2y becomes independent since
the eqn. it is in restricts 1x to be smallest.
Simplex Method
15071429.028571.0 212 xyy
1x
)23,0(
)0,5.34(
We pivot to express dependent
)30,0(
)0,24(
)15,12(
2x
12057143.042857.0 112 xyy
02 x
01 x
01 y
02 y
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
zxx ,, 12
functions of 12 , yyas
independent at
750714286.014286.2 12 zyy
Simplex Method
15071429.028571.0 212 xyy
1x
)23,0(
)0,5.34(
We stop since in the equation for objective
)30,0(
)0,24(
)15,12(
2x
12057143.042857.0 112 xyy
02 x
01 x
01 y
02 y
Start at 120,690;0 2121 yyxx
Move to 28,23;0 2211 yxyxMove to 12,15;0 1212 xxyy
variable
750714286.014286.2 12 zyy
z all coeff. are nonnegative.
Tableau Format
Study Example 1 on pages 268-271
Observe that
Tableau 0 represents the 3 equations in vufoil 13
Tableau 1 represents the 3 equations in vufoil 15
Tableau 2 represents the 3 equations vufoil in 18
Suggested Reading
Linear Programming 2: Algebraic Solutions p. 259-263
Linear Programming 3: The Simplex Method p. 263-273
Tutorial 9 Due Week 27–31 Oct
Problem 1. Modify the Bus Stop Waiting program to write a program to compute the waiting time histogram assuming that people form a queue so that people who arrive first are the first to board a bus. Then RUN this program and compare the waiting times with those without queuing.
Problem 2. Do problem 5 on page 259. Suggestion: study how to formulate a Chebyshev approximation problem into a linear program on pages 107-109.
Homework 3 Due Friday 31 October
7.4 Project 1 Write a computer program to perform the basic simplex algorithm. The program should print out the sequence of all tableaus from the initial to the final one. It should print out the coordinates of all of the extreme points and the values of the objective function at these points. Then use the program to solve problem 3 on p. 258.
