Assignment #3 (80 Points) – COSC 5390 –Dr. Leonard Brown Due: April 4, 2013 (atthe beginning of class) Problem Description Write a java program that will implement the Simplex algorithm discussed in class that will solve a systemof equationsin the following form. Maximize Z = c 1x 1 +…+ c nx n Subjectto a 11x 1 +…+ a 1nx n ≤ b1 a 21x 1 +…+ a 2nx n ≤ b2 … a m1x 1 +…+ a mnx n ≤ b m and xi ≥ 0 fori=1,…,n Input The input programwill be a systemof inequalities augmented with slack variablesso thatitisin a formsuitable forthe “table”method. So, ifthe goal wasto solve the following system: Maximize Z = 3×1 + 5×2 x 1 ≤ 4 2×2 ≤ 12 3×1 + 2×2 ≤ 18 xj ≥ 0,forj = 1, 2 Then the input to the program will be represented by the following system of equations (with the slack variablesincluded). Z – 3x 1 – 5x 2 = 0 x 1 + x 3 = 4 2×2 + x 4 = 12 3×1 + 2×2 + x 5 = 18 More specifically,the inputto the programwould be the following lines oftext 1 -3 -5 0 0 0 0 0 1 0 1 0 0 4 0 0 2 0 1 0 12 0 3 2 0 0 1 18 Which comesfromthe shaded area in the following table: Basic Var Eq Z x 1 x 2 x 3 x 4 x 5 Right Side Z 0 1 ‐3 ‐5 0 0 0 0 x 3 1 0 1 0 1 0 0 4 x 4 2 0 0 2 0 1 0 12 x 5 3 0 3 2 0 0 1 18
Assignment #3 (80 Points) – COSC 5390 – Dr. Leonard Brown
Due: April 4, 2013 (at the beginning of class)
Problem Description
Write a java program that will implement the Simplex algorithm discussed in class that will
solve a system of equations in the following form.
Maximize
Z = c1x1 + … + cnxn
Subject to
a11x1 + … + a1nxn ≤ b1
a21x1 + … + a2nxn ≤ b2
…
am1x1 + … + amnxn ≤ bm
and
xi ≥ 0 for i=1,…,n
Input
The input program will be a system of inequalities augmented with slack variables so that it is in
a form suitable for the “table” method. So, if the goal was to solve the following system:
Maximize Z = 3×1 + 5×2
x1 ≤ 4
2×2 ≤ 12
3×1 + 2×2 ≤ 18
xj ≥ 0, for j = 1, 2
Then the input to the program will be represented by the following system of equations (with
the slack variables included).
Z – 3×1 – 5×2 = 0
x1 + x3 = 4
2×2 + x4 = 12
3×1 + 2×2 + x5 = 18
More specifically, the input to the program would be the following lines of text
1 -3 -5 0 0 0 0
0 1 0 1 0 0 4
0 0 2 0 1 0 12
0 3 2 0 0 1 18
Which comes from the shaded area in the following table:
Basic Var Eq Z x1 x2 x3 x4 x5 Right Side
Z 0 1 ‐3 ‐5 0 0 0 0
x3 1 0 1 0 1 0 0 4
x4 2 0 0 2 0 1 0 12
x5 3 0 3 2 0 0 1 18
Output
The program should display the simplex table after each iteration of the algorithm. There
should be at least one blank line between each pair of tables. After the optimal solution is
found, the program should print at least one blank link followed by the optimal solution. In the
given example, the optimal solution is (2, 6, 2, 0, 0).
Assumptions
All constraints will be of the form a11x1 + … + a1nxn ≤ b1. Thus, you will not have to solve
greater than or greater than inequality constraints.
In addition to the objective function, there will be three inequalities in the system.
Thus, the input will be a grid with four rows.
There will be at most 10 total columns in the input grid.
The last column in the input grid represents the right hand side. The three columns
immediately preceding this last column represent the slack variables.
Submission
Submit your assignment through Blackboard. If your assignment contains multiple files, zip
them into a single folder before submitting.
Notes
Points can be deducted from your assignment based on the quality of its presentation.
Handwritten assignments will not be accepted.