+Xl6 < 1.01:
int x9, x10, x11, X12, x13, x14, x15, x16;
c. Graphic View of the Transportation Problem
W
a. Result:
There are 6 nodesl and 8 arcs in this network.
Nodes with Nonzero Demands or Supplies
Node # Supply(er demand if positive)
1 -60.
2 -40
3

Simplex Method and Linear Programming
1. What is Simplex Method?
Linear Programming Problem may have infinite feasible solutions, while the finite
vertexes correspond to the basic feasible solutions. In Simplex Method, we start from
one of the basic feasi

Simplex Method and Linear Programming
1. What is Simplex Method?
Linear Programming Problem may have infinite feasible solutions, while
the finite vertexes correspond to the basic feasible solutions. In
Simplex Method, we start from one of the basic feasi

Assignment 3
W
a. Result:
There are 6 nodes, and 8 arcs in this networkl
Nodes with Nonzero Demands or Supplies.
Node # Supply(or demand if positive)
1 60
4 10'
5 30.
6 20.
From To
Arc# Node# Node# Cost FixedCost Capacity Multiplier Solution
1 1 2 0 100.

7. (3 points) Suppose we have three projects: F, M, D. The value 0 means not selecting
the project, and 1 means selecting. Write down logic conditions to each of the following:
(i) At least one project is undertaken;
(ii) If you make the project M, then y

4. (2 points) Consider the following system:
Maximize 7x1+ 9x2
Subject to xl +x2 s 6
5x1 + 9x2 S 44
xnxz 2 0 integer.
To nd Gomory cuts, we change the two inequalities as:
x1 + x2 + sl = 6
5x1+9x2 +52 =44
51,32 2 0 integer.
By eliminating x2, we obtain
4x

Family Name First Name
Student #
SYS 5130 Midterm-October 14, 19:00-20:30
Closed Book
(In total you have 8 questions, 20 marks in total.)
1. (2 points) True/False:
(a) Any optimization problem has at least one variable. T
(b) Optimization problem may

5. (3 points) Use Fourier Motzkin Method (not software) to solve the
following question:
Max 5x +y
s.t. 5x +2y S14
y 23
x+3y S 13
36,322 0
Step 1: Let 2=5x+y.
Step 2: Eliminate x:
2 +32 514 y 514-2
y 23 i.e., y 23
z+l4y < 65 l4y<65-z
2,372 0 2,322 0
Step

8. (4 points) A bank makes four kinds of loans L1, L2, L3 and L4 to its
personal customers. These loans yield the following annual interest rates to
the bank:
I L]: 6%,
0 L2: 5%,
I L3: 5%,
I L4: 3%.
The bank has a maximum lending capability of 250 thousan

6. (2 points) We are going to solve the following system using branch and bound
method:
Max 15x1+24x2
SI. 18X1+10X2<=90
20X1+20x2<=118
x] , X2>=0 integer
Look at the following branch and bound:
Optimum=141 .6
where x1=0, x2=5 .9
NODE3
Optimum=120 Optimu

SIMPLEX
METHOD
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LINEAR
PROGRAMMING
Zhijie Cui, Ci Lin, Xiaojin
Shan
2
Introduction For Simplex Method
WhatNeed
Why
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Simplex
Method?
Method?
In practice,
To
solve thereal-world
Linear Programming
problems commonly are complex that
Problem,
wit