1
CHAPTER 3 MP REVIEW PROBLEMS
1. Let x1 = barrels of beer produced
x2 = barrels of ale produced
Then we should solve
max z = 5x1 + 2x2
s.t.
5x1 + 2x260
2x1 + x225
x1, x20
Graphically we find the optimal solution to be z = 60,
x1 = 10, x2 = 5, x1 = 12, x2

Introduction to Linear Programming
Linear programming (LP) is a tool for solving optimization problems. In 1947, George Dantzig developed an efcient method, the simplex algorithm, for solving linear programming problems (also
called LP). Since the develo

Nonlinear Programming
In previous chapters, we have studied linear programming problems. For an LP, our goal was to maximize or minimize a linear function subject to linear constraints. But in many interesting maximization and minimization problems, the o

Review of Calculus and Probability
We review in this chapter some basic topics in calculus and probability, which will be useful in later chapters.
12.1
Review of Integral Calculus
In our study of random variables, we often require a knowledge of the basi

Deterministic EOQ Inventory Models
In this chapter, we begin our formal study of inventory modeling. In earlier chapters, we described how linear programming can be used to solve certain inventory problems. Our study of inventory will continue in Chapters

Sensitivity Analysis: An Applied Approach
In this chapter, we discuss how changes in an LPs parameters affect the optimal solution. This is called sensitivity analysis. We also explain how to use the LINDO output to answer questions of managerial interest

Game Theory
In previous chapters, we have encountered many situations in which a single decision maker chooses an optimal decision without reference to the effect that the decision has on other decision makers (and without reference to the effect that the

Decision Making under Uncertainty
We have all had to make important decisions where we were uncertain about factors that were relevant to the decisions. In this chapter, we study situations in which decisions are made in an uncertain environment. The foll

Markov Chains
Sometimes we are interested in how a random variable changes over time. For example, we may want to know how the price of a share of stock or a rms market share evolves. The study of how a random variable changes over time includes stochasti

Winston Chapter 5 Review, Page 226, Number 4 (Sensitivity Analysis)
1
Winston Chapter 5 Review, Page 226, Number 4 (Sensitivity Analysis)
Problem Statement: Zales Jewelers uses rubies and sapphires to produce two types of
rings. A type 1 ring requires 2 r

Spanning Tree
OPL
CPM
Integer Solution
Lecture 11:
Network Programming II
Seeronk Prichanont1
Oran Kittithreerapronchai1
1
OR I v1.2
Department of Industrial Engineering, Chulalongkorn University
Bangkok 10330 THAILAND
1/ 22
Spanning Tree
OPL
CPM
Integer

Network
Min-Cost
Shortest Path
Max-Flow
Bipartite
Lecture 10:
Network Programming I
Seeronk Prichanont1
Oran Kittithreerapronchai1
1
OR I v1.0
Department of Industrial Engineering, Chulalongkorn University
Bangkok 10330 THAILAND
1/ 27
Network
Min-Cost
Sho

Advanced Topics in Linear Programming
In this chapter, we discuss six advanced linear programming topics: the revised simplex method, the product form of the inverse, column generation, the DantzigWolfe decomposition algorithm, the simplex method for uppe

Transportation, Assignment, and Transshipment Problems
In this chapter, we discuss three special types of linear programming problems: transportation, assignment, and transshipment. Each of these can be solved by the simplex algorithm, but specialized alg

Assignment 2
1) Use the simplex algorithm to find the optimal solution to the following
LP :
min z = 4x1 x2
s.t. 2x1 + x2
8
x2
5
x1 x2
4
x1 , x2
0
2) Use simplex method to find all optimal solutions to the following LP:
max z = 3x1 + 3x2
s.t. x1 + x2
1
Al

1
Winston Chapter 7.6, Page 383, Number 2 (Transshipment)
Problem Statement: Sunco Oil produces oil at two wells. Well 1 can produce up to
150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is
possible to ship oil directly

IE 311 - Operations Research I
Spring 2005
Solved Exercises on LP Modeling
1.
[Problem 3 on page 64 in the 3rd edition of the textbook] Leary chemical manufactures three chemicals: A, B, and C. These
chemicals are produced via two production processes: 1

1. Firm want to sell Product A and B. Product A must sell at least 80% of the total
sale of A and B. Market demand of product A not exceed 100. Raw material used
to product A and B is the same type and 240kg of raw material are available.
Product A use 2k

An Introduction to Model Building
1.1
An Introduction to Modeling
Operations research (often referred to as management science) is simply a scientific approach to decision making that seeks to best design and operate a system, usually under conditions req

Basic Linear Algebra
In this chapter, we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra: matrices and vectors. Then we use our knowledge of matrices and vectors

Spring, 2007
Spring,
ISE 102
Introduction to Linear
Programming (LP)
Asst. Prof. Dr. Mahmut Ali GKE
Industrial Systems Engineering Dept.
zmir University of Economics
Asst. Prof. Dr. Mahmut Ali GKE, Izmir University of Economics
www.izmirekonomi.edu.tr
1
S

Integer Programming
Recall that we dened integer programming problems in our discussion of the Divisibility Assumption in Section 3.1. Simply stated, an integer programming problem (IP) is an LP in which some or all of the variables are required to be non

Network Models
Many important optimization problems can best be analyzed by means of a graphical or network representation. In this chapter, we consider four specic network modelsshortest-path problems, maximum-ow problems, CPMPERT project-scheduling mode

Assignment 1
1) Finco must determine how much investment and debt to undertake during
the next year. Each dollar invested reduces the NPV of the company by 10
cents, and each dollar of debt increases the NPV by 50 cents (due to
deductibility of interest p