IE528 Lec 4
1. (Bertsimas 10.6) A company produces a set of K products at I plants. It then ships these
products to J market zones. For k = 1, ., K i = 1, ., I and j = 1, ., J the following data is
given:
vik = variable cost of producing one unit of produ
IE528 Mathematical Programming and Modeling
HW1
Due March 18 (midnight)
Spring 2012
1. (Bertsimas 10.1) Suppose that we are given m constraints ai x bi for all i = 1.m,
but without the restriction ai 0. Model the requirement that at least k of them are
sa
Sensitivity Analysis and Duality
Two of the most important topics in linear programming are sensitivity analysis and duality. After studying these important topics, the reader will have an appreciation of the beauty and logic of linear programming and be
Advanced Linear Programming, May 18, 2009, 10:00 - 13.00
Please notice my mistake in Exercise 5 for which my sincere
apologises
(1) (1 pt.) Formulate Farkas Lemma.
Answer.
Theorem. Given m n matrix A and b I m , exactly one of the following
R
two alternat
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
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
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
APPENDIX 2
Cases
Jeffrey B. Goldberg
UNIVERSITY OF ARIZONA
CASE
1 2 3 4 5 6 7 8 9 10 11
Help, Im Not Getting Any Younger! Solar Energy for Your Home
1351
1351
CASE
CASE
Golf-Sport: Managing Operations
1352 1355 1356
CASE
Vision Corporation: Production Pla
APPENDIX 1
@Risk Crib Sheet
@Risk Icons Once you are familiar with the function of the @Risk icons, you will nd @Risk easy to learn. Here is a description of the icons.
Opening an @Risk Simulation
This icon allows you to open up a saved @Risk simulation.
Simulation with Process Model
In Chapter 21, we learned how to build simulation models of many different situations. In this chapter, we will explain how the powerful, user-friendly simulation package Process Model can be used to simulate queuing systems.
Simulation
Simulation is a very powerful and widely used management science technique for the analysis and study of complex systems. In previous chapters, we were concerned with the formulation of models that could be solved analytically. In almost all of
Probabilistic Dynamic Programming
Recall from our study of deterministic dynamic programming that many recursions were of the following form: ft (current state) min
all feasible decisions
(or max)cfw_costs during current stage
ft
1
(new state)
For all the
Deterministic Dynamic Programming
Dynamic programming is a technique that can be used to solve many optimization problems. In most applications, dynamic programming obtains solutions by working backward from the end of a problem toward the beginning, thus
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
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
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
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
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
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
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 develop
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
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
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
15.053 Introduction to Optimization (Spring 2005)
Recitation 8, March 8th 2005
Problem 1 Formulating a Piece Wise Linear IP
Formulate the following as an IP. Do not solve your IP:
A petrochemical company wants to maxi
Manufacturing Planning and Control
Stephen C. Graves
Massachusetts Institute of Technology
November 1999
Manufacturing planning and control entails the acquisition and allocation of limited
resources to production activities so as to satisfy customer dema
Advanced Linear Programming: The Exercises
The answers are sometimes not written out completely.
1.5 a)
min cT x + dT y
s.t. Ax + By b
y = |x|
(1)
First reformulation, using z smallest number satisfying x z and x z :
min cT x + dT z
s.t. Ax + Bz b
xz
x z.
IE528 Mathematical Programming and
Modeling
Lecture 6
Dr. Zeliha Akca
March 2012
1 / 21
Production Planning Problems in General
Production planning problems are the problems focused on
allocation of limited resources to the production activities so as to
IE528 Mathematical Programming and
Modeling
Lecture 5
Dr. Zeliha Akca
March 2012
1 / 12
General Case of Disjunctive Constraints
Let a x b and d x f be two constraints.
a x and d x are not necessarily 0.
We want to satisfy at least one of these constraints