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

Mathematical Programming II (MP2)
Network Optimization Models
Xuan Vinh Doan, [email protected]
ORMS Group
Warwick Business School
October 10-15, 2013
Learning Objectives
Agenda
Minimum-cost network ow problems
Shortest-path problem
Maximum-ow problem
D

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

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 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

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

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 no

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

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

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

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

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

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

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.

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

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

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

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

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

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

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.

Lectures by Steve Alpern,
slides prepared by Dr Arne K. Strauss
Warwick Business School
IB207 - Mathematical Programming II (MP2)
Todays topic
Modelling:
Dijkstras algorithm
Knapsack problem
Solving / Concept:
Dynamic programming algorithms
References:
W

Mathematical Programming II (MP2)
Seminar 4
Xuan Vinh Doan, [email protected]
ORMS Group
Warwick Business School
October 24, 2013
Problem 1
Given a directed network G = (N , A) with arc capacities,
formulate the minimum-cut problem as a binary integer p

Mathematical Programming II (MP2)
Seminar 1
Xuan Vinh Doan, [email protected]
ORMS Group
Warwick Business School
October 4, 2013
Problem 1
Consider the following linear optimization formulation
m
n
min
m
cij xij +
i=1 j=1
m
s.t.
xij = 1,
fi yi
i=1
j =

Mathematical Programming II (MP2)
Seminar 2
Xuan Vinh Doan, [email protected]
ORMS Group
Warwick Business School
October 11, 2013
Problem 1
Derive the dual problem for the following linear program:
n
max
x
ri xi
i=1
n
s.t.
xi = k,
i=1
0 xi 1,
i = 1, .

IB207 Mathematical Programming II
DIFFERENTIAL CONDITIONS FOR
CONVEX FUNCTIONS
PRINCIPAL MINORS
Let A be a symmetrical n n matrix.
The i-th leading principal minor of A is any
determinant of the ii matrix obtained from A by
deleting the last n i rows and

IB207 Mathematical Programming II
CONVEX SETS AND FUNCTIONS
CONVEX SETS
Geometric definition:
X is called a convex set if, for any two points x1
and x2 in X, the line segment [x1, x2] X.
Algebraic definition:
X is called a convex set if, for any two poi

Steve Alpern, Lecturer for Weeks 6-10
Slides by Dr. Arne K. Strauss (who taught this last year)
Warwick Business School
Contact Details
Steve Alpern
Professor of OperaHonal Research, ORMS Group
Room E0.17 (Social