Presentation Lecture 1 INDU 6111
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Transcript of Presentation Lecture 1 INDU 6111
INDU 6111: Theory of Operations ResearchCourse Introduction and Mathematical Modeling
Ivan Contreras
Mechanical and Industrial Engineering DepartmentConcordia University
Interuniversity Research Centre on Enterprise Networks,Logistics and Transportation (CIRRELT)
Montreal, Canada
September 5th, 2012
Operations Research: The Science of Better
Operations Research (OR) is the discipline of applying advancedanalytical methods to help make better decisions.
By using techniques such as mathematical modeling to analyzecomplex situations, OR gives executives the power to make moreeffective decisions and build productive systems based on:
The latest decision tools and techniquesConsideration of all available optionsCareful predictions of outcomes and estimates of riskMore complete data
Extracted from: http://www.scienceofbetter.org/
Operations Research: The Science of Better
To achieve these results, OR professionals draw upon the latest analyticaltechnologies, including:
Simulation: giving the ability to try out approaches and test ideasfor improvement
Optimization: narrowing choices to the very best when there arevirtually innumerable feasible options and comparing them is difficult
Probability and Statistics: helping measure risk, mine data to findvaluable connections and insights, test conclusions, and makereliable forecasts
Extracted from: http://www.scienceofbetter.org/
Operations Research: The Science of Better
OR can help executives with many challenges they face:
Locating a warehouse or distributions center to deliver materials overshorter distances at reduced cost
Solving complex scheduling problems
Deciding when to discount, and how much
Optimizing a portfolio of investments
Planting crops in the face of uncertainty about weather andconsumer demand
Forecasting sales for a new kind of product that has never beenmarketed before
Figuring out the best way to run a call center
For success stories in OR go to:http://www.scienceofbetter.org/can do/success stories.htm
Operations Research: The Science of Better
Operations Research encompasses a wide range of problem-solvingmethods and techniques such as:
Optimization
Simulation
Game theory
Queueing theory
Stochastic processes
Expert systems
Decision analysis
. . .
Applications in Operations Research
Several areas within Industrial Engineering in which OR can be used:
Design and analysis of manufacturing systems
Production planning and inventory control
Machine and personnel scheduling
Transportation and logistics
Product design and development
Quality control
. . .
There are other fields of knowledge in which OR can be employed:
Economics
Computer Science
Statistics
Mathematics
Computational Biology
. . .
Applications in Operations Research
Several areas within Industrial Engineering in which OR can be used:
Design and analysis of manufacturing systems
Production planning and inventory control
Machine and personnel scheduling
Transportation and logistics
Product design and development
Quality control
. . .
There are other fields of knowledge in which OR can be employed:
Economics
Computer Science
Statistics
Mathematics
Computational Biology
. . .
Optimization
Main fields in Optimization (or Mathematical Programming):
Convex programmingLinear programmingSemidefinite programmingConic programming
Quadratic programmingNonlinear programmingNetwork optimizationInteger and combinatorial optimizationStochastic programmingRobust optimizationConstraint programmingDynamic programmingSemi-infinite programmingMulti-objective optimizationOptimal control. . .
OR Institutes and Associations
Canada:Canadian Operational Research Society (CORS)http://www.cors.ca/
North America:Institute for Operations Research and Management Sciences (INFORMS)http://www.informs.org/
Europe:The Association of European Operational Research Societies (EURO)http://www.euro-online.org/
The Entire Globe:International Federation of Operational Research Societies (IFORS)http://ifors.org/
Course Overview
This course focuses on both the theory and applications of OR. Inparticular, it covers topics from:
Mathematical Modeling
Computational Complexity
Convex Analysis
Linear Programming
Network Optimization
Course Overview
Mathematical ModelingLinear, nonlinear, and integer programming models
Convex Analysis
Convex sets, polyhedral sets and polyhedral conesExtreme points and extreme directionsRepresentation of polyhedral sets
Linear ProgrammingMotivation of the simplex method and the revised simplex methodFarkas’ lemma and the Karush-Kuhn-Tucker optimality conditionsDuality and sensitivity analysisInterior point methods
Computational Complexity TheoryComplexity issues, polynomial-time algorithmsDecision problems and classes NP and P
Network OptimizationNetwork simplex methodMatching and assignment problemsMin-cost, max-flow problems
Optimization Solvers
Linear and Integer Programming Software:
CPLEXhttps://www-304.ibm.com/support/docview.wss?uid=swg21419058
XPRESShttp://optimization.fico.com/student-version-of-fico-xpress.html
GUROBIhttp://www.gurobi.com/
Linear Algebra
Concepts in linear algebra that are assumed to be know are:
Vectors: addition, scalar multiplication, inner product, and norms
Vector spaces (in particular the Euclidean space)
Linear and convex combinations, linear independence
Spanning set and basis
Matrices, partitioned matrices, inverse, and transpose
Elementary matrix operations
Solving a system of linear equations
Rank of a matrix
Mathematical Modeling in Optimization
The modeling and analysis of models evolves though several stages:
1 Problem definition: study of the system; data collection;identification of specific problem that needs to be analyzed.
2 Devising a mathematical formulation: construction of a modelthat satisfactorily represents the system while keeping the modeltractable.
3 Solving the formulation: use/develop a proper technique thatexploits any special structure of the model
4 Testing, analysis, and restructuring the model: examination ofmodel solution and its sensitivity to system parameters; study ofvarious what-if types of scenarios; enrich the model further orsimplify the model.
5 Implementation: development of a decision support system to aidin the decision-making process.
The interested reader is referred to: Brown and Rosenthal, Optimization Tradecraft:
Hard-Won Insights from Real-Worls Decision Support, Interfaces, 38(5), 2008.
Mathematical Modeling in Optimization
Translating a problem description into a formulation should be donesystematically.
1 Define what appears to be the necessary decision variables
2 Use these variables to define a set of constraints so that the feasiblepoints correspond to the feasible solutions of the problem
3 Use these variables to define the objective function
If difficulties arise, define an additional or alternative set of variables anditerate.
Important: A clear distinction should be made between the data of theproblem and the decision variables used in the model.
Note: This modeling recipe has been provided by Lawrence Wolsey in Integer
Programming, Wiley, 1998.
Mathematical Modeling in Optimization
Translating a problem description into a formulation should be donesystematically.
1 Define what appears to be the necessary decision variables
2 Use these variables to define a set of constraints so that the feasiblepoints correspond to the feasible solutions of the problem
3 Use these variables to define the objective function
If difficulties arise, define an additional or alternative set of variables anditerate.
Important: A clear distinction should be made between the data of theproblem and the decision variables used in the model.
Note: This modeling recipe has been provided by Lawrence Wolsey in Integer
Programming, Wiley, 1998.
Mathematical Modeling in Optimization
Translating a problem description into a formulation should be donesystematically.
1 Define what appears to be the necessary decision variables
2 Use these variables to define a set of constraints so that the feasiblepoints correspond to the feasible solutions of the problem
3 Use these variables to define the objective function
If difficulties arise, define an additional or alternative set of variables anditerate.
Important: A clear distinction should be made between the data of theproblem and the decision variables used in the model.
Note: This modeling recipe has been provided by Lawrence Wolsey in Integer
Programming, Wiley, 1998.
Mathematical Modeling in Optimization
We will first study five classical problems in optimization:
Transportation problem
p-median problem
Cutting stock problem
Uncapacitated lot-sizing problem
Traveling salesman problem
Example 1: The Transportation ProblemPlants Warehouses
a1
a2
a3
a4
a5
a6
b1
b2
b3
b4
b5
c65
c11
Input:
M : set of plants
N : set of warehouses
cij : cost between i and j
bj : demand at warehouse j
ai: capacity at plant i
The transportation problem:
Decide the production quantity at each plant
Find a shipping pattern form plants to warehouses to satisfy demand
Objective: Minimize the total shipping cost
Unit shipping cost from plant i to warehouse j: cij
Example 1: The Transportation Problem
Production-shipping variables
xij = amount of product produced at plant i shipped to warehouse j
Plants Warehouses
a1
a2
a3
a4
a5
a6
b1
b2
b3
b4
b5
x54
xij variables (continuous)
Example 1: The Transportation Problem
The transportation problem can be formulated as the following linearprogram:
minimize∑i∈M
∑j∈N
cijxij
subject to∑i∈M
xij = bj j ∈ N∑j∈N
xij ≤ ai i ∈M
xij ≥ 0 i ∈M, j ∈ N
Example 2: The p-Median Problem
Input:
J : set of customers
I: set of candidate locations
cij : cost between i and j
dj : demand of customer j
p: number of facilities to open
The p-median problem:
The location of p facilities
The assignment of each customer to its closest facility
Assume that each facility has unlimited capacity
Objective: Minimize the total transportation cost
Transportation cost from i to j: djcij
Example 2: The p-Median Problem
Location variables
zi =
{1 if a facility is located at node i;0 otherwise
Allocation variables
xij =
{1 if customer j is assigned to facility located at node i;0 otherwise.
Example 2: The p-Median Problem
The p-median problem can be formulated as the following integer linearprogram:
minimize∑i∈I
∑j∈J
djcijxij
subject to∑i∈I
zi = p∑i∈I
xij = 1 j ∈ J
xij ≤ zi i ∈ I, j ∈ J
zi, xij ∈ {0, 1} i ∈ I, j ∈ J
Example 3: The Cutting Stock Problem
Len
gth
(L)
Width (W)
Standard Metal Sheet Roll A cutting pattern
Width (W)
l1
l2
l2
l3
Example 3: The Cutting Stock Problem
Input:
M : set of sheet orders of different size (|M | = m)
L: standard length of sheet rolls
bi: demand of sheets with length li
The cutting stock problem:
Cut the standard rolls in such a way as to satisfy the orders
We assume that scrap pieces are useless
Objective: Minimize the waste by minimizing number of rolls needed
Example 3: The Cutting Stock Problem
Observe that given a standard sheet of length L, there are manyways of cutting it.
Each such way is called a cutting pattern.
The jth cutting pattern is characterized by the vector:
aj =
a1ja2j. . .amj
,
where the ith component aij is a nonnegative integer denoting thenumber of sheets of length li in the jth pattern.
Note that aj denotes a feasible cutting pattern if and only if∑i∈M
aij li ≤ L.
Let P denotes the set of feasible cutting patterns.
Example 3: The Cutting Stock Problem
Cutting pattern variables:
xj : number of standard rolls cut according to the jth pattern
The cutting stock problem can be formulated as the following integerlinear program:
minimize∑j∈P
xj
subject to∑j∈P
aijxi ≥ bi i ∈M
xj ∈ N+ j ∈ P
Example 4: The Uncapacitated Lot-Sizing Problem
Input:
T : set of time periods
dt: demand at period t
pt: unit production cost at period t
qt: fixed production cost at period t
ht: unit inventory cost at period t
The uncapacitated lot-sizing problem:
Decide the production lot size and inventory level in every period
Capacity in each period is assumed to be unlimited
We assume that demand must be satisfied in every time period
Objective: Minimize total production, set-up and inventory cost
Example 4: The Uncapacitated Lot-Sizing Problem
Set-up variables
yt =
{1 if there is a positive production in period t;0 otherwise
Production variables
xj : production lot size in period t
Inventory variables
sj : inventory level at the end of period t
Example 4: The Uncapacitated Lot-Sizing Problem
The uncapacitated lot-sizing problem can be formulated as the followinginteger linear program:
minimize∑t∈T
(ptxt + qtyt + htst)
subject to st−1 + xt − st = dt t ∈ T
xt ≤Mtyt t ∈ T
xt, st ≥ 0 t ∈ T
yt ∈ {0, 1} t ∈ T,
where Mt is a large positive number, denoting an upper bound on themaximum lot size in period t.
Example 5: The Traveling Salesman Problem
Input:
N : set of cities (|N | = n)
cij : travel time between i and j
The traveling salesman problem (TSP):
A salesman must visit each city exactly once and then return to hisstarting point
Objective: Minimize the total travel time of the tour
Example 5: The Traveling Salesman Problem
Input:
N : set of cities (|N | = n)
cij : travel time between i and j
The traveling salesman problem (TSP):
A salesman must visit each city exactly once and then return to hisstarting point
Objective: Minimize the total travel time of the tour
Example 5: The Traveling Salesman Problem
Routing variables
xij =
{1 if the salesman goes directly from city i to city j;0 otherwise
Using these variables, we could model the TSP as:
minimize∑i∈N
∑j∈N
cijxij
subject to∑
j∈N :j 6=i
xij = 1 j ∈ N
∑i∈N :i 6=j
xij = 1 i ∈ N
xij ∈ {0, 1} i ∈ N, j ∈ N
Is this enough to model the problem?
Example 5: The Traveling Salesman Problem
We have to eliminate subtours = obtain one single connected component!
Example 5: The Traveling Salesman Problem
To achieve so we need additional constraints that guarantee connectivity.For that, we have at least two options:
1 Cut-set constraints:∑i∈S
∑j /∈S
xij ≥ 1 ∀S ⊂ N,S 6= ∅.
2 Subtour elimination constraints:
∑i∈S
∑j∈S
xij ≤ |S| − 1 ∀S ⊂ N, 2 ≤ |S| ≤ n− 1.