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Pesquisa Operacional Aplicada à Logística
Prof. Fernando Augusto Silva Marins
fmarins@feg.unesp.br
www.feg.unesp.br/~fmarins
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Sumário
Introdução à Pesquisa Operacional (P.O.)
Impacto da P.O. na Logística
Modelagem e Softwares
Exemplos
Cases em Logística
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Pesquisa Operacional
Operations Research
Operational Research
Management Sciences
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A P.O. e o Processo de Tomada de Decisão
Tomar decisões é uma tarefa básica da gestão.
Decidir: optar entre alternativas viáveis.
Papel do Decisor :
Identificar e Definir o Problema
Formular objetivo (s)
Analisar Limitações
Avaliar Alternativas Escolher a “melhor”
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PROCESSO DE DECISÃO
Abordagem Qualitativa: Problemas simples e experiênciado decisor
Abordagem Quantitativa: Problemas complexos, ótica
científica e uso de métodos quantitativos.
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Pesquisa Operacional faz diferença nodesempenho de organizações?
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Resultados - finalistas do Prêmio Edelman
INFORMS 2007
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FINALISTAS EDELMAN 1984-2007Ano Empresa Título do Trabalho
1996 South African National Defense Force* "Guns or Butter: Decision Support for Determining the Size and Shape of the
South African National Defense Force (SANDF)"
1996 The Finance Ministry of Kuwait "The Use of Linear Programming in Disentangling the Bankruptcies of al-Manakh
Stock Market Crash
1996 AT&T Capital "Credit and Collections Decision Automation in AT&T Capital's Small-TicketBusiness"
1996 British National Health Service "A New Formula for Distributing Hospital Funds in England"
1996 National Car Rental System, Inc. "Revenue Management Program"
1996 Procter and Gamble "North American Product Supply Restructuring at Procter & Gamble"
1996 Federal Highway Administration/California Department
of Transportation
"PONTIS: A System for Maintenance Optimization and Improvement of U.S.
Bridge Networks "
1995 Harris Corporation/Semiconductor Sector* "IMPReSS: An Automated Production-Planning and Delivery-Quotation System at
Harris Corporation - Semiconductor Sector"
1995 Israeli Air Force "Air Power Multiplier Through Management Excellence"
1995 KeyCorp "The Teller Productivity System and Customer Wait Time Model"
1995 NYNEX "The Arachne Network Planning System"1995 Sainsbury's "An Information Systems Strategy for Sainsbury’s"
1995 SADIA "Integrated Planning for Poultry Production"
1994 Tata Iron & Steel Company, Ltd.* "Strategic and Operational Management with Optimization at Tata Steel"
1994 Bellcore "SONET Toolkit: A Decision Support System for the Design of Robust and Cost-
Effective Fiber-Optic Networks"
1994 Chinese State Planning Commission and the World "Investment Planning for China’s Coal and Electricity Delivery System"
1994 Digital Equipment Corp. "Global Supply Chain Management at Digital Equipment Corp."
1994 Hanshin Expressway Publ ic Corporation "Traffic Control System on the Hanshin Expressway"
1994 U.S. Army "An Analytical Approach to Reshaping the Army"
1993 AT&T* "AT&T's Call Processing Simulator (CAPS) Operational Design for Inbound Call
Centers"1993 Frank Russell Company & The Yasuda Fire and Marine
Insurance Co. Ltd.
"An Asset/Liability Model for a Japanese Insurance Company Using Multistage
Stochastic Programming"
1993 North Carolina Department of Public Instruction "Data Envelopment Analysis of Nonhomogeneous Units: Improving Pupil
Transportation in North Carolina"
1993 National Aeronautic and Space Administration (NASA) "Management of the Heat Shield of the Space Shuttle Orbiter: Priorities and
Recommendations Based on Risk Analysis"
1993 Delta Airlines "COLDSTART: Daily Fleet Assignment Model"
1993 Bellcore "An Optimization Approach to Analyzing Price Quotations Under Business Volume
Discounts"
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FINALISTAS EDELMAN 1984-2007Ano Empresa Título do Trabalho
1985 Weyerhaeuser Company* Weyerhaeuser Decision Simulator Improves Timber Profits
1985 Canadian National Railways "Cost Effective Strategies for Expanding Rail-Line Capacity Using Simulation and
Parametric Analysis"
1985 Pacific Gas and Electric Company "PG&E's State-of-the-Art Scheduling Tool for Hydro Systems"
1985 New York, NY, Department of Sanitation "Polishing the Big Apple"
1985 Eletrobras and CEPEL, Brazil Coordinating the Energy Generation of the Brazilian System1985 United Airlines United Airlines Station Manpower Planning System
1984 Blue Bell, Inc.* Blue Bell Trims Its Inventory
1984 The Netherlands Rijkswaterstaat and the Rand Planning the Netherlands' Water Resources
1984 Austin, Texas, Emergency Medical Services Determining Emergency Medical Service Vehicle Deployment1984 Pfizer, Inc. "Inventory Management at Pfizer Pharmaceuticals"
1984 Monsanto Corporation "Chemical Production Optimization"
1984 U.S. Air Force "Improving Utilization of Air Force Cargo Aircraft"
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Como construir Modelos Matemáticos?
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Classification of Mathematical Models
Classification by the model purpose
– Optimization models
– Prediction models
Classification by the degree of certainty of the data in themodel
– Deterministic models
– Probabilistic (stochastic) models
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Mathematical Modeling
A constrained mathematical model consists of
– An objective: Function to be optimised with one or moreControl /Decision Variables
Example: Max 2x – 3y; Min x + y
– One or more constraints: Functions (“”, “”, “=”) with one
or more Control /Decision Variables
Examples: 3x + y 100; x - 4y 100; x + y = 10;
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New Office Furniture Example
Products
Desks
Chairs
Molded Steel
Profit
$50
$30
$6 / pound
Raw Steel Used
7 pounds (2.61 kg.)
3 pounds (1.12 kg.)
1.5 pounds (0.56 kg.)
1 pound (troy) = 0.373242 kg.
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Defining Control/Decision Variables
Ask, “Does the decision maker have the authorityto decide the numerical value (amount) of theitem?”
If the answer “yes” it is a control/decision variable.
By very precise in the units (and if appropriate, thetime frame) of each decision variable.
D: amount of desks (number)C: amount of chairs (number)M: amount of molded steel (pound)
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Objective Function
The objective of all optimization models, is to
figure out how to do the best you can with whatyou’ve got.
“The best you can” implies maximizing something
(profit, efficiency...) or minimizing something (cost,time...).
Total Profit = 50 D + 30 C + 6 M
Products
Desks
Chairs
Molded Steel
Profit
$50
$30
$6 / pound
D: amount of desks (number)C: amount of chairs (number)M: amount of molded steel (pound)
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Writing Constraints Create a limiting condition for each scarce resource :
(amount of a resource required) (“
”, “
”, “=”) (resource availability) Make sure the units on the left side of the relation are the same as those on
the right side.
Use mathematical notation with known or estimated values for theparameters and the previously defined symbols for the decision/controlvariables.
Rewrite the constraint, if necessary, so that all terms involving the decision
variables are on the left side of the relationship, with only a constant valueon the right side
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New Office Furniture Example
If New Office has only 2000 pounds (746.5 kg) of raw steel available for
production.
7 D + 3 C + 1.5 M 2000
Products
Desks
Chairs
Molded Steel
Raw Steel Used
7 pounds (2.61 kg.)
3 pounds (1.12 kg.)
1.5 pounds (0.56 kg.)
D: amount of desks (number)C: amount of chairs (number)M: amount of molded steel (pound)
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Special constraints or Variable Constraint
Variable Constraint
Non negativity constraintLower bound constraintUpper bound constraint
Integer constraintBinary constraint
Mathematical Expression
X 0X L (a number other than 0)X U
X = integerX = 0 or 1
Writing Constraints
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No production can be negative;
D 0, C 0, M 0
To satisfy contract commitments;• at least 100 desks, and
• due to the availability of seat cushions, no more than500 chairs must be produced.
D 100, C 500
Quantities of desks and chairs produced during theproduction must be integer valued.
D, C integers
New Office Furniture Example
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Example Mathematical ModelMAXIMIZE Z = 50 D + 30 C + 6 M (Total Profit)
SUBJECT TO: 7 D + 3 C + 1.5 M 2000 (Raw Steel)
D 100 (Contract)
C 500 (Cushions)
D 0, C 0, M 0 (Nonnegativity)
D and C are integers
Best or Optimal Solution:
100 Desks, 433 Chairs,0.67 pounds Molded SteelTotal Profit: $17,994
t
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xamp e - e ta ar wareStores
Problem Statement
Delta Hardware Stores is a
regional retailer withwarehouses in three cities inCalifornia
San Jose Fresno
Azusa
D lt H d
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Delta HardwareStores
Problem Statement
Each month, Deltarestocks itswarehouses with
its own brand ofpaint.
Delta has its ownpaintmanufacturing
plant inPhoenix
, Arizona.
San Jose
Fresno
Azusa
Phoenix
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Although the plant’s production capacity is sometime inefficient to
meet monthly demand, a recent feasibility study commissioned byDelta found that it was not cost effective to expand production
capacity at this time.
To meet demand, Delta subcontracts with a national paintmanufacturer to produce paint under the Delta label and deliver it
(at a higher cost) to any of its three California warehouses.
Delta Hardware StoresProblem Statement
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Given that there is to be no expansion of plant capacity,the problem is to determine a least cost distributionscheme of paint produced at its manufacturing plant andshipments from the subcontractor to meet the demands of
its California warehouses.
Delta Hardware Stores
Problem Statement
D lt H d St
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Decision maker has no control over demand, production capacities, or unitcosts.
The decision maker is simply being asked,
“How much paint should be shipped this month (note the time frame) fromthe plant in Phoenix to San Jose, Fresno, and Asuza”
and
“How much extra should be purchased from the subcontractor and sent toeach of the three cities to satisfy their orders?”
Delta Hardware StoresVariable Definition
D lt H d St Decision/Control Variables
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X1 : amount of paint shipped this month from Phoenix to San Jose
X2 : amount of paint shipped this month from Phoenix to Fresno
X3 : amount of paint shipped this month from Phoenix to Azusa
X4 : amount of paint subcontracted this month for San Jose
X5 : amount of paint subcontracted this month for Fresno
X6 : amount of paint subcontracted this month for Azusa
Delta Hardware Stores: Decision/Control Variables
N t k M d l
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NationalSubcontractor
San Jose
Fresno
Azusa Phoenix
Network Model
D lt H d St
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The objective is to minimize the total overall monthly costs of
manufacturing, transporting and subcontracting paint,
The constraints are (subject to):
The Phoenix plant cannot operate beyond its capacity;
The amount ordered from subcontractor cannot exceed a
maximum limit;
The orders for paint at each warehouse will be fulfilled.
Delta Hardware Stores
D lt H d St
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To determine the overall costs:
The manufacturing cost per 1000 gallons of paint at the plant inPhoenix
- (M)
The procurement cost per 1000 gallons of paint from National
Subcontractor- (C)
The respective truckload shipping costs form Phoenix to San Jose,Fresno, and Azusa- (T1, T2, T3)
The fixed purchase cost per 1000 gallons from the subcontractor toSan Jose, Fresno, and Azusa(S1, S2, S3)
Delta Hardware Stores
Delta Hardware Stores: Objective Function
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MINIMIZE (M + T1) X1 + (M + T2) X2 + (M + T3) X3 +
(C + S1) X4 + (C + S2) X5 + (C + S3) X6
Delta Hardware Stores: Objective Function
Where:
Manufacturing cost at the plant in Phoenix: M
Procurement cost from National Subcontractor: C
Truckload shipping costs from Phoenix to San Jose, Fresno, and Azusa: T1, T2, T3
Fixed purchase cost from the subcontractor to San Jose, Fresno, and Azusa: S1, S2, S3
X1 : amount of paint shipped this month from Phoenix to San Jose
X2 : amount of paint shipped this month from Phoenix to Fresno
X3 : amount of paint shipped this month from Phoenix to Azusa
X4 : amount of paint subcontracted this month for San Jose
X5 : amount of paint subcontracted this month for Fresno
X6 : amount of paint subcontracted this month for Azusa
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To write to constraints, we need to know:
The capacity of the Phoenix plant(Q1)
The maximum number of gallons available from thesubcontractor
(Q2) The respective orders for paint at the warehouses in San Jose,
Fresno, and Azusa(R1, R2, R3)
Delta Hardware Stores
Constraints
Delta Hardware Stores
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The number of truckloads shipped out from Phoenix cannot exceed the
plant capacity:X1 + X2 + X3 Q1
The number of thousands of gallons ordered from the subcontratorcannot exceed the order limit:X4 + X5 + X6 Q2
The number of thousands of gallons received at each warehouse equalsthe total orders of the warehouse:X1 + X4 = R1X2 + X5 = R2
X3 + X6 = R3
All shipments must be nonnegative and integer:X1, X2, X3, X4, X5, X6 0X1, X2, X3, X4, X5, X6 integer
Delta Hardware StoresConstraints
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Respective Orders: R1 = 4000, R2 = 2000, R3 = 5000 (gallons)
Capacity: Q1 = 8000, Q2 = 5000 (gallons)
Subcontractor price per 1000 gallons: C = $5000
Cost of production per 1000 gallons: M = $3000
Delta Hardware StoresData Collection and Model Selection
Delta Hardware Stores
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Transportation costs per 1000 gallons
Subcontractor: S1 = $1200; S2 = $1400; S3 = $1100
Phoenix Plant: T1 = $1050; T2 = $750; T3 = $650
Delta Hardware StoresData Collection and Model Selection
Delta Hardware Stores
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Min (3000+1050)X1+(3000+750)X2+(3000+650)X3+(5000+1200)X4+(5000+1400)X5+(5000+1100)X6
Ou
MIN 4050 X1 + 3750 X2 + 3650 X3 + 6200 X4 + 6400 X5 + 6100 X6
SUBJECT TO: X1 + X2 + X3 8000 (Plant Capacity)X4 + X5 + X6 5000 (Upper Bound - order from subcontracted)
X1 + X4 = 4000 (Demand in San Jose)
X2 + X5 = 2000 (Demand in Fresno)
X3 + X6 = 5000 (Demand in Azusa)
X1, X2, X3, X4, X5, X6 0 (non negativity)X1, X2, X3, X4, X5, X6 integer
Delta Hardware StoresOperations Research Model
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X1 = 1,000 gallons
X2 = 2,000 gallonsX3 = 5,000 gallons
X4 = 3,000 gallons
X5
= 0
X6 = 0
Cost = $48,400
Delta Hardware Stores
Solutions
Case em Logística – Encontrar um Modelo de Pesquisa
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Uma empresa está planejando expandir suas atividades abrindo doisnovos CD’s, sendo que há três Locais sob estudo para a instalação
destes CD’s (Figura 1 adiante). Quatro Clientes devem ter atendidassuas Demandas (Ci): 50, 100, 150 e 200.
As Capacidades de Armazenagem (A j) em cada local são: 350, 300 e 200.Os Investimentos Iniciais em cada CD são: $50, $75 e $90. Os CustosUnitários de Operação em cada CD são: $5, $3 e $2.
Admita que quaisquer dois locais são suficientes para atender toda ademanda existente, mas o Local 1 só pode atender Clientes 1, 2 e 4; oLocal 3 pode atender Clientes 2, 3 e 4; enquanto o Local 2 podeatender todos os Clientes. Os Custos Unitários de Transporte do CDque pode ser construído no Local i ao Cliente j (Cij) estão dados naFigura 1.
Deseja-se selecionar os locais apropriados para a instalação dos CD’s deforma a minimizar o custo total de investimento, operação edistribuição.
Case em Logística Encontrar um Modelo de PesquisaOperacional para a Expansão de Centros de Distribuição - CD
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Rede Logística, com Demandas (Clientes), Capacidades (Armazéns) eCustos de Transporte (Armazém-Cliente)
A1=350
C2 = 100
C1 = 50
A2
=300
C3=150
A3=200
C4=200
C12
=9
C14=12
C24
=4
C34
=7
C23
=11
C33=13
C32
=2
C22
=7
C21
=10
C11
=13
Figura 1
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Variáveis de Decisão/Controle:
Xij = Quantidade enviada do CD i ao Cliente j
Li é variável binária, i {1, 2, 3} sendo
Li =
1, se o CD i for instalado
0, caso contrário
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Modelagem
Função Objetivo: Minimizar CT = Custo Total de Investimento+ Operação + Distribuição
CT = 50L1 + 5(X11 + X12 + X14) + 13X11 + 9X12 + 12X14 +
+ 75L2 + 3(X21+X22+X23+X24) + 10X21+7X22+11X23+4X24 ++ 90L3 + 2(X32 + X33 + X34) + 2X32 + 13X33 + 7X34
Cancelando os termos semelhantes, tem-se
CT = 50L1 + 75L2 + 90L3 + 18X11 + 14X12 + 17X14 + 13X21+
+ 10X22+14X23+7X24 + 4X32 + 15X33 + 9X34
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Restrições: sujeito a
X11 + X12 + X14 350L1
X21 + X22 + X23 + X24 300L2
X32 + X33 + X34 200L3
L1 + L2 + L3 = 2 Instalar 2 CD’s
X11 + X21 = 50X12 + X22 + X32 = 100
X23 + X33 = 150
X14 + X24 + X34 = 200Xij 0
Li {0, 1}
Produção
Demanda
Não - Negatividade
Integralidade