Pesquisa Operacional Aplicada à Logística

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Pesquisa Operacional Aplicada à Logística

Prof. Fernando Augusto Silva Marins

[email protected]

www.feg.unesp.br/~fmarins

mailto:[email protected]

http://www.feg.unesp.br/~fmarins

http://www.feg.unesp.br/~fmarins

mailto:[email protected]



Sumário

Introdução à Pesquisa Operacional (P.O.)

Impacto da P.O. na Logística

Modelagem e Softwares

Exemplos

Cases em Logística



Pesquisa Operacional

Operations Research

Operational Research

Management Sciences



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”



PROCESSO DE DECISÃO

Abordagem Qualitativa: Problemas simples e experiênciado decisor

Abordagem Quantitativa: Problemas complexos, ótica

científica e uso de métodos quantitativos.



Pesquisa Operacional faz diferença nodesempenho de organizações?



Resultados - finalistas do Prêmio Edelman

INFORMS 2007



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"



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"



Como construir Modelos Matemáticos?



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



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;



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.



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)



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




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




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.)




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



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




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



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



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



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



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



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



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



NationalSubcontractor

San Jose

Fresno

Azusa Phoenix

Network Model

D lt H d St



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.


D lt H d St



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: Objective Function



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



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)


Constraints




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



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




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




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



X1 = 1,000 gallons

X2 = 2,000 gallonsX3 = 5,000 gallons

X4 = 3,000 gallons

X5

= 0

X6 = 0

Cost = $48,400


Solutions

Case em Logística – Encontrar um Modelo de Pesquisa



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



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



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



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



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

Documents

Pesquisa Operacional Aplicada à Logística