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. Sumário Introdução à Pesquisa Operacional (P.O.) Impacto da P.O. na Logística Modelagem e Softwares Exemplos Cases em Logística. Pesquisa Operacional

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Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Pesquisa Operacional Aplicada à LogísticaProf. Fernando Augusto Silva Marins [email protected]/~fmarins


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Sumário

  • Introdução à Pesquisa Operacional (P.O.)

    • Impacto da P.O. na Logística

    • Modelagem e Softwares

      • Exemplos

      • Cases em Logística


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Pesquisa Operacional

Operations Research

Operational Research

Management Sciences


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

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”


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

PROCESSO DE DECISÃO

Abordagem Qualitativa: Problemas simples e experiência do decisor

Abordagem Quantitativa: Problemas complexos, ótica científica e uso de métodos quantitativos.


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Pesquisa Operacional faz diferença no desempenho de organizações?


Resultados finalistas do pr mio edelman

Resultados - finalistas do Prêmio Edelman

INFORMS 2007


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

FINALISTAS EDELMAN 1984-2007


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

FINALISTAS EDELMAN 1984-2007


Como construir modelos matem ticos

Como construir Modelos Matemáticos?


Classification of mathematical models

Classification of Mathematical Models

  • Classification by the model purpose

    • Optimization models

    • Prediction models

  • Classification by the degree of certainty of the data in the model

    • Deterministic models

    • Probabilistic (stochastic) models


Mathematical modeling

Mathematical Modeling

A constrained mathematical model consists of

  • An objective: Function to be optimised with one or more Control /Decision Variables

    Example: Max 2x – 3y; Min x + y

  • One or more constraints: Functions (“£”, “³”, “=”) with one or more Control /Decision Variables

    Examples: 3x + y £ 100; x - 4y ³ 100; x + y = 10;


New office furn i ture example

New Office Furniture Example

Raw Steel Used

7 pounds (2.61 kg.)

3 pounds (1.12 kg.)

1.5 pounds (0.56 kg.)

Products

Desks

Chairs

Molded Steel

Profit

$50

$30

$6 / pound

1 pound (troy) = 0.373242 kg.


Defining control decision variables

Defining Control/Decision Variables

  • Ask, “Does the decision maker have the authority to decide the numerical value (amount) of the item?”

  • If the answer “yes” it is a control/decision variable.

  • By very precise in the units (and if appropriate, the time frame) of each decision variable.

D: amount of desks (number)

C: amount of chairs (number)

M: amount of molded steel (pound)


Objective function

Objective Function

  • The objective of all optimization models, is to figure out how to do the best you can with what you’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)


Writing constraints

Writing Constraints

  • Create a limiting condition for each scarce resource:(amount of a resource required) (“£”, “³”, “=”) (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 the parameters and the previously defined symbols for the decision/control variables.

  • 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 value on the right side


New office furn i ture example1

New Office Furniture Example

If NewOffice has only 2000 pounds (746.5 kg) of raw steel available for production.

£

7D + 3C + 1.5M

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)


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Writing Constraints

  • Special constraints or Variable Constraint

Variable Constraint

Nonnegativity constraint

Lower bound constraint

Upper bound constraint

Integer constraint

Binary constraint

Mathematical Expression

X ³ 0

X ³ L (a number other than 0)

X £ U

X = integer

X = 0 or 1


New office furn i ture example2

New Office Furniture Example

  • To satisfy contract commitments;

  • at least 100 desks, and

  • due to the availability of seat cushions, no more than 500 chairs must be produced.

  • D ³ 100, C £ 500

  • Quantities of desks and chairs produced during the production must be integer valued.

  • D, C integers

  • No production can be negative;

    D ³ 0, C ³ 0, M ³ 0


Example mathematical model

Example Mathematical Model

MAXIMIZE Z = 50D + 30C + 6M(Total Profit)

SUBJECT TO: 7D + 3C + 1.5M £ 2000(Raw Steel)

D³ 100(Contract)

C£ 500(Cushions)

D ³ 0, C ³ 0, M ³ 0(Nonnegativity)

D and C are integers

Best or Optimal Solution:

100 Desks, 433 Chairs,

0.67 pounds Molded Steel

Total Profit: $17,994


Example delta hardware stores problem statement

Example - Delta Hardware StoresProblem Statement

Delta Hardware Stores is aregional retailer withwarehouses in three cities in California

San Jose

Fresno

Azusa


Delta hardware stores problem statement

Delta Hardware StoresProblem Statement

  • Each month, Delta restocks its warehouses with its own brand of paint.

  • Delta has its own paint manufacturing plant in Phoenix, Arizona.

San Jose

Fresno

Phoenix

Azusa


Delta hardware stores problem statement1

Delta Hardware StoresProblem Statement

  • Although the plant’s production capacity is sometime inefficient to meet monthly demand, a recent feasibility study commissioned by Delta found that it was not cost effective to expand production capacity at this time.

  • To meet demand, Delta subcontracts with a national paint manufacturer to produce paint under the Delta label and deliver it (at a higher cost) to any of its three California warehouses.


Delta hardware stores problem statement2

Delta Hardware StoresProblem Statement

  • Given that there is to be no expansion of plant capacity, the problem is to determine a least cost distribution scheme of paint produced at its manufacturing plant and shipments from the subcontractor to meet the demands of its California warehouses.


Delta hardware stores variable definition

Delta Hardware StoresVariable Definition

Decision maker has no control over demand, production capacities, or unit costs.

The decision maker is simply being asked,

“How much paint should be shipped this month (note the time frame) from the plant in Phoenix to San Jose, Fresno, and Asuza”

and

“How much extra should be purchased from the subcontractor and sent to each of the three cities to satisfy their orders?”


Delta hardware stores decision control variables

Delta Hardware Stores: 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


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Network Model

National

Subcontractor

X4

San Jose

X5

X6

Fresno

X2

X1

X3

Azusa

Phoenix


Delta hardware stores

Delta Hardware Stores

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 stores1

Delta Hardware Stores

To determine the overall costs:

  • The manufacturing cost per 1000 gallons of paint at the plant in Phoenix

    - (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 to San Jose, Fresno, and Azusa(S1, S2, S3)


Delta hardware stores objective function

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

MINIMIZE(M + T1) X1 +(M + T2) X2 +(M + T3) X3 +

(C + S1) X4 +(C + S2) X5 + (C + S3) X6

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 constraints

Delta Hardware StoresConstraints

To write to constraints, we need to know:

  • The capacity of the Phoenix plant(Q1)

  • The maximum number of gallons available from the subcontractor(Q2)

  • The respective orders for paint at the warehouses in San Jose, Fresno, and Azusa(R1, R2, R3)


Delta hardware stores constraints1

Delta Hardware StoresConstraints

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 subcontrator cannot exceed the order limit:X4 + X5 + X6 £Q2

The number of thousands of gallons received at each warehouse equals the total orders of the warehouse: X1 + X4 = R1 X2 + X5 = R2 X3 + X6 = R3

All shipments must be nonnegative and integer: X1, X2, X3, X4, X5, X6 ³ 0 X1, X2, X3, X4, X5, X6 integer


Delta hardware stores data collection and model selection

Delta Hardware StoresData Collection and Model Selection

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 stores data collection and model selection1

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 stores operations research model

Delta Hardware StoresOperations ResearchModel

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, X6³ 0 (non negativity)

X1, X2, X3, X4, X5, X6 integer


Delta hardware stores solutions

Delta Hardware StoresSolutions

X1 = 1,000 gallons

X2 = 2,000 gallons

X3 = 5,000 gallons

X4 = 3,000 gallons

X5 = 0

X6 = 0

Cost = $48,400


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Case em Logística – Encontrar um Modelo de Pesquisa Operacional para a Expansão de Centros de Distribuição - CD

Uma empresa está planejando expandir suas atividades abrindo dois novos CD’s, sendo que há três Locais sob estudo para a instalação destes CD’s (Figura 1 adiante). Quatro Clientes devem ter atendidas suas Demandas (Ci): 50, 100, 150 e 200.

As Capacidades de Armazenagem (Aj) em cada local são: 350, 300 e 200. Os Investimentos Iniciais em cada CD são: $50, $75 e $90. Os Custos Unitários de Operação em cada CD são: $5, $3 e $2.

Admita que quaisquer dois locais são suficientes para atender toda a demanda existente, mas o Local 1 só pode atender Clientes 1, 2 e 4; o Local 3 pode atender Clientes 2, 3 e 4; enquanto o Local 2 pode atender todos os Clientes. Os Custos Unitários de Transporte do CD que pode ser construído no Local i ao Cliente j (Cij) estão dados na Figura 1.

Deseja-se selecionar os locais apropriados para a instalação dos CD’s de forma a minimizar o custo total de investimento, operação e distribuição.


Rede log stica com demandas clientes capacidades armaz ns e custos de transporte armaz m cliente

Figura 1

Rede Logística, com Demandas (Clientes),Capacidades (Armazéns) e Custos de Transporte (Armazém-Cliente)

A1=350

C2 = 100

C12=9

C11=13

C22=7

C21=10

A2 =300

C14=12

C1 = 50

C32=2

C23=11

C3=150

C24=4

C33=13

C34=7

C4=200

A3=200


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

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

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


Sum rio introdu o pesquisa operacional p o impacto da p o na log stica

Produção

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 = 50

X12 + X22 + X32 = 100

X23 + X33 = 150

X14 + X24 + X34 = 200

Xij 0

Li {0, 1}

Demanda

Não - Negatividade

Integralidade


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