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Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins

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Apresentação em tema: "Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins"— Transcrição da apresentação:

1 Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins

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

3 Pesquisa Operacional Operations Research Operational Research Management Sciences

4 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

5 PROCESSO DE DECISÃO Abordagem Qualitativa: Problemas simples e experiência do decisor Abordagem Quantitativa: Problemas complexos, ótica científica e uso de métodos quantitativos.

6 Pesquisa Operacional faz diferença no desempenho de organizações?

7 Resultados - finalistas do Prêmio Edelman INFORMS 2007

8 FINALISTAS EDELMAN

9

10 Como construir Modelos Matemáticos?

11 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

12 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 (,, =) with one or more Control /Decision Variables Examples: 3x + y 100; x - 4y 100; x + y 10;

13 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) = kg.

14 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)

15 Objective Function The objective of all optimization models, is to figure out how to do the best you can with what youve 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)

16 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 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

17 New Office Furniture Example If New Office has only 2000 pounds (746.5 kg) of raw steel available for production. 7 D + 3 C M2000 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)

18 Special constraints or Variable Constraint Variable Constraint Non negativity constraint Lower bound constraint Upper bound constraint Integer constraint Binary constraint Mathematical Expression X 0 X L (a number other than 0) X U X = integer X = 0 or 1 Writing Constraints

19 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 than 500 chairs must be produced. D 100, C 500 Quantities of desks and chairs produced during the production must be integer valued. D, C integers New Office Furniture Example

20 Example Mathematical Model MAXIMIZE Z = 50 D + 30 C + 6 M(Total Profit) SUBJECT TO: 7 D + 3 C 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 Steel Total Profit: $17,994

21 Example - Delta Hardware Stores Problem Statement Delta Hardware Stores is a regional retailer with warehouses in three cities in California San Jose Fresno Azusa

22 Delta Hardware Stores Problem 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 Azusa Phoenix

23 Although the plants 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 Statement

24 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 Problem Statement

25 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 Variable Definition

26 X 1 : amount of paint shipped this month from Phoenix to San Jose X 2 : amount of paint shipped this month from Phoenix to Fresno X 3 : amount of paint shipped this month from Phoenix to Azusa X 4 : amount of paint subcontracted this month for San Jose X 5 : amount of paint subcontracted this month for Fresno X 6 : amount of paint subcontracted this month for Azusa Delta Hardware Stores: Decision/Control Variables

27 National Subcontractor X4X4 X5X5 X6X6 X1X1 X2X2 X3X3 San Jose Fresno Azusa Phoenix Network Model

28 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

29 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 - (T 1, T 2, T 3 ) The fixed purchase cost per 1000 gallons from the subcontractor to San Jose, Fresno, and Azusa (S 1, S 2, S 3 ) Delta Hardware Stores

30 MINIMIZE(M + T 1 ) X 1 + (M + T 2 ) X 2 + (M + T 3 ) X 3 + (C + S 1 ) X 4 + (C + S 2 ) X 5 + (C + S 3 ) X 6 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: T 1, T 2, T 3 Fixed purchase cost from the subcontractor to San Jose, Fresno, and Azusa: S 1, S 2, S 3 X 1 : amount of paint shipped this month from Phoenix to San Jose X 2 : amount of paint shipped this month from Phoenix to Fresno X 3 : amount of paint shipped this month from Phoenix to Azusa X 4 : amount of paint subcontracted this month for San Jose X 5 : amount of paint subcontracted this month for Fresno X 6 : amount of paint subcontracted this month for Azusa

31 To write to constraints, we need to know: The capacity of the Phoenix plant (Q 1 ) The maximum number of gallons available from the subcontractor (Q 2 ) The respective orders for paint at the warehouses in San Jose, Fresno, and Azusa (R 1, R 2, R 3 ) Delta Hardware Stores Constraints

32 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 0 X1, X2, X3, X4, X5, X6 integer Delta Hardware Stores Constraints

33 Respective Orders: R 1 = 4000, R 2 = 2000, R 3 = 5000 (gallons) Capacity: Q 1 = 8000, Q 2 = 5000 (gallons) Subcontractor price per 1000 gallons: C = $5000 Cost of production per 1000 gallons: M = $3000 Delta Hardware Stores Data Collection and Model Selection

34 Transportation costs per 1000 gallons Subcontractor: S 1 = $1200;S 2 = $1400;S 3 = $1100 Phoenix Plant: T 1 = $1050;T 2 = $750;T 3 = $650 Delta Hardware Stores Data Collection and Model Selection

35 Min ( )X 1 +( )X 2 +( )X 3 +( )X 4 +( )X 5 +( )X 6 Ou MIN 4050 X X X X X X 6 SUBJECT TO: X 1 + X 2 + X (Plant Capacity) X 4 + X 5 + X (Upper Bound - order from subcontracted) X 1 + X 4 = 4000 (Demand in San Jose) X 2 + X 5 = 2000 (Demand in Fresno) X 3 + X 6 = 5000 (Demand in Azusa) X 1, X 2, X 3, X 4, X 5, X 6 0 (non negativity) X 1, X 2, X 3, X 4, X 5, X 6 integer Delta Hardware Stores Operations Research Model

36 X 1 = 1,000 gallons X 2 = 2,000 gallons X 3 = 5,000 gallons X 4 = 3,000 gallons X 5 = 0 X 6 = 0 Cost = $48,400 Delta Hardware Stores Solutions

37 Uma empresa está planejando expandir suas atividades abrindo dois novos CDs, sendo que há três Locais sob estudo para a instalação destes CDs (Figura 1 adiante). Quatro Clientes devem ter atendidas suas Demandas (C i ): 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 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 (C ij ) estão dados na Figura 1. Deseja-se selecionar os locais apropriados para a instalação dos CDs de forma a minimizar o custo total de investimento, operação e distribuição. Case em Logística – Encontrar um Modelo de Pesquisa Operacional para a Expansão de Centros de Distribuição - CD

38 Rede Logística, com Demandas (Clientes), Capacidades (Armazéns) e Custos de Transporte (Armazém-Cliente) A 1 =350 C 2 = 100 C 1 = 50 A 2 =300 C 3 =150 A 3 =200 C 4 =200 C 12 =9 C 14 =12 C 24 =4 C 34 =7 C 23 =11 C 33 =13 C 32 =2 C 22 =7 C 21 =10 C 11 =13 Figura 1

39 Variáveis de Decisão/Controle: X ij = Quantidade enviada do CD i ao Cliente j L i é variável binária, i {1, 2, 3} sendo L i = 1, se o CD i for instalado 0, caso contrário

40 Modelagem Função Objetivo: Minimizar CT = Custo Total de Investimento + Operação + Distribuição CT = 50L 1 + 5(X 11 + X 12 + X 14 ) + 13X X X L 2 + 3(X 21 +X 22 +X 23 +X 24 ) + 10X 21 +7X X 23 +4X L 3 + 2(X 32 + X 33 + X 34 ) + 2X X X 34 Cancelando os termos semelhantes, tem-se CT = 50L L L X X X X X X 23 +7X X X X 34

41 Restrições: sujeito a X 11 + X 12 + X L 1 X 21 + X 22 + X 23 + X L 2 X 32 + X 33 + X L 3 L 1 + L 2 + L 3 = 2 Instalar 2 CDs X 11 + X 21 = 50 X 12 + X 22 + X 32 = 100 X 23 + X 33 = 150 X 14 + X 24 + X 34 = 200 X ij 0 L i {0, 1} Produção Demanda Não - Negatividade Integralidade


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