Sumário Introdução à Pesquisa Operacional (P.O.) Impacto da P.O. na Logística

1. Pesquisa Operacional Aplicada à LogísticaProf. Fernando Augusto Silva Marins fmarins@feg.unesp.brwww.feg.unesp.br/~fmarins

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 1984-2007

9. FINALISTAS EDELMAN 1984-2007

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

13. 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.

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 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)

16. 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

17. 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)

18. 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

19. 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

20. Example Mathematical Model MAXIMIZE Z = 50D + 30C + 6M(Total Profit) SUBJECT TO: 7D + 3C + 1.5M £ 2000(Raw Steel) D ³ 100(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

21. Example - Delta Hardware StoresProblem Statement Delta Hardware Stores is aregional retailer withwarehouses in three cities in California San Jose Fresno Azusa

22. 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

23. 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.

24. 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.

25. 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?”

26. 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

27. Network Model National Subcontractor X4 San Jose X5 X6 Fresno X2 X1 X3 Azusa Phoenix

28. 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.

29. 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)

30. 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

31. 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)

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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.

38. 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

39. 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

40. 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

41. 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