Pesquisa Operacional faz diferença no desempenho de organizações?

1. Pesquisa Operacional Aplicada à LogísticaProf. Fernando Augusto Silva Marinswww.feg.unesp.br/~fmarinsfmarins@feg.unesp.br

2. Pesquisa Operacional faz diferença no desempenho de organizações?

3. Finalistas do Prêmio Edelman INFORMS 2007

4. Questões Logísticas (Pesquisa Operacional)

5. Delta Hardware StoresProblem Statement • Delta Hardware Stores is a regional retailer withwarehouses in three cities in California San Jose Fresno Azusa

6. 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

7. 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.

8. 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.

9. 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?”

10. 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

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

12. Mathematical Model • 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.

13. Mathematical Model 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)

14. Mathematical Model: 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

15. Mathematical Model:Constraints 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)

16. Mathematical Model: 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 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

17. Mathematical Model: Data Collection 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

18. Mathematical Model: Data Collection Transportation costs $ per 1000 gallons Subcontractor: S1=$1200; S2=$1400; S3= $1100 Phoenix Plant: T1 = $1050;T2 = $750; T3 = $650

19. 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 subcont.) 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 (nonnegativity) X1, X2, X3, X4, X5, X6 integer

20. Delta Hardware StoresSolutions X1 = 1,000 gallons X2 = 2,000 gallons X3 = 5,000 gallons X4 = 3,000 gallons X5 = 0 X6 = 0 Optimum Total Cost = $48,400

21. CARLTON PHARMACEUTICALS • Carlton Pharmaceuticals supplies drugs and other medical supplies. • It has three plants in: Cleveland, Detroit, Greensboro. • It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis. • Management at Carlton would like to ship cases of a certain vaccine as economically as possible.

22. CARLTON PHARMACEUTICALS • Data • Unit shipping cost, supply, and demand • Assumptions • Unit shipping costs are constant. • All the shipping occurs simultaneously. • The only transportation considered is between sources and destinations. • Total supply equals total demand. To From Boston Richmond Atlanta St. Louis Supply Cleveland $35 30 40 32 1200 Detroit 37 40 42 25 1000 Greensboro 40 15 20 28 800 Demand 1100 400 750 750

23. CARLTON PHARMACEUTICALSNetwork presentation

24. Destinations Boston Sources 35 Cleveland 30 Richmond 40 S1=1200 32 37 40 Detroit 42 25 S2=1000 Atlanta 35 15 20 St.Louis Greensboro 28 S3= 800 D1=1100 D2=400 D3=750 D4=750

25. CARLTON PHARMACEUTICALS – Linear Programming Model • The structure of the model is: Minimize Total Shipping Cost ST [Amount shipped from a source]  [Supply at that source] [Amount received at a destination] = [Demand at that destination] • Decision variables Xij = the number of cases shipped from plant i to warehouse j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

26. The supply constraints Supply from Cleveland X11+X12+X13+X14  1200 Supply from Detroit X21+X22+X23+X24  1000 Supply from Greensboro X31+X32+X33+X34  800 X11 Cleveland X12 X31 S1=1200 X21 X13 X14 X22 X32 Detroit X23 S2=1000 X24 X33 Greensboro S3= 800 X34 Boston D1=1100 Richmond D2=400 Atlanta D3=750 St.Louis D4=750

27. Total shipment out of a supply node cannot exceed the supply at the node. Total shipment received at a destination node, must equal the demand at that node. CARLTON PHARMACEUTICAL – The complete mathematical model Minimize 35X11 + 30X12 + 40X13 + 32X14 + 37X21 + 40X22 + 42X23 + 25X24+ + 40X31+15X32 + 20X33 + 38X34 ST Supply constraints: £ £ £ = = = = X11+ X12+ X13+ X14 1200 X21+ X22+ X23+ X24 1000 X31+ X32+ X33+ X34 800 Demand constraints: X11+ X21+ X31 1100 X12+ X22+ X32 400 X13+ X23+ X33 750 X14+ X24+ X34 750 All Xij are nonnegative

28. =SUMPRODUCT(B7:E9,B15:E17) =SUM(B7:E7) Drag to cells G8:G9 =SUM(B7:B9) Drag to cells C11:E11 CARLTON PHARMACEUTICALS Spreadsheet

29. CARLTON PHARMACEUTICALS Spreadsheet MINIMIZE Total Cost SHIPMENTS Demands are met Supplies are not exceeded

30. CARLTON PHARMACEUTICALS Spreadsheet - solution

31. CARLTON PHARMACEUTICALS Sensitivity Report • Reduced costs • The unit shipment cost between Cleveland and Atlanta must be reduced by at least $5, before it would become economically feasible to utilize it • If this route is used, the total cost will increase by $5 for each case shipped between the two cities.

32. CARLTON PHARMACEUTICALS Sensitivity Report • Allowable Increase/Decrease • This is the range of optimality. • The unit shipment cost between Cleveland and Boston may increase up to $2 or decrease up to $5 with no change in the current optimal transportation plan.

33. CARLTON PHARMACEUTICALS Sensitivity Report • Shadow prices • For the plants, shadow prices convey the cost savings realized for each extra case of vaccine produced.For each additional unit available in Cleveland the total cost reduces by $2.

34. CARLTON PHARMACEUTICALS Sensitivity Report • Shadow prices • For the warehouses demand, shadow prices represent the cost savings for less cases being demanded.For each one unit decrease in demanded in Richmond, the total cost decreases by $32. • Allowable Increase/Decrease • This is the range of feasibility. • The total supply in Cleveland may increase up to $250, but doesn´t may decrease up, with no change in the current optimal transportation plan.

35. Modifications to the Transportation Problem Cases may arise that require modifications to the basic model: • Blocked Routes • Minimum shipment • Maximum shipment

36. Shipments on a Blocked Route = 0 Cases may arise that require modifications to the basic model: Blocked routes - shipments along certain routes are prohibited Remedies: • Assign a large objective coefficient to the route of the form Cij = 1,000,000 • Add a constraint to Excel solver of the form Xij = 0

37. Cases may arise that require modifications to the basic model: Blocked routes - shipments along certain routes are prohibited Remedy: - Do not include the cell representing the route in the Changing cells Shipments from Greensboro to Cleveland are prohibited Only Feasible Routes Included in Changing Cells Cell C9 is NOT Included

38. Cases may arise that require modifications to the basic model: • Minimum shipment - the amount shipped along a certain route must not fall below a pre-specified level. • Remedy: Add a constraint to Excel of the form Xij B • Maximum shipment - an upper limit is placed on the amount shipped along a certain route. • Remedy: Add a constraint to Excel of the form Xij  B

39. Uma empresa tem 3 fábricas e 4 clientes, referentes a um determinado produto, e conhece-se os dados abaixo: Problema (Desbalanceado) de Max Lucro com possibilidade de estoque remanescente 39

40. Conhecem-se os custos de se manter o produto em estoque ($/unidade estocada) nas Fábricas 1 e 2: $1 para estocagem na Fábrica 1, $2 para estocagem na Fábrica 2. Sabe-se que a Fábrica 3 não pode ter estoques. Os custos de transporte ($/unidade) são: Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo. Problema (Desbalanceado) de Max de Lucro com possibilidade de estoque remanescente 40

41. Problema Desafio Problema (Desbalanceado) de Maximização de Lucro com possibilidade de multa devido a falta de produto Uma empresa tem fábricas onde fabrica o mesmo produto. Existem depósitos regionais e os preços pagos pelos consumidores são diferentes em cada caso. Tendo em vista os dados das tabelas a seguir, qual o melhor programa de produção e distribuição? Sabe-se que o Cliente 3 é preferencial (tem que ser atendido totalmente). Além disso, não é economicamente viável entregar o produto da Fábrica A ao Cliente 4.

42. *M = valor muito grande, pois C3 é preferencial Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto 42

43. Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto *M = valor muito grande, pois não é viável a entrega Local de Local de Locais de Venda Locais de Venda Fabrica Fabrica çã çã o o C C C C C C C C 1 1 2 2 3 3 4 4 F F 3 9 5 *M 1 1 F F 1 1 7 4 6 2 2 F F 5 8 3 4 3 3 Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo.

44. Modelo de PO para a Expansão de Centros de Distribuição • 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 (slide 67). • Deseja-se selecionar os locais apropriados para a instalação dos CD’s de forma a minimizar o custo total de investimento, operação e distribuição.

45. 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 Figura 1

46. 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

47. 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

48. 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