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

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Apresentação em tema: "1 Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins"— Transcrição da apresentação:

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

2 2 Pesquisa Operacional faz diferença no desempenho de organizações?

3 3 Finalistas do Prêmio Edelman INFORMS 2007

4 4 FINALISTAS EDELMAN

5 5

6 6 Delta Hardware Stores Problem Statement Delta Hardware Stores is a regional retailer with warehouses in three cities in California San Jose Fresno Azusa

7 7 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

8 8 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

9 9 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

10 10 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

11 11 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 Decision/Control Variables:

12 12 National Subcontractor X4X4 X5X5 X6X6 X1X1 X2X2 X3X3 San Jose Fresno Azusa Phoenix Network Model

13 13 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. Mathematical Model

14 14 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 ) Mathematical Model

15 15 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 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: 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

16 16 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 ) Mathematical Model: Constraints

17 17 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 Mathematical Model: Constraints

18 18 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 Mathematical Model: Data Collection

19 19 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 Mathematical Model: Data Collection

20 20 Min ( )X 1 +( )X 2 +( )X 3 +( )X ( )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 subcont.) 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 (nonnegativity) X 1, X 2, X 3, X 4, X 5, X 6 integer Delta Hardware Stores Operations Research Model

21 21 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 Optimum Total Cost = $48,400 Delta Hardware Stores Solutions

22 22 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.

23 23 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 FromBostonRichmondAtlantaSt. LouisSupply Cleveland $ Detroit Greensboro Demand CARLTON PHARMACEUTICALS

24 24 CARLTON PHARMACEUTICALS Network presentation

25 25 Boston Richmond Atlanta St.Louis Destinations Sources Cleveland Detroit Greensboro S 1 =1200 S 2 =1000 S 3 = 800 D 1 =1100 D 2 =400 D 3 =750 D 4 =

26 26 –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 X ij = 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) CARLTON PHARMACEUTICALS – Linear Programming Model

27 27 Boston Richmond Atlanta St.Louis D 1 =1100 D 2 =400 D 3 =750 D 4 =750 The supply constraints Cleveland S 1 =1200 X11 X12 X13 X14 Supply from Cleveland X11+X12+X13+X Detroit S 2 =1000 X21 X22 X23 X24 Supply from Detroit X21+X22+X23+X Greensboro S 3 = 800 X31 X32 X33 X34 Supply from Greensboro X31+X32+X33+X34 800

28 28 CARLTON PHARMACEUTICAL – The complete mathematical model Minimize 35X X X X X X X X X31+15X X X34 ST Supply constraints: X11+X12+X13+X X21+X22+X23+X X31+X32+X33+X34800 Demand constraints: X11+X21+X X12+ X22+X32400 X13+ X23+X33750 X14+X24+X34750 All Xij are nonnegative = 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.

29 29 CARLTON PHARMACEUTICALS Spreadsheet =SUM(B7:E9) Drag to cells C11:E11 =SUMPRODUCT(B7:E9,B15:E17) =SUM(B7:E7) Drag to cells G8:G9

30 30 MINIMIZE Total Cost SHIPMENTS Demands are met Supplies are not exceeded CARLTON PHARMACEUTICALS Spreadsheet

31 31 CARLTON PHARMACEUTICALS Spreadsheet - solution

32 32 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.

33 33 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.

34 34 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.

35 35 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.

36 36 Cases may arise that require modifications to the basic model: -Blocked Routes -Minimum shipment -Maximum shipment Modifications to the Transportation Problem

37 37 Blocked routes - shipments along certain routes are prohibited Remedies: –Assign a large objective coefficient to the route of the form C ij = 1,000,000 –Add a constraint to Excel solver of the form X ij = 0 Cases may arise that require modifications to the basic model: Shipments on a Blocked Route = 0

38 38 Blocked routes - shipments along certain routes are prohibited Remedy: - Do not include the cell representing the route in the Changing cells Cases may arise that require modifications to the basic model: Only Feasible Routes Included in Changing Cells Cell C9 is NOT Included Shipments from Greensboro to Cleveland are prohibited

39 39 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 X ij 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 X ij B Cases may arise that require modifications to the basic model:

40 40 Problema (Desbalanceado) de Max Lucro com possibilidade de estoque remanescente 40 Uma empresa tem 3 fábricas e 4 clientes, referentes a um determinado produto, e conhece-se os dados abaixo: Fábrica Capacidade mensal da produção Custo de produção ($/unidade) Cliente Demanda mensal Preço de venda ($/unidade) F1F1 8550C1C1 100 F2F2 9030C2C F3F3 7540C3C C4C Total250Total240

41 41 Problema (Desbalanceado) de Max de Lucro com possibilidade de estoque remanescente 41 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: Local deLocais de Venda FabricaçãoC1C1 C2C2 C3C3 C4C4 F1F F2F F3F Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo.

42 42 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.

43 43 Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto 43 Fábrica Capacidade mensal da produção Cliente Multas por falta ($/unidade) Demanda mensal Preço de venda ($/unidade) F1F1 80 C1C F2F2 200 C2C F3F3 100 C3C3 *M F4F4 100 C4C Total480Total490 *M = valor muito grande, pois C 3 é preferencial

44 44 Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto Local deLocais de Venda Fabricação C 1 C 2 C 3 C 4 F 1 39*M F F Local deLocais de Venda Fabricação C 1 C 2 C 3 C 4 F 1 5 F 2 14 F 3 F4F *M = valor muito grande, pois não é viável a entrega Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo.

45 45 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 (slide 67). Deseja-se selecionar os locais apropriados para a instalação dos CDs de forma a minimizar o custo total de investimento, operação e distribuição. Modelo de PO para a Expansão de Centros de Distribuição

46 46 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

47 47 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

48 48 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 X 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

49 49 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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