Pesquisa Operacional Aplicada à Logística

Transcrição

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

2 Pesquisa Operacional faz diferença no desempenho de organizações? 2

3 Finalistas do Prêmio Edelman 3 INFORMS 2007

4 Questões Logísticas (Pesquisa Operacional) 4

5 Delta Hardware Stores Problem Statement San Jose Fresno Delta Hardware Stores is a regional retailer with warehouses in three cities in California Azusa 5

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 Azusa Phoenix 6

7 Delta Hardware Stores Problem 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. 7

8 Delta Hardware Stores Problem 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. 8

9 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 Delta Hardware Stores Variable Definition How much extra should be purchased from the subcontractor and sent to each of the three cities to satisfy their orders? 9

10 Decision/Control Variables: 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 10

11 Network Model San Jose National Subcontractor Fresno Azusa Phoenix 11

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; 12 The orders for paint at each warehouse will be fulfilled.

13 Mathematical Model 13 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 )

14 Mathematical Model: Objective Function 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 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 14

15 Mathematical Model: Constraints 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 ) 15

16 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: 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 = $

18 Mathematical Model: Data Collection 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 18

19 Delta Hardware Stores Operations Research Model Ou Min ( )X 1 +( )X 2 +( )X 3 +( )X ( )X 5 + ( )X 6 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 19

20 Delta Hardware Stores Solutions 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 20

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. 21

22 CARLTON PHARMACEUTICALS Data Unit shipping cost, supply, and demand To From Boston Richmond Atlanta St. Louis Supply Cleveland $ Detroit Greensboro 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. 22

23 CARLTON PHARMACEUTICALS Network presentation 23

24 Sources Cleveland S 1 =1200 Detroit S 2 =1000 Destinations D 1 =1100 D 2 =400 Boston Richmond Atlanta D 3 =750 Greensboro 24 S 3 = 800 D 4 =750 St.Louis

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 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) 25

26 Supply from Cleveland X11+X12+X13+X Supply from Detroit X21+X22+X23+X Supply from Greensboro X31+X32+X33+X Cleveland S 1 =1200 Detroit S 2 =1000 X11 X12 X21 X13 X14 X22 X23 The supply constraints X31 X32 D 1 =1100 Boston Richmond D 2 =400 Atlanta X24 X33 D 3 =750 Greensboro 26 S 3 = 800 X34 D 4 =750 St.Louis

27 CARLTON PHARMACEUTICAL The complete mathematical model Minimize 35X X X X X X X X X31+15X X X34 ST Supply constraints: Total shipment out of a supply node cannot exceed the supply at the node. X11+ X12+ X13+ X X21+ X22+ X23+ X X31+ X32+ X33+ X Demand constraints: X11+ X21+ X31 = 1100 X12+ X22+ X32 = 400 X13+ X23+ X33 = 750 X14+ X24+ X34 = All Xij are nonnegative Total shipment received at a destination node, must equal the demand at that node.

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

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

30 CARLTON PHARMACEUTICALS Spreadsheet - solution SOLUTION MINIMUM COST CARLTON PHARMACEUTICALS SHIPMENTS (CASES) BOSTON RICHMOND ATLANTA ST. LOUIS SHIPPED CLEVELAND DETROIT GREENSBORO RECEIVED INPUT COST (PER CASE) BOSTON RICHMOND ATLANTA ST. LOUIS SUPPLY CLEVELAND DETROIT GREENSBORO DEMAND

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. Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$7 CLEVELAND BOSTON $C$7 CLEVELAND RICHMOND $D$7 CLEVELAND ATLANTA E+30 5 $E$7 CLEVELAND ST. LOUIS E+30 9 $B$8 DETROIT BOSTON $C$8 DETROIT RICHMOND E+30 8 $D$8 DETROIT ATLANTA E+30 5 $E$8 DETROIT ST. LOUIS E+30 $B$9 GREENSBORO BOSTON E $C$9 GREENSBORO RICHMOND $D$9 GREENSBORO ATLANTA E+30 $E$9 GREENSBORO ST. LOUIS E

32 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. Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$7 CLEVELAND BOSTON $C$7 CLEVELAND RICHMOND $D$7 CLEVELAND ATLANTA E+30 5 $E$7 CLEVELAND ST. LOUIS E+30 9 $B$8 DETROIT BOSTON $C$8 DETROIT RICHMOND E+30 8 $D$8 DETROIT ATLANTA E+30 5 $E$8 DETROIT ST. LOUIS E+30 $B$9 GREENSBORO BOSTON E $C$9 GREENSBORO RICHMOND $D$9 GREENSBORO ATLANTA E+30 $E$9 GREENSBORO ST. LOUIS E+30 20

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. Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$7 CLEVELAND SHIPPED $G$8 DETROIT SHIPPED E+30 0 $G$9 GREENSBORO SHIPPED $B$11 RECEIVED BOSTON $C$11 RECEIVED RICHMOND $D$11 RECEIVED ATLANTA $E$11 RECEIVED ST. LOUIS

34 CARLTON PHARMACEUTICALS Sensitivity Report 34 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$7 CLEVELAND SHIPPED $G$8 DETROIT SHIPPED E+30 0 $G$9 GREENSBORO SHIPPED $B$11 RECEIVED BOSTON $C$11 RECEIVED RICHMOND $D$11 RECEIVED ATLANTA $E$11 RECEIVED ST. LOUIS 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 35

36 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 C ij = 1,000,000 Add a constraint to Excel solver of the form X ij = 0 Shipments on a Blocked Route = 0 36

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 37 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 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 38

39 Problema (Desbalanceado) de Max Lucro com possibilidade de estoque remanescente Uma empresa tem 3 fábricas e 4 clientes, referentes a um determinado produto, e conhece-se os dados abaixo: Capacidade Custo de Preço de Demanda mensal da produção venda Fábrica Cliente mensal produção ($/unidade) ($/unidade) F C F C F C C Total 250 Total

40 Problema (Desbalanceado) de Max de Lucro com possibilidade de estoque remanescente 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 de Locais de Venda 40 Fabricação C 1 C 2 C 3 C 4 F F F Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para 40 resolvê-lo.

41 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 (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 CD s de forma a minimizar o custo total de investimento, operação e distribuição. 41

42 Rede Logística, com Demandas (Clientes), Capacidades (Armazéns) e Custos de Transporte (Armazém-Cliente) A 1 =350 C 2 = 100 C 12 =9 C 11 =13 C 22 =7 C 14 =12 C 1 = 50 C 21 =10 A 2 =300 C 23 =11 C 32 =2 C 24 =4 C 3 =150 C 4 =200 C 34 =7 C 33 =13 A 3 =200 Figura 1 42

43 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 43

44 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 44

45 Restrições: sujeito a X 11 + X 12 + X L 1 X 21 + X 22 + X 23 + X L 2 Produção X 32 + X 33 + X L 3 L 1 + L 2 + L 3 = 2 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} Não - Negatividade Integralidade Instalar 2 CD s Demanda 45

46 Check Point 10 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. 46

47 Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto Fábrica Capacidade mensal da produção Cliente Multas por falta ($/unidade) Demanda mensal Preço de venda ($/unidade) F 1 80 C F C F C 3 *M F C Total 480 Total *M = valor muito grande, pois C 3 é preferencial 47

48 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 Locais de Venda Fabrica çã o C 1 C 2 C 3 C 4 F *M F F F 3 F Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo. 48