Pesquisa Operacional Aplicada à Logística Prof Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins www.feg.unesp.br/~fmarins fmarins@feg.unesp.br

Pesquisa Operacional faz diferença no desempenho de organizações? 2

Finalistas do Prêmio Edelman INFORMS 2007 3

FINALISTAS EDELMAN 1984-2007 4

FINALISTAS EDELMAN 1984-2007 5

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

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

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.

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.

Delta Hardware Stores Variable 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?”

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

Network Model San Jose National Subcontractor Fresno Azusa Phoenix X4

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.

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)

Mathematical Model: Objective Function Minimize (M + T1) X1 + (M + T2) X2 + (M + T3) X3 + (C + S1) X4 + (C + S2) X5 + (C + S3) X6 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 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

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)

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

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

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

Delta Hardware Stores Operations Research Model 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

Delta Hardware Stores Solutions 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

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.

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

CARLTON PHARMACEUTICALS Network presentation

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

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)

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

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

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

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

CARLTON PHARMACEUTICALS Spreadsheet - solution

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.

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.

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.

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.

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

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 Shipments on a Blocked Route = 0

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

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

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: Fábrica Capacidade mensal da produção Custo de produção ($/unidade) Cliente Demanda mensal Preço de venda ($/unidade) F1 85 50 C1 100 F2 90 30 C2 80 110 F3 75 40 C3 20 105 C4 125 Total 250 240 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 Fabricação C1 C2 C3 C4 F1 43 57 33 60 F2 30 49 25 47 F3 44 58 64 Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo. 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.

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) F1 80 C1 4 90 30 F2 200 C2 5 150 32 F3 100 C3 *M 36 F4 C4 2 34 Total 480 490 *M = valor muito grande, pois C3 é preferencial 43

*M = valor muito grande, pois não é viável a entrega 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 F4 7 3 8 2 Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo.

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.

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

Li = 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

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

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} Produção Demanda Não - Negatividade Integralidade