Aula 9 - PO Aplicada à Logística

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Pesquisa Operacional

Aplicada à Logística

Prof. Fernando Augusto Silva Marinswww.feg.unesp.br/~fmarins

[email protected]

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Pesquisa Operacional faz diferença no desempenho de organizações?

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Finalistas do Prêmio Edelman

INFORMS 2007

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• Questões Logísticas

• (Pesquisa Operacional)

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Delta Hardware StoresProblem Statement

• Delta Hardware Stores is a regional retailer with warehouses in three cities in California

San JoseFresno

Azusa

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• 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

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• 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.


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• 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.


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• 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 StoresVariable Definition

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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

Decision/Control Variables:

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NationalSubcontractor

X4

X 5

X 6

X1

X2

X3

San Jose

Fresno

Azusa Phoenix

Network Model

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• 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

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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

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Minimize (M + T1) X1 + (M + T2) X2 + (M + T3) X3 +

(C + S1) X4 + (C + S2) X5 + (C + S3) X6

Mathematical Model: Objective Function

Where:Manufacturing cost at the plant in Phoenix: MProcurement cost from National Subcontractor: CTruckload 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

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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

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• 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

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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

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Transportation costs $ per 1000 gallons

Subcontractor: S1=$1200; S2=$1400;

S3= $1100

Phoenix Plant: T1 = $1050;T2 = $750;

T3 = $650

Mathematical Model: Data Collection

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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 StoresOperations Research Model

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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

Delta Hardware StoresSolutions

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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.

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• 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


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CARLTON PHARMACEUTICALSNetwork presentation

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Boston

Richmond

Atlanta

St.Louis

Destinations

Sources

Cleveland

Detroit

Greensboro

S1=1200

S2=1000

S3= 800

D1=1100

D2=400

D3=750

D4=750

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40

42

32

35

40

30

25

3515

20

28

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– The structure of the model is:

Minimize Total Shipping CostST[Amount shipped from a source] [Supply at that source][Amount received at a destination] = [Demand at that destination]

– Decision variablesXij = 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

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Boston

Richmond

Atlanta

St.Louis

D1=1100

D2=400

D3=750

D4=750

The supply constraints

Cleveland S1=1200

X11

X12

X13

X14

Supply from Cleveland X11+X12+X13+X14 1200

DetroitS2=1000

X21

X22

X23

X24

Supply from Detroit X21+X22+X23+X24 1000

GreensboroS3= 800

X31

X32

X33

X34

Supply from Greensboro X31+X32+X33+X34 800

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CARLTON PHARMACEUTICAL –

The complete mathematical modelMinimize 35X11 + 30X12 + 40X13 + 32X14 + 37X21 + 40X22 + 42X23 + 25X24+ + 40X31+15X32

+ 20X33 + 38X34ST

Supply constraints:X11+ X12+ X13+ X14 1200

X21+ X22+ X23+ X24 1000X31+ X32+ X33+ X34 800

Demand constraints: X11+ X21+ X31 1100

X12+ X22+ X32 400X13+ X23+ X33 750

X14+ X24+ X34 750

All Xij are nonnegative

====

Total shipment out of a supply nodecannot exceed the supply at the node.

Total shipment received at a destinationnode, must equal the demand at that node.

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CARLTON PHARMACEUTICALS Spreadsheet

=SUM(B7:B9)Drag to cells

C11:E11

=SUMPRODUCT(B7:E9,B15:E17) =SUM(B7:E7)Drag to cells

G8:G9

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MINIMIZE Total Cost

SHIPMENTS

Demands are metSupplies are not exceeded

CARLTON PHARMACEUTICALS Spreadsheet

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SOLUTIONMINIMUM COST 84000

BOSTON RICHMOND ATLANTA ST. LOUIS SHIPPEDCLEVELAND 850 350 1200DETROIT 250 750 1000GREENSBORO 50 750 800

RECEIVED 1100 400 750 750

INPUTBOSTON RICHMOND ATLANTA ST. LOUIS SUPPLY

CLEVELAND 35 30 40 32 1200DETROIT 37 40 42 25 1000GREENSBORO 40 15 20 28 800

DEMAND 1100 400 750 750


COST (PER CASE)

SHIPMENTS (CASES)

CARLTON PHARMACEUTICALS Spreadsheet - solution

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CARLTON PHARMACEUTICALS Sensitivity Report

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$7 CLEVELAND BOSTON 850 0 35 2 5$C$7 CLEVELAND RICHMOND 350 0 30 5 17$D$7 CLEVELAND ATLANTA 0 5 40 1E+30 5$E$7 CLEVELAND ST. LOUIS 0 9 32 1E+30 9$B$8 DETROIT BOSTON 250 0 37 5 2$C$8 DETROIT RICHMOND 0 8 40 1E+30 8$D$8 DETROIT ATLANTA 0 5 42 1E+30 5$E$8 DETROIT ST. LOUIS 750 0 25 9 1E+30$B$9 GREENSBORO BOSTON 0 20 40 1E+30 20$C$9 GREENSBORO RICHMOND 50 0 15 17 5$D$9 GREENSBORO ATLANTA 750 0 20 5 1E+30$E$9 GREENSBORO ST. LOUIS 0 20 28 1E+30 20

– 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.

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Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$7 CLEVELAND BOSTON 850 0 35 2 5$C$7 CLEVELAND RICHMOND 350 0 30 5 17$D$7 CLEVELAND ATLANTA 0 5 40 1E+30 5$E$7 CLEVELAND ST. LOUIS 0 9 32 1E+30 9$B$8 DETROIT BOSTON 250 0 37 5 2$C$8 DETROIT RICHMOND 0 8 40 1E+30 8$D$8 DETROIT ATLANTA 0 5 42 1E+30 5$E$8 DETROIT ST. LOUIS 750 0 25 9 1E+30$B$9 GREENSBORO BOSTON 0 20 40 1E+30 20$C$9 GREENSBORO RICHMOND 50 0 15 17 5$D$9 GREENSBORO ATLANTA 750 0 20 5 1E+30$E$9 GREENSBORO ST. LOUIS 0 20 28 1E+30 20

– 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.

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ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$G$7 CLEVELAND SHIPPED 1200 -2 1200 250 0$G$8 DETROIT SHIPPED 1000 0 1000 1E+30 0$G$9 GREENSBORO SHIPPED 800 -17 800 250 0$B$11 RECEIVED BOSTON 1100 37 1100 0 250$C$11 RECEIVED RICHMOND 400 32 400 0 250$D$11 RECEIVED ATLANTA 750 37 750 0 250$E$11 RECEIVED ST. LOUIS 750 25 750 0 750

– 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.

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ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$G$7 CLEVELAND SHIPPED 1200 -2 1200 250 0$G$8 DETROIT SHIPPED 1000 0 1000 1E+30 0$G$9 GREENSBORO SHIPPED 800 -17 800 250 0$B$11 RECEIVED BOSTON 1100 37 1100 0 250$C$11 RECEIVED RICHMOND 400 32 400 0 250$D$11 RECEIVED ATLANTA 750 37 750 0 250$E$11 RECEIVED ST. LOUIS 750 25 750 0 750

– 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.

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

Modifications to the Transportation Problem

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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

Cases may arise that require modifications to the basic model:

Shipments on a Blocked

Route = 0

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Blocked routes - shipments along certain routes are prohibited

Remedy:

- Do not include the cell representing the route in the Changing cells


Only Feasible Routes Included in Changing Cells

Cell C9 is NOT Included

Shipments from Greensboroto Cleveland are prohibited

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• 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


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Problema (Desbalanceado) de Max Lucro com possibilidade de estoque remanescente

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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 100 100

F2 90 30 C2 80 110

F3 75 40 C3 20 105

C4 40 125

Total 250 Total 240

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Problema (Desbalanceado) de Max de Lucro

com possibilidade de estoque remanescente

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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 33 64

Encontrar o programa de distribuição que proporcione lucro máximo. Formule o modelo de PL e aplique o Solver do Excel para resolvê-lo.

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Problema DesafioProblema (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.

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Problema (Desbalanceado) de Max Lucro com

possibilidade de multa devido a falta de produto

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Fábrica

Capacidade

mensal da

produçãoCliente

Multas por

falta

($/unidade)

Demanda

mensal

Preço de

venda

($/unidade)

F180 C1 4 90 30

F2200 C2 5 150 32

F3100 C3 *M 150 36

F4100 C4 2 100 34

Total 480 Total 490

*M = valor muito grande, pois C3 é preferencial

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Problema (Desbalanceado) de Max Lucro com possibilidade de multa devido a falta de produto


Fabrica çã o C 1 C 2 C 3 C 4

F 1 3 9 *MF 2 1 7 6F 3 5 8 3 4


Fabrica çã o C 1 C 2 C 3 C 4

F 1 5F 2 1 4F 3

F4 7 3 8 2

*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.

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• 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.

Modelo de PO para a Expansão de Centros

de Distribuição

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Rede Logística, com Demandas (Clientes), Capacidades (Armazéns) e Custos de Transporte (Armazém-Cliente)•

A1=350 C2 = 100

C1 = 50A2 =300

C3=150

A3=200C4=200

C12=9

C14=12

C24=4

C34=7

C23=11

C33=13

C32=2

C22=7

C21=10

C11=13

Figura 1

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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

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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

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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

Documents

Aula 9 - PO Aplicada à Logística