Aula 9 - PO Aplicada à Logística
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Transcript of Aula 9 - PO Aplicada à Logística
1
Pesquisa Operacional
Aplicada à Logística
Prof. Fernando Augusto Silva Marinswww.feg.unesp.br/~fmarins
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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.
Delta Hardware StoresProblem Statement
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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.
Delta Hardware StoresProblem Statement
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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
CARLTON PHARMACEUTICALS
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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
CARLTON PHARMACEUTICALS
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 $5 for each case shipped
between the two cities.
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CARLTON PHARMACEUTICALS Sensitivity Report
Adjustable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$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
– 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 $5 with no change in the current optimal transportation plan.
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CARLTON PHARMACEUTICALS Sensitivity Report
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$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 $2.
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CARLTON PHARMACEUTICALS Sensitivity Report
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$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 $250, but doesn´t may decrease up, with no change in the current optimal transportation plan.
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Cases may arise that require modifications to the basic model:- Blocked Routes- Minimum shipment- Maximum shipment
Modifications to the Transportation Problem
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Blocked routes - shipments along certain routes are prohibited
Remedies:– Assign a large objective coefficient to the route
of the form Cij = 1,000,000
– Add a constraint to Excel solver of the form Xij = 0
Cases may arise that require modifications to the basic model:
Shipments on a Blocked
Route = 0
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Blocked routes - shipments along certain routes are prohibited
Remedy:
- Do not include the cell representing the route in the Changing cells
Cases may arise that require modifications to the basic model:
Only Feasible Routes Included in Changing Cells
Cell C9 is NOT Included
Shipments from Greensboroto Cleveland are prohibited
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• Minimum shipment - the amount shipped along a certain route must not fall below a pre-specified level.
–Remedy: Add a constraint to Excel of the form Xij B
• Maximum shipment - an upper limit is placed on the amount shipped along a certain route.
–Remedy: Add a constraint to Excel of the form Xij B
Cases may arise that require modifications to the basic model:
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Problema (Desbalanceado) de Max Lucro com possibilidade de estoque remanescente
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Uma empresa tem 3 fábricas e 4 clientes, referentes a um determinado produto, e conhece-se os dados abaixo:
Fábrica
Capacidade
mensal da
produção
Custo de
produção
($/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
Local de Locais de Venda
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
Local de Locais de Venda
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