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

2. Introdução à Pesquisa Operacional (P.O.)
Sumário
Introdução à Pesquisa Operacional (P.O.)
Impacto da P.O. na Logística
Modelagem e Softwares
Exemplos
Cases em Logística

3. Pesquisa Operacional (P.O.)
Operations Research
Operational Research
Management Sciences

4. A P.O. e o Processo de Tomada de Decisão
Tomar decisões é uma tarefa básica da gestão.
Decidir: optar entre alternativas viáveis.
Papel do Decisor:
Identificar e Definir o Problema
Formular objetivo (s)
Analisar Limitações
Avaliar Alternativas  Escolher a “melhor”

5. PROCESSO DE DECISÃO
Abordagem Qualitativa: Problemas simples e experiência do decisor
Abordagem Quantitativa: Problemas complexos, ótica científica e uso de métodos quantitativos.

6. Pesquisa Operacional faz diferença no desempenho de organizações?6

7. Resultados - Finalistas do Prêmio Edelman
INFORMS 2007 7

8. FINALISTAS EDELMAN 8

9. FINALISTAS EDELMAN 9

10. Como construir Modelos Matemáticos?

11. Classification of Mathematical Models
Classification by the model purpose
Optimization models
Prediction models
Classification by the degree of certainty of the data in the model
Deterministic models
Probabilistic (stochastic) models

12. Mathematical Modeling
A constrained mathematical model consists of
An objective: Function to be optimised with one or more Control /Decision Variables
Example: Max 2x – 3y; Min x + y
One or more constraints: Functions (“£”, “³”, “=”) with one or more Control /Decision Variables
Examples: 3x + y £ 100; x - 4y ³ 100; x + y = 10;

13. New Office Furniture Example
Raw Steel Used
7 pounds (2.61 kg.)
3 pounds (1.12 kg.)
1.5 pounds (0.56 kg.)
Products
Desks
Chairs
Molded Steel
Profit
$50
$30
$6 / pound
1 pound (troy) = kg.

14. Defining Control/Decision Variables
Ask, “Does the decision maker have the authority to decide the numerical value (amount) of the item?”
If the answer “yes” it is a control/decision variable.
By very precise in the units (and if appropriate, the time frame) of each decision variable.
D: amount of desks (number)
C: amount of chairs (number)
M: amount of molded steel (pound)

15. Objective Function
The objective of all optimization models, is to figure out how to do the best you can with what you’ve got.
“The best you can” implies maximizing something (profit, efficiency...) or minimizing something (cost, time...).
Total Profit =
50 D + 30 C + 6 M
Products
Desks
Chairs
Molded Steel
Profit
$50
$30
$6 / pound
D: amount of desks (number)
C: amount of chairs (number)
M: amount of molded steel (pound)

16. Writing Constraints
Create a limiting condition for each scarce resource : (amount of a resource required) (“£”, “³”, “=”) (resource availability)
Make sure the units on the left side of the relation are the same as those on the right side.
Use mathematical notation with known or estimated values for the parameters and the previously defined symbols for the decision/control variables.
Rewrite the constraint, if necessary, so that all terms involving the decision variables are on the left side of the relationship, with only a constant value on the right side

17. New Office Furniture Example
If New Office has only 2000 pounds (746.5 kg) of raw steel available for production.
7 D + 3 C M
2000
Products
Desks
Chairs
Molded Steel
Raw Steel Used
7 pounds (2.61 kg.)
3 pounds (1.12 kg.)
1.5 pounds (0.56 kg.)
D: amount of desks (number)
C: amount of chairs (number)
M: amount of molded steel (pound)

18. Writing Constraints Special constraints or Variable Constraint
Non negativity constraint
Lower bound constraint
Upper bound constraint
Integer constraint
Binary constraint
Mathematical Expression
X ³ 0
X ³ L (number  0)
X £ U (number  0)
X = integer  {0, 1, 2, 3, 4 ,…}
X = 0 or 1

19. New Office Furniture Example
No production can be negative;
D ³ 0, C ³ 0, M ³ 0
To satisfy contract commitments;
at least 100 desks, and
due to the availability of seat cushions, no more than 500 chairs must be produced.
Quantities of desks and chairs produced during the production must be integer valued.
D ³ 100, C £ 500
D, C integers

20. Example Mathematical Model
MAXIMIZE Z = 50 D + 30 C + 6 M (Total Profit)
SUBJECT TO: 7 D C M £ (Raw Steel)
D ³ 100 (Contract)
C £ 500 (Cushions)
D ³ 0, C ³ 0, M ³ 0 (Nonnegativity)
D and C are integers
Best or Optimal Solution:
100 Desks, 433 Chairs,
0.67 pounds Molded Steel
Total Profit: $17,994
Ver Modelo no Lindo

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

22. Delta Hardware Stores Problem Statement
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

23. 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.

24. 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.

25. 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?”

26. 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

27. Network Model and Decision Variables
National
Subcontractor X4
San Jose X5 X6
Fresno X2 X1 X3
Azusa
Phoenix

28. Model Structure
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.

29. Costs 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)

30. 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

31. 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)

32. Constraints (Cont.)
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 ³ 0 X1, X2, X3, X4, X5, X6 integer

33. Data Collection and Model Selection
Respective 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

34. Data Collection and Model Selection (Cont.)
Transportation costs per 1000 gallons
Subcontractor: S1 = $1200; S2 = $1400; S3 = $1100
Phoenix Plant: T1 = $1050; T2 = $750; T3 = $650

35. Delta Hardware Stores Operations Research Model
Min ( )X1+( )X2+( )X3+( )X4+( )X5+( )X6 Ou
MIN 4050 X X X X X X6
SUBJECT TO: X1 + X2 + X3 £ (Plant Capacity)
X4 + X5 + X6 £ (Upper Bound - order from subcontracted)
X1 + X4 = (Demand in San Jose)
X2 + X5 = (Demand in Fresno)
X3 + X6 = (Demand in Azusa)
X1, X2, X3, X4, X5, X6 ³ 0 (non negativity)
X1, X2, X3, X4, X5, X6 integer

36. Delta Hardware Stores Solutions
X1 = 1,000 gallons
X2 = 2,000 gallons
X3 = 5,000 gallons
X4 = 3,000 gallons
X5 = 0
X6 = 0
Cost = $48,400
Ver Modelo no Excel

37. 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.

38. 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

39. CARLTON PHARMACEUTICALS Network presentation
Boston
Richmond
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

40. CARLTON PHARMACEUTICALS – Linear Programming Model
The structure of the model is:
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)
Minimize Total Shipping Cost ST
[Amount shipped from a source] £ [Supply at that source]
[Amount received at a destination] = [Demand at that destination]

41. Supply constraints Boston Cleveland Richmond Detroit Atlanta St.Louis
X11
X12
X13
X14
Supply from Cleveland X11+X12+X13+X14 
Supply constraints
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

42. CARLTON PHARMACEUTICAL – The complete mathematical model
Min 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

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

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

45. CARLTON PHARMACEUTICALS Spreadsheet - solution

46. 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.

47. 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.

48. 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

49. 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.
CARLTON PHARMACEUTICALS Sensitivity Report
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.

50. Modifications to the transportation problem
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 (Cij = 1,000,000)

51. Modifications to the transportation problem
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 blocked route (like as Cij = 1,000,000)
Add a constraint to Excel solver of the form Xij = 0
Shipments on a Blocked Route = 0

52. Modifications to the transportation problem
Only Feasible Routes Included in Changing Cells
Cell C9 is NOT Included
Modifications to the transportation problem
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 (Cij = 1,000,000)
Add a constraint to Excel solver of the form Xij = 0
Do not include the cell representing the rout in the Changing cells
Shipments from Greensboro
to Cleveland are prohibited

53. Modifications to the transportation problem
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.
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 ³ B.
Remedy: Add a constraint to Excel of the form Xij £ B.

54. The Capacitated Transshipment Model
Sometimes shipments to destination nodes are made through transshipment nodes.
Transshipment nodes may be
Independent intermediate nodes with no supply or demand
Supply or destination points themselves.
Transportation on arcs may be bounded by given bounds

55. The Capacitated Transshipment Model
The Linear Programming Model of this problem consists of:
Flow on arcs decision variables
Cost minimization objective function
Balance constraints on each node as follows:
Supply node – net flow out does not exceed the supply
Intermediate node – flow into the node is equal to the flow out
Demand node – net flow into the node is equal to the demand
Bound constraints on each arc. Flow cannot exceed the capacity on the arc

56. DEPOT MAX A General Network Problem
Depot Max has six stores located in the Washington D.C. area.

57. DEPOT MAX
The stores in Falls Church (FC) and Bethesda (BA) are running low on the model 5A Arcadia workstation.
DATA:
-12 5 FC
-13 6 BA

58. DEPOT MAX
The stores in Alexandria (AA) and Chevy Chase (CC) have an access of 25 units.
DATA:
-12
+10 1 5 FC AA
-13
+17 2 6 BA CC

59. DEPOT MAX
The stores in Fairfax and Georgetown are transshipment nodes with no access supply or demand of their own.
DATA:
-12 FX
+10 1 3 5 FC AA
Depot Max wishes to transport the available workstations to FC and BA at minimum total cost. GN
-13
+17 2 4 6 BA CC

60. DEPOT MAX
Data
There is a maximum limit for quantities shipped on various routes.
There are different unit transportation costs for different routes.

61. DEPOT MAX
The possible routes and the maximum flow limits are shown.
DATA: 6
-12 FX 10 8
+10 1 3 FC FC AA 3 7 8 5 7 GN
-13
+17 2 4 BA BA 10 17 CC

62. DEPOT MAX
The possible routes and the shipping unit costs are shown.
DATA: 20
-12 FX 10 7
+10 1 3 FC FC AA 5 6 12 11 7 GN
-13
+17 2 4 BA BA 15 15 CC

63. DEPOT MAX – Types of constraints
Supply nodes: Net flow out of the node]  [Supply at the node]
X12 + X13 + X15 - X21  10 (Node 1) X21 + X24 - X  17 (Node 2)
Intermediate transshipment nodes: [Total flow out of the node] = [Total flow into the node]
X34+X35 = X (Node3) X46 = X24 + X34 (Node 4) 20
-12 10 7
+10 1 3 5 7 5 6 12 11 7 15 15 2 4 6
-13
+17
Demand nodes: [Net flow into the node] = [Demand for the node]
X15 + X35 +X65 - X56 = 12 (Node 5) X46 +X56 - X65 = 13 (Node 6)

64. All variables are non-negative
DEPOT MAX
The Complete mathematical model
Min 5X X X X X X34 + 7X X X56 + 7X65
S.T. X12 + X13 + X15 – X £ 10
- X X21 + X £ 17
– X X34 + X = 0
– X24 – X X = 0
– X – X35 + X56 - X65 = -12
-X46 – X56 + X65 = -13
X12 £ 3; X15 £ 6; X21 £ 7; X24 £ 10; X34 £ 8; X35 £ 8; X46 £ 17; X56 £ 7; X65 £ 5
All variables are non-negative

65. DEPOT MAX - spreadsheet
Usando o Template Network
DEPOT MAX - spreadsheet

66. Modelo de Pesquisa Operacional
Case em Logística
Modelo de Pesquisa Operacional
Expansão de Centros de Distribuição - CD
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 do slide seguinte.
Deseja-se selecionar os locais apropriados para a instalação dos CD’s de forma a minimizar o custo total de investimento, operação e distribuição.

67. Figura 1
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

68. 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

69. Modelagem
Função Objetivo: Minimizar CT = Custo Total de Investimento + Operação + Distribuição
CT = 50L1 + 5(X11 + X12 + X14) + 13X11 + 9X X14 +
+ 75L2 + 3(X21+X22+X23+X24) + 10X21+7X22+11X23+4X24 +
+ 90L3 + 2(X32 + X33 + X34) + 2X X33 + 7X34
Cancelando os termos semelhantes, tem-se
CT = 50L1 + 75L2 + 90L3 + 18X X X X21+
+ 10X22+14X23+7X24 + 4X X33 + 9X34

70. Restrições: sujeito a X11 + X12 + X14  350L1
Ver Modelo no Lindo
Restrições: sujeito a
X11 + X12 + X14  350L1
X21 + X22 + X23 + X24  300L2
X32 + X33 + X34  200L3
L1 + L2 + L3 = 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

71. Solução do Case em Logística
OBJECTIVE FUNCTION VALUE =
VARIABLE VALUE REDUCED COST L L L X X X X X X X X X X