1. Chapter 4
Network Models

2. A network problem is one that can be represented by...
4.1 Introduction
A network problem is one that can be represented by... 6 8 9 10
Nodes 7
Arcs 10
Function on Arcs

3. 4.1 Introduction The importance of network models
Many business problems lend themselves to a network formulation.
Optimal solutions of network problems are guaranteed integer solutions, because of special mathematical structures. No special restrictions are needed to ensure integrality.
Network problems can be efficiently solved by compact algorithms due to their special mathematical structure, even for large scale models.

4. Network Terminology Flow Directed/undirected arcs
the amount sent from node i to node j, over an arc that connects them. The following notation is used:
Xij = amount of flow
Uij = upper bound of the flow
Lij = lower bound of the flow
Directed/undirected arcs
when flow is allowed in one direction the arc is directed (marked by an arrow). When flow is allowed in two directions, the arc is undirected (no arrows).
Adjacent nodes
a node (j) is adjacent to another node (i) if an arc joins node i to node j.

5. Network Terminology Path / Connected nodes
Path :a collection of arcs formed by a series of adjacent nodes.
The nodes are said to be connected if there is a path between them.
Cycles / Trees / Spanning Trees
Cycle : a path starting at a certain node and returning to the same node without using any arc twice.
Tree : a series of nodes that contain no cycles.
Spanning tree : a tree that connects all the nodes in a network ( it consists of n -1 arcs).

6. 4.2 The Transportation Problem
Transportation problems arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods.

7. The Transportation Problem
Problem definition
There are m sources. Source i has a supply capacity of Si.
There are n destinations. The demand at destination j is Dj.
Objective:
Minimize the total shipping cost of supplying the
destinations with the required demand from the available
supplies at the sources.

8. 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.

9. 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

10. CARLTON PHARMACEUTICALS Network presentation

11. Destinations Boston Sources 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

12. 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)

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

14. CARLTON PHARMACEUTICAL – The complete mathematical model
Total shipment out of a supply node
cannot exceed the supply at the node. =
Total shipment received at a destination
node, must equal the demand at that node.

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

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

17. CARLTON PHARMACEUTICALS Spreadsheet - solution

18. 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.

19. 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.

20. 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.

21. 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 Boston, the total cost decreases by $37.

22. 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)

23. 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
Shipments on a Blocked Route = 0

24. 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
Only Feasible Routes Included in Changing Cells
Cell C9 is NOT Included
Shipments from Greensboro
to Cleveland are prohibited

25. 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.
Remedy: Add a constraint to Excel of the form Xij ³ 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

26. MONTPELIER SKI COMPANY Using a Transportation model for production scheduling
Montpelier is planning its production of skis for the months of July, August, and September.
Production capacity and unit production cost will change from month to month.
The company can use both regular time and overtime to produce skis.
Production levels should meet both demand forecasts and end-of-quarter inventory requirement.
Management would like to schedule production to minimize its costs for the quarter.

27. MONTPELIER SKI COMPANY
Data:
Initial inventory = 200 pairs
Ending inventory required =1200 pairs
Production capacity for the next quarter = 400 pairs in regular time.
= 200 pairs in overtime.
Holding cost rate is 3% per month per ski.
Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)

28. MONTPELIER SKI COMPANY
Analysis of demand:
Net demand in July = = 200 pairs
Net demand in August = 600
Net demand in September = = 2200 pairs
Analysis of Supplies:
Production capacities are thought of as supplies.
There are two sets of “supplies”:
Set 1- Regular time supply (production capacity)
Set 2 - Overtime supply
Initial inventory
In house inventory
Forecasted demand

29. MONTPELIER SKI COMPANY
Analysis of Unit costs
Unit cost = [Unit production cost] +
[Unit holding cost per month][the number of months stays in inventory]
Example: A unit produced in July in regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50

30. Network representation
Production
Month/period
Network representation
Month
sold
July
R/T
July
R/T 25
25.75
26.50
1000
200
July
July
O/T
500 30
30.90
31.80 +M 37 +M 26
26.78 +M 29
Aug.
R/T +M 32
32.96
800
Aug.
600
Demand
Production Capacity
Aug.
O/T
400
Sept.
2200
Sept.
R/T
400
300
Sept.
O/T
200

31. MONTPELIER SKI COMPANY - Spreadsheet

32. MONTPELIER SKI COMPANY
Summary of the optimal solution
In July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store = 1300 at the end of July.
In August, produce 800 pairs in R/T, and 300 in O/T. Store additional = 500 pairs.
In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be
( ) = 1200 pairs available for shipment to Ski Chalet.
Inventory +
Production -
Demand

33. 4.3 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

34. 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

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

36. 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

37. 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
+15 2 6 BA CC

38. 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
+15 2 4 6 BA CC

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

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

41. DEPOT MAX – Types of constraints5 10 20 6 15 12 7 11
-12
+10 1 3 5
Supply nodes: Net flow out of the node] = [Supply at the node]
X12 + X13 + X15 - X21 = 10 (Node 1) X21 + X24 - X = 15 (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) 7
Demand nodes: [Net flow into the node] = [Demand for the node]
X15 + X35 +X65 - X56 = 12 (Node 5) X46 +X56 - X65 = 13 (Node 6) 2 4 6
-13
+15

42. 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

43. DEPOT MAX - spreadsheet

44. 4.4 The Assignment Problem
Problem definition
m workers are to be assigned to m jobs
A unit cost (or profit) Cij is associated with worker i performing job j.
Minimize the total cost (or maximize the total profit) of assigning workers to job so that each worker is assigned a job, and each job is performed.

45. BALLSTON ELECTRONICS
Five different electrical devices produced on five production lines, are needed to be inspected.
The travel time of finished goods to inspection areas depends on both the production line and the inspection area.
Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized.

46. BALLSTON ELECTRONICS
Data: Travel time in minutes from assembly lines to inspection areas.

47. BALLSTON ELECTRONICS- NETWORK REPRESENTATION

48. 1 2 3 4 5
Assembly Line
Inspection Areas A B C D E
S1=1
S2=1
S3=1
S4=1
S5=1
D1=1
D2=1
D3=1
D4=1
D5=1

49. BALLSTON ELECTRONICS – The Linear Programming Model
Min 10X11 + 4X12 + … X X55
S.T. X11 + X12 + X13 + X14 + X15 = 1
X21 + X … X25 = 1
… … … …
X51 + X52+ X53 + X54 + X55 = 1
All the variables are non-negative

50. BALLSTON ELECTRONICS – Computer solutions
A complete enumeration is not an efficient procedure even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate).
The Hungarian method provides an efficient solution procedure.

51. BALLSTON ELECTRONICS – Transportation spreadsheet
=SUMPRODUCT(B7:F11,B17:F217)
=SUM(B7:F7)
Drag to cells H8:H12
=SUM(B7:B11)
Drag to cells C13:F13

52. BALLSTON ELECTRONICS – Transportation spreadsheet
Each Area is Served
Each Line is Assigned

53. BALLSTON ELECTRONICS – Assignment spreadsheet

54. The Assignments Model - Modifications
Unbalanced problem: The number of supply nodes and demand nodes is unequal.
Prohibitive assignments: A supply node should not be assigned to serve a certain demand node.
Multiple assignments: A certain supply node can be assigned for more than one demand node .
A maximization assignment problem.

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