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Network Flows
Based on the book: Introduction to Management Science. Hillier & Hillier. McGraw-Hill
2Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Minimum Cost FlowDistribution Unlimited Co. Problem The Distribution Unlimited Co. has two factories
producing a product that needs to be shipped to two warehouses Factory 1 produces 80 units. Factory 2 produces 70 units. Warehouse 1 needs 60 units. Warehouse 2 needs 90 units.
There are rail links directly from Factory 1 to Warehouse 1 and Factory 2 to Warehouse 2.
Independent truckers are available to ship up to 50 units from each factory to the distribution center, and then 50 units from the distribution center to each warehouse.
Question: How many units (truckloads) should be shipped along each shipping lane?
3Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
The Distribution Network
F1
DC
F2 W2
W180 unitsproduced
70 units produced
60 unitsneeded
90 units needed
4Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Data for Distribution Network
1
2
4
5
3
700
900
200300
400400
50
50 50
50
80
70
60
90
5Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Transportation costs for each unit of product and max capacity of each road is given below
From To cost/ unit Max capacity
1 4 700 No limit
1 3 300 50
2 3 400 50
2 5 900 No limit
3 4 200 50
3 5 400 50
There is no other link between any pair of points
Minimum Cost Flow Problem: Narrative representation
6Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Minimum Cost Flow Problem: decision variables
x14 = Volume of product sent from point 1 to 4
x13 = Volume of product sent from point 1 to 3
x23 = Volume of product sent from point 2 to 3
x25 = Volume of product sent from point 2 to 5
x34 = Volume of product sent from point 3 to 4
x35 = Volume of product sent from point 3 to 5
We want to minimize
Z = 700 x14 +300 x13 + 400 x23 + 900 x25 +200 x34 + 400 x35
7Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Minimum Cost Flow Problem: constraints
Supply
x14 + x13 = 80
x23 + x25 = 70
Demand
x14 + x34 = 60
x25 + x35 = 90
Transshipment
x13 + x23 = x34 + x35 (Move all variables to LHS)
x13 + x23 - x34 - x35 =0
Supply
x14 + x13 ≤ 80
x23 + x25 ≤ 70
Demand
x14 + x34 ≥ 60
x25 + x35 ≥ 90
8Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Minimum Cost Flow Problem: constraints
Capacity
x13 50
x23 50
x34 50
x35 50
Nonnegativity
x14, x13 , x23 , x25 , x34 , x35 0
9Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
The SUMIF Function
The SUMIF formula can be used to simplify the node flow constraints. =SUMIF(Range A, x, Range B)
For each quantity in (Range A) that equals x, SUMIF sums the corresponding entries in (Range B).
The net outflow (flow out – flow in) from node x is then=SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)ARCS Shipment Capacity Cost Nodes Netflow Suply/ Demand
1 3 1 <= 50 300 1 3 = 80
1 4 2 700 2 9 = 70
2 3 5 <= 50 400 3 4 = 0
2 5 4 900 4 -7 = -60
3 4 5 <= 50 200 5 -9 = -90
3 5 5 <= 50 400
Total Cost 10300
Out Flow 10
Node 3 InFlow 6
NetFlow 4
10Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Excel Implementation
ARCS Shipment Capacity Cost Nodes Netflow Suply/ Demand
1 3 <= 50 300 1 0 = 80
1 4 700 2 0 = 70
2 3 <= 50 400 3 0 = 0
2 5 900 4 0 = -60
3 4 <= 50 200 5 0 = -90
3 5 <= 50 400
Total Cost 0
11Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Excel Implementation
ARCS Shipment Capacity Cost Nodes Netflow Suply/ Demand
1 3 50 <= 50 300 1 80 = 80
1 4 30 700 2 70 = 70
2 3 30 <= 50 400 3 0 = 0
2 5 40 900 4 -60 = -60
3 4 30 <= 50 200 5 -90 = -90
3 5 50 <= 50 400
Total Cost 110000
12Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Terminology for Minimum-Cost Flow Problems1. The model for any minimum-cost flow problem is represented
by a network with flow passing through it.
2. The circles in the network are called nodes.
3. Each node where the net amount of flow generated (outflow minus inflow) is a fixed positive number is a supply node.
4. Each node where the net amount of flow generated is a fixed negative number is a demand node.
5. Any node where the net amount of flow generated is fixed at zero is a transshipment node. Having the amount of flow out of the node equal the amount of flow into the node is referred to as conservation of flow.
6. The arrows in the network are called arcs.
7. The maximum amount of flow allowed through an arc is referred to as the capacity of that arc.
13Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Assumptions of a Minimum-Cost Flow Problem1. At least one of the nodes is a supply node.
2. At least one of the other nodes is a demand node.
3. All the remaining nodes are transshipment nodes.
4. Flow through an arc is only allowed in the direction indicated by the arrowhead, where the maximum amount of flow is given by the capacity of that arc. (If flow can occur in both directions, this would be represented by a pair of arcs pointing in opposite directions.)
5. The network has enough arcs with sufficient capacity to enable all the flow generated at the supply nodes to reach all the demand nodes.
6. The cost of the flow through each arc is proportional to the amount of that flow, where the cost per unit flow is known.
7. The objective is to minimize the total cost of sending the available supply through the network to satisfy the given demand. (An alternative objective is to maximize the total profit from doing this.)
14Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Typical Applications of Minimum-Cost Flow Problems
Kind ofApplication
SupplyNodes
Transshipment Nodes
DemandNodes
Operation of a distribution network
Sources of goods
Intermediate storage facilities
Customers
Solid waste management
Sources of solid waste
Processing facilities
Landfill locations
Operation of a supply network
Vendors Intermediate warehouses
Processing facilities
Coordinating product mixes at plants
Plants Production of a specific product
Market for a specific product
Cash flow management
Sources of cash at a specific time
Short-term investment options
Needs for cash at a specific time
15Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Data for Distribution Network
1
2
4
5
3
700
900
200300
400400
50
50 50
50
80
70
60
90
16Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Transportation problem II : Formulation
D1 D2 DT31 DT32 SupplyO1 700 10000 300 10000 80O2 10000 900 10000 400 70
OT34 200 10000 0 0 50OT35 10000 400 0 0 50
Demand 60 90 50 50
17Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Transportation problem II : Solution
D1 D2 DT31 DT32 SupplyO1 700 10000 300 10000 80O2 10000 900 10000 400 70OT34 200 10000 0 0 50OT35 10000 400 0 0 50
Demand 60 90 50 50
D1 D2 DT31 DT32 SupplyO1 80O2 70OT34 50OT35 50
Demand 60 90 50 50
18Ardavan Asef-Vaziri Jan. 2014Network Flow Problems
Transportation problem II : Solution
D1 D2 DT31 DT32 Supply
O1 700 10000 300 10000 80
O2 10000 900 10000 400 70
OT34 200 10000 0 0 50
OT35 10000 400 0 0 50
Demand 60 90 50 50
D1 D2 DT31 DT32 Allocation Supply
O1 30 0 50 0 80 80
O2 0 40 0 30 70 70
OT34 30 0 0 20 50 50
OT35 0 50 0 0 50 50
Allocation 60 90 50 50
Demand 60 90 50 50