case5.2 energy

Chapter 5
Network Modeling

1. If supplies are represented by positive numbers and demands are represented numbers, the balance-of-flow rule would be stated as follows:
For Minimum Cost Network
Flow Problems Where:
Apply This Balance-of-Flow Rule
At Each Node:
Total Supply > Total Demand
Outflow - Inflow  Supply or Demand
Total Supply < Total Demand
Outflow - Inflow  Supply or Demand
Total Supply = Total Demand
Outflow - Inflow = Supply or Demand

2. Multiply both sides of the constraint by -1 and reverse the sign of the inequality.

3. See file: Prb5_3.xlsm

a. Total cost = $3,398, All materials are used, All demand is met.
Node
Net Flow
Supply/Demand
1
Newspaper
-80.0
-80
2
Mixed Paper
-50.0
-50
3
White Office
-30.0
-30
4
Cardboard
-40.0
-40
5
Process 1
0.0
0
6
Process 2
0.0
0
7
Newsprint
60.0
60
8
Packaging
40.0
40
9
Print Stock
50.0
50

b. Total cost = $3,129, All materials are NOT used, All demand is met.
Node
Net Flow
Supply/Demand
1
Newspaper
-80.0
-80
2
Mixed Paper
-50.0
-50
3
White Office
-30.0
-30
4
Cardboard
-25.4
-40
5
Process 1
0.0
0
6
Process 2
0.0
0
7
Newsprint
60.0
60
8
Packaging
40.0
40
9
Print Stock
50.0
50

c. In part a, if we assume supply is inadequate the demand (when, in fact, it is adequate) we require all the supply to be used – even if it is not all needed. This results in higher than necessary costs. In part b, assuming supply is adequate to meet demand (when, in fact, it is) resulted in a smaller total cost as this solution does not require all the supply to be used.
d. The shortage on print stock pulp can be reduced to 4 units (without sacrificing newspaper or packing paper pulp) at an additional cost of $307.

4. Because there is a 10% loss of flow on all arcs going to node 4, a total of 702/0.9 = 780 units must flow into node 4. Thus, we can simply increase the demand at node 4 to 780 and assume no loss of flow occurs on arcs leading into this node. Similarly, only 608/1.05 = 579.05 units must flow into node 5. Thus,