2011TransportationHandsOnWinQSB
IE 416
Fall 2011
Understanding Shadow Price Using WinQSB
Last updated: Oct 5, 2011
Here is an example on learning the concept and application of “Shadow Price (SP)” using
WinQSB. Shadow Price indicates the amount that the minimum total cost will change if we
ADD one unit to the capacity of a supply (or demand) point.
Consider the Foster example (from another handout). Below is a copy of the input information,
solution, and the range of feasibility obtained using WinQSB.
The range of feasibility indicates that the capacity of Clev can change from a minimum 5000 to a
maximum 7500 and the shadow price is $-3. This means if you add one unit to the capacity of
Clev, to make it 5001, then the total minimum cost will reduce by $3, that is $39497. For
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example, add 4 unit to Clev’s capacity and make it 5004. This is shown in the figure below and
its solution is shown in the next figure. We expect the total cost to decrease by 4*$3= $12.
Now let’s check the changes to the solution. First of all we expect to have 4 extra supply
capacity (unbalanced transportation problem) not used.
From
To
Clev
Clev
Bed
Bed
Bed
Bed
York
Bos
Chi
Chi
Stlo
Lex
Lou
Bos
Capacity not used
Total cost
Amount shipped
originally
3500
1500
2500
2000
1500
2500
39500
Amount shipped
recently
3500
1504
2496
1500
2000
2500
4 at Bed
39488
Comment
changed
changed
changed
changed
changed
changed
What did you learn from this exercise?

The concept and application of shadow price.
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
If capacity is changed the shipment plan will change in terms of amount of shipment and
routes for shipment.
Notice that you cannot add to the capacity of Bos to see the effect of its Shadow Price. The
reason is that the capacity will be more than the allowable maximum value. For Bos you can
reduce its capacity and observe that the total cost will decrease as much as its shadow price.
Learning the Effect of Changes in Unit Cost Using WinQSB
Using WinQSB you can better understand the effect of changing the unit cost. Perform the
following exercises and learn the changes.
Cost of shipping from Bedf to Chic is 5 (is a BV), change it to 3 (within range), then change it to
6 (within range), then change it to 8(outside range). What happens to the total cost? What
happens to the shipment plan in terms of the amount of shipment (amount of Xij, change in basic
feasible solution)? What happens to the shipment plan in terms of the shipment routes (same ij
or different ij, change in basis)?
Cost of shipping from Bedf to Bost is 7 (is a NBV), change it to 6 (within range), then change it
to 10 (within range), then change it to 4(outside range). What happens to the total cost? What
happens to the shipment plan in terms of the amount of shipment (amount of Xij, change in basic
feasible solution)? What happens to the shipment plan in terms of the shipment routes (same ij
or different ij, change in basis)?
For Foster example (another handout) we used Parametric analysis of WinQSB for changes of
unit cost of Clev to Bost and obtained the following graph. The original unit cost was $3.
Change the unit cost to $3.5. What happens to the total cost? What happens to the shipment
plan in terms of the amount of shipment (amount of Xij, change in basic feasible solution)?
What happens to the shipment plan in terms of the shipment routes (same ij or different ij,
change in basis)?
Change the unit cost to $7. What happens to the total cost? What happens to the shipment plan
in terms of the amount of shipment (amount of Xij, change in basic feasible solution)? What
happens to the shipment plan in terms of the shipment routes (same ij or different ij, change in
basis)?
Change the unit cost to $12. What happens to the total cost? What happens to the shipment plan
in terms of the amount of shipment (amount of Xij, change in basic feasible solution)? What
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happens to the shipment plan in terms of the shipment routes (same ij or different ij, change in
basis)?
Notice that I selected the changes to be from different line segments (with different slope) in the
figure below. What does the horizontal line segment mean?
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