Page 124 -

104 CHAPTER 3 LINEAR PROGRAMMING: SENSITIVITY ANALYSIS AND INTERPRETATION OF SOLUTION

shows, as usual, the change in the objective function for a change in the RHS of each
constraint while the Allowable Increase and Allowable Decrease show how much
the RHS of each constraint can change and the current solution still remain as the
optimal solution.
So, let’s pull this together into a brief report for management.
The company can minimize its cost by producing 250 litres of A and 100
litres of B. Cost will be E800. Thecostper litreofA(currently E2per litre)
could rise by as much as E1 and the optimal combination of A and B would
not change (although the actual total cost clearly would). The unit cost of A
could decrease indefinitely and again the optimal combination would not
change. For product B, the equivalent changes are an infinite increase and a
decrease of E1.
Turning to the constraints, we must produce a minimum of 125 litres of A
and we do so in fact producing a further 125 litres. For the second constraint, we
were required to limit total production to no more than 350 litres in total. We
have kept within this limit. The shadow price tells us that if we change this
constraint – for example allowing a unit increase in production to 351 – then
total costs will increase by E4 for each unit change in the RHS. This would apply
for changes in the RHS up to 475 litres and as low as 300 litres. Finally, looking
at constraint 3, we only had 600 hours of production time available. With the
optimal combination of A and B we are using all of these 600 hours. In fact, we
know that if we could obtain additional hours then we could actually reduce total
Try problem 14 to test costs (the shows price is negative). An extra production hour would allow us to
your ability to interpret
the computer output for a reduce total cost by E1, up to a maximum of an additional 100 hours of
minimization problem. production time.

Cautionary Note on the Interpretation of Dual Prices
As stated previously, the dual price is the change in the value of the optimal
solution per unit change in the right-hand side of a constraint. When the right-
hand side of the constraint represents the amount of a resource available, the
associated dual price is often interpreted as the maximum amount one should be
willing to pay for one additional unit of the resource. However, such an inter-
pretation is not always correct. To see why, we need to understand the differ-
ence between sunk and relevant costs. A sunk cost is one that is not affected by
the decision made. It will be incurred no matter what values the decision
variables assume. A relevant cost is one that depends on the decision made.
The amount of a relevant cost will vary depending on the values of the decision
variables.
Let us reconsider the GulfGolf problem. The amount of cutting and dyeing
time available is 630 hours. The cost of the time available is a sunk cost if it
must be paid regardless of the number of standard and deluxe golf bags pro-
duced. It would be a relevant cost if the company only had to pay for the
number of hours of cutting and dyeing time actually used to produce golf bags.
All relevant costs should be reflected in the objective function of a linear
programme. Sunk costs should not be reflected in the objective function. For
GulfGolf we have been assuming that the company must pay its employees’
wages regardless of whether their time on the job is completely utilized. There-
fore, the cost of the labour-hours resource for the company is a sunk cost and
has not been reflected in the objective function.
When the cost of a resource is sunk, the dual price can be interpreted as the
maximum amount the company should be willing to pay for one additional unit of

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.