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

management on the optimum production mix for the extra unit of CTY45 produced. We also see from
next production period. This is: the Allowable increase that we could profitably
produce an extra 700 units. Clearly, we do not
BFX1 600 units know from the information given why this con-
AHS15 700 units straint is there. It may be that Sales and Marketing
in the company have advised that realistically we
CTY45 200 units
cannot sell more than 200 units. However, the
This production mix will generate a profit contribu- sensitivity information allows management to con-
tion of E10 800. However we can go further and sider the consequences of this. Our third binding
provide information from the binding/non-binding constraint related to the requirement to produce at
constraints. We can advise management that the least 600 units of BFX1 because of an existing
profit contribution is being prevented from increas- order. In this case the Shadow price is negative,
ing further by the available amount of Shaping at E1. This indicates that any increase in this
time; the maximum sales of CTY45; and the exist- constraint will actually cost us money and
ing order for BFX1. If management wish to decrease profit contribution for each extra unit
increase profit contribution further then these three above the 600 that we produce. Conversely, if
areas should be examined further. Looking at we could relax this constraint and produce less
Shaping time first, the company currently has an than 600 units in the next period then profit con-
anticipated 7200 minutes available (120 hours). All tribution would increase. Again, it may be that
of this is needed for the optimum production mix. management are unable to alter this constraint,
We can advise management that any shortfall in we may already have signed a contract with the
the anticipated time availability of 120 hours will customer. Again, however, the sensitivity analysis
affect production and hence profit. The Shadow information allows management to see the conse-
price of E1.50 tells us that each minute shortfall will quences of this and maybe to consider alterna-
reduce profit contribution by E1.50. Conversely, if tives. Could we persuade the customer to delay
management can somehow increase the amount of the order? We could even offer a financial incen-
available shaping time beyond 120 hours this will tiveforthemtodothis, up to E1for eachunit we
allow us to produce more and increase profit con- didn’t produce for them.
tribution by E1.50 for each extra minute of Shaping Is there anything else we can tell management?
time we can obtain. Looking at the Available Well, we can advise that we do not actually need all
increase information, we can advise management the available grinding time, since five hours (300
that it would be worthwhile trying to increase the minutes) remains unused at the optimum produc-
available shaping time by an extra four hours (240 tion mix. We can also inform them if there is any
minutes). After this point, we are no longer sure uncertainty about the product profit contributions
from the existing sensitivity analysis what would (of E8, E6and E9) we can advise on the effects
happen to the optimum solution. of this. For example, that BFX1’s profit contribution
If we look at the second binding constraint, could increase from E8upto E9 without affecting
maximum sales of CTY45, we are producing the the optimum production mix; and that its profit
maximum quantity permitted at the optimum sol- contribution could effectively fall to zero and still
ution. Again, the Shadow price of E3 indicated the not affect the optimum solution. The latter point
opportunity cost of this constraint. Other things may need further explaining. Even if BFX1 has a
being equal, if we were allowed to relax this con- zero profit contribution we would still have to pro-
straint (i.e., produce more than 200 units) then duce a minimum of 600 units because of the rele-
profit contribution would increase by E3for each vant constraint.

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