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

the problem will provide a maximum profit of $48 450. The optimal values of the
decision variables are given by A ¼ 25, B ¼ 425, C ¼ 150 and D ¼ 0. Thus, the optimal
strategy is to concentrate on agriculture and forestry with B ¼ 425 units. In addition,
the firm should allocate 25 units to the oil rig market (A ¼ 25) and meet its 150-unit
commitment to the national retail chain store (C ¼ 150). With D ¼ 0, the optimal
solution indicates that the firm should not use the Internet market.
Looking at the four constraints, we see that those for advertising, the produc-
tion level and the retail stores requirement are binding. We are using all the
available advertising budget; we are matching the required production level of
600 units in total; and we are meeting the minimum requirement for supplying
retail stores, although for this last constraint there is no surplus either. The
constraint relating to salesforce availability, however, is non-binding and has a
slack value of 25. That is, we do not require all the available salesforce time; 25
hours of the available 1800 hours is unused. Looking now at the sensitivity
information output, we see that for the reduced cost for the four decision variables,
three are zero and one nonzero, that for variable D at 45. Recall that reduced costs
indicate by how much each objective function coefficient would have to change
before the corresponding decision variable took a positive value in the optimal
solution.Fairlyobviously,thefirst threedecision variables,A,B,C,havezero
reduced costs since they are already taking positive values in the solution. For
variable D, however, we see a reduced cost value of 45. Recollect that variable D
indicates the number of radios sold through the Internet. At the optimal solution,
this is set to zero – none of our sales will be through this channel. Effectively, since
we are seeking to maximize profit, we are being told that D is not sufficiently
profitable at $60 per unit, other things being equal. The reduced costs figure of $45
for D tells us how much more profitable D needs to be to take a nonzero value.
Radios sold through the Internet would need to make $105 ($60 + $45) profit per
unit sold to be viable. Let us also look at the other sensitivity information about the
objective function. We can summarize the range of optimality for each of the
objective function coefficients as follows:
A 84 C A No upper limit
B 50 C b 90
C No lower limit < C c 87
D No lower limit < C d 105

So,wesee that thecurrent solutionwill remainoptimalaslongastheobjective
function coefficients remain in the given ranges of optimality. For radios sold to
oil rigs, A, the profit per unit can fall to $84 and A will still remain in the
solution. For radios sold to agriculture and forestry, the profit contribution could
be between $50 and $90 and B remains in the solution. For radios sold through
local distributors, C, there is no lower limit but an upper limit of $87 per unit.
For D, the figures confirm the reduced cost value that we discussed earlier.
Let us now look at the sensitivity information for the four constraints. We already
know that three constraints are binding. Looking at the shadow price for each
constraint and the allowable increase/decrease we can provide the following man-
agement information. The advertising budget of $5000 is all spent at the optimal
solution. The shadow price of $3 indicates that for each extra dollar spent on
advertising over and above the current budget of $5000 total profit will increase by
$3. This will be valid up to an increase of $850. After that we cannot tell from the
sensitivity information what will happen to profit. To do so, we would need to
reformulate and re-solve the problem. Other things being equal, then, management
should give serious consideration to increasing the advertising budget to $5850. Our

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