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

MANAGEMENT SCIENCE IN ACTION

The Nutricia Dairy and Drinks Group, Hungary
ince the early 1990s, much of what used to be central collection points; transportation costs
S called Eastern Europe has been undergoing involved in shipping the milk from the central col-
major economic, political and social transformation lection points to the plants; production costs;
moving from the old Soviet Union command econ- inter-plant costs incurred when semi-finished prod-
omy into market economies. This has meant major ucts are shipped from one plant to another; trans-
efforts to improve performance and economic effi- portation costs to the distribution centres; ware-
ciency. The Nutricia Dairy and Drinks group (NDDG) housing costs involved in storing products at the
started acquiring dairy companies in Hungary in distribution centres. NDDG developed an LP
1995 and has been using linear programming to model to minimize total costs but was particularly
help it improve efficiency and also to undertake sce- interested in using the model to undertake sce-
nario planning using sensitivity analysis. nario planning – looking at a variety of strategic
The overall structure of NDDG’s business is as options for rationalizing the supply side of the
follows. Its nine plants are supplied with raw milk business. The sensitivity findings include the
by over 400 farmers throughout Hungary. The following:
plants produce a range of dairy and dairy-related
• A change in production costs of up to 10 per cent
products – over 300 different products in total.
had no impact on the optimal solution.
Some products are produced at individual plants.
Other products are semi-finished at one plant and • An increase of 5 per cent in milk transportation
then shipped to another plant to be turned into costs would make the opening of three new
fully finished products. The plant output is sent to plants cost-effective.
17 distribution centres which in turn supply over • Changes in inter-plant costs had no effect on the
17 000 shops. NDDG had identified a number of optimal solution.
major costs in its operations. First, milk collection
Based on FHE. Wouda, P van beck, JGAJ. van der Vorst and H. Tacke,
costs incurred in collecting from individual farms ‘An application of mixed-integer linear programming models on the
(which range from supplying only 20 000 litres a redesign of the supply network of Nutricia Dairy and Drinks Group in
year to those supplying around 11 million litres) to Hungary’, OR Spectrum 24/4 (Nov. 2002): 449–465.

Suppose that management is willing to reconsider their position regarding the
maximum weight of the daily diet. The dual price of 91.92 for constraint 4 shows that
a one-unit increase in the right-hand side of constraint 4 will reduce total cost by
91.92 sh. The RIGHT HAND SIDE RANGES section of the output shows that this
interpretation is valid for increases in the right-hand side up to a maximum of 8.478
kilos. Thus, the effect of increasing the right-hand side of constraint 4 from six to
eight kilos is a decrease in the total daily cost of 2 91.92 or 183.84 sh. Keep in mind
that if this change were made, the feasible region would change, and we would
obtain a new optimal solution.
The OBJECTIVE COEFFICIENT RANGES section of the computer output
shows a lower limit of 39.29 for S. Clearly, in a real problem, the objective function
coefficient of S (the cost of the standard product) cannot take on a negative value.
So, from a practical point of view, we can think of the lower limit for the objective
function coefficient of S as being zero. We can thus conclude that no matter how
much the cost of the standard mix were to decrease, the optimal solution would not
change. Even if KCC could obtain the standard product for free, the optimal
solution would still specify a daily diet of 3.51 kilos of the standard product, 0.95
kilos of the enriched product and 1.54 kilos of the vitamin and mineral feed additive.
However, any decrease in the per-unit cost of the standard feed would result in a
decrease in the total cost for the optimal daily diet.

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