2.4 solving the problem – using simultaneous equations You may consider that the whole process would be easier by solving the constraints as sets of simultaneous equations and not bothering with a graph. This is possible and you may get the right answer, but such technique should be used with caution and is not recommended until you have determined graphically which constraints are effective in determining the optimal solution. Furthermore if the question asks for a graphical solution, then a graph must be used.
The technique can, however, be used as a check, or to establish the exact quantities for the optimal solution when the graph does not give sufficient accuracy.
Illustration 5 – Finding the optimal solution
For example, using the Hebrus example the optimal solution can be checked by solving the two simultaneous equations for the two constraint boundaries.
Point Q is the intersection of the lines:
Constraint考试用书
6x + 3y = 36 (ⅰ)
4x + 8y = 48 (ⅱ)
3 × (ⅱ) – 2 × (ⅰ) gives:
18y = 72
Substituting into (ⅰ)
x = 4
Thus, the maximum contribution is obtained when four summerhouses and four sheds per week are produced, and the maximum contribution is (4 × $50) + (4 × $40) = $360.
Test your understanding 5 – Alfred Co – part Ⅲ
Using the Alfred Co example again you are required to solve for the optimal solution.
Test your understanding 6 – Minimising costs
J Farms Ltd can buy two types of fertiliser which contain the following percentage of chemicals:
NitratesPhosphatesPotash
Type X1852
Type Y325
For a certain crop the following minimum quantities (kg) are required:转自:考试网 - [Examw.Com]
Nitrates 100 Phosphates 50 Potash 40
Type X costs ￡10 per kg and type Y costs ￡5 kg. J Farms Ltd currently buys 1,000 kg of each type and wishes to minimise its expenditure on fertilisers.
(a) Write down the objective function and the constraints for J Farms Ltd.
(b) Draw a graph to illustrate all the constraints (equations/inequalities), shading the feasible region.
(c) Recommend the quantity of each type of fertiliser which should be bought and the cost of these amounts.
(d) Find the saving J Farms Ltd can make by switching from its current policy to your recommendation. |
发表于 2012-2-23 17:09:42
