# Linear Programming Problem - Help with Constraints Please!

I have been struggling with Part 3 of this question for some time:

Computico Limited, currently operating in the UK, assembles electronic components at its two factories, located at Manchester and London, and sells these components to three major customers. Next month the customers, in units, orders are:

Customer A|Customer B|Customer C
3000|4200|5300

As Computicos three customers are in different industries it allows them to charge slightly different prices to different customers. For each component, Customer A pays £108, Customer B pays £105 and Customer C pays £116. The variable costs of assembling the components at the two factories vary because of different labour costs and power costs; these are £28 per unit at Manchester and £36 per unit at London.

The transportation costs per unit between each factory and customer are:

From| To Customer| Cost (£)
Manchester| A| 45
Manchester| B| 40
Manchester| C| 55
London| A| 42
London| B| 43
London| C| 44

1. Set up a table showing the contribution to profit of supplying one unit to each customer.

Table 2.1 below shows the contributions available and the profit gained when one unit is supplied to each customer. It can be seen that this contribution is different dependent upon the factory the goods are supplied from:

Factory| Customer| Transport Cost| Labour Cost| Price to Customer| Profit
Manchester| A| -£28| -£45| £108| £35
Manchester| B| -£28| -£40| £105| £37
Manchester| C| -£28| -£55| £116| £33
London| A| -£36| -£42| £108| £30
London| B| -£36| -£43| £105| £26
London| C| -£36| -£44| £116| £36

Table 2.1 – Contribution to Profit

2. Assuming no restrictions on capacity at the two factories calculate the maximum total contribution from next months orders and the associated capacities for the two factories.

Table 2.2 below shows the profits available from each customer if they are supplied from either factory:

Customer| Factory| Profit (£)| Units Ordered| Total Profit (£)
Customer A| Manchester| 35| 3000| 105,000
Customer A| London| 30| 3000| 90,000
Customer B| Manchester| 37| 4200| 155,400
Customer B| London| 26| 4200| 109,200
Customer C| Manchester| 33| 5300| 174,900
Customer C| London| 36| 5300| 190,800

From this it can be seen that should there be no limit on productions, the maximum profit available for the company would be to supply:

Customer A – Manchester - £105,000 profit
Customer B – Manchester - £155,400 profit
Customer C – London - £190,800

Maximum Profit available with NO restrictions = £451,200

This would require a total of:

7200 units to be produced in Manchester and 5300 units to be produced in London.

Next months orders from the three customers does in fact exceed the total capacity for the two factories. The maximum output for Manchester is 4000 units and for London it is 6000 units.

3. Calculate the total contribution and indicate which customer will have unsatisfied demand.

With the restrictions stated above, it is clear that the company will be unable to meet the needs of all customers, however the company will still wish to maximise profit. Let Customer A = X1, Customer B = X2 and Customer C = X3.

The objective function becomes:

MAXIMISE 35X1 + 37X2 + 36X3

Subject to:

3000X1¬ + 4200X2 + 5300X3 ≤ 10,000 - Maximum units which can be produced.

3000X1 + 4200X2 ≤ 4000 - Units produced in Manchester

5300X3 ≤ 6000 - Units produced in London

X1, X2 and X3 ≥ 0

With 3 variables, the Simplex Method will be used to calculate the optimal solution, the first step of which requires the introduction of Slack Variables. The constraints therefore become:

3000X1¬ + 4200X2 + 5300X3 + S1 = 10,000

3000X1 + 4200X2 + S2 = 4000

5300X3 + S3 = 6000

These are then entered into a Simplex table

When this is progressed, using the simplex method, through the iterations the answer achieved is meaningless. I have calculated it using both pen and paper and some software, both of which arrive at the same answer. This would indicate to me that my constraints are wrong - any help appreciated!
1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

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Hey

I think you have missunderstood one point here. You have for example assumed what all production for the customer A is performed in Manchester (and similarly for B and C).
This is true if you don't have any constraints, but with constraints the function f you should maximize is:

$$f=35A_{M}+30A_{L}+37B_{M}+26B_{L}+33C_{M}+36C_{L}$$
$$A_{M}+B_{M}+C_{M}\leq4000$$
$$A_{L}+B_{L}+C_{L}\leq6000$$
$$A_{M},A_{L},B_{M},B_{L},C_{M},C_{L}\geq0$$

Where $$A_{M}$$ is the number of units produced for the customer A in Manchester and $$A_{L}$$ is the number of units produced for the customer A in London, etc.

SOLVED! Thank you

Many thanks for that!

After you pointed it out it was so blindingly obvious.... doh!

Anyway, I hav enow been able to use the simplex method to obtain figures that agree with the mathematically obtained ones, so happy days.

Thanks for the help.