Manufacturing problems

What you will be able to do

• turn a manufacturing story into an objective function and constraints;
• draw the feasible region and find the best plan at a corner;
• say the answer in words, with units, and check it with linprog.

Four steps from a story to a plan

Every word problem of this chapter is solved in the same four steps.

1. Name the unknowns. Let $x$ and $y$ be the numbers of the two things we choose. Say what they count and in what unit.
2. Write the objective. This is the profit, or the cost, as a formula in $x$ and $y$.
3. Write the constraints. Each limit on a resource gives one inequality. Add $x \ge 0$ and $y \ge 0$, since we cannot make a negative number of tables.
4. Solve and read. Draw the region, find the corners, work out the objective at each, choose the best, and write the answer in the words of the story.

A worked example

A workshop makes tables and chairs. A table needs $2$ hours of cutting and $2$ hours of finishing. A chair needs $3$ hours of cutting and $1$ hour of finishing. The cutting machine is available for $24$ hours a week and the finishing bench for $16$ hours a week. A table earns a profit of $50$ pounds and a chair earns $40$ pounds. How many of each should be made each week to earn the most?

1. The unknowns. Let $x$ be the number of tables and $y$ the number of chairs made in a week.
2. The objective. The profit in pounds is $Z = 50x + 40y$. We want it as large as possible.
3. The constraints. Cutting: $2x + 3y \le 24$. Finishing: $2x + y \le 16$. Also $x \ge 0$ and $y \ge 0$.
4. The corners. The lines $2x + 3y = 24$ and $2x + y = 16$ meet where $2y = 8$, so $y = 4$ and $x = 6$. The other corners are $(0, 0)$, $(8, 0)$ and $(0, 8)$.

5. The values. At $(0, 0)$, $Z = 0$. At $(8, 0)$, $Z = 400$. At $(6, 4)$, $Z = 300 + 160 = 460$. At $(0, 8)$, $Z = 320$.
6. The answer. The best plan is $6$ tables and $4$ chairs a week. The profit is $460$ pounds.

Check that the plan fits the limits. Cutting uses $2(6) + 3(4) = 24$ hours, and finishing uses $2(6) + 4 = 16$ hours. Both machines are used for their full time. That is why the best corner is where the two lines meet.

Practise with new numbers

Try it in Python

The program lists the profit at each corner. Then it asks linprog for the best plan as a check. Remember that linprog minimises, so it is given the negative profit.

Common mistakes

• Writing the constraints the wrong way round. A table needs $2$ hours of cutting, so the cutting rule has $2x$, not $3x$.
• Forgetting $x \ge 0$ and $y \ge 0$. They are part of the problem.
• Mixing up the objective and a constraint. Profit is what we maximise. Hours are what we are limited by.
• Answering with numbers only. Say $6$ tables and $4$ chairs, and give the profit in pounds.

Remember these

• Name the unknowns, write the objective, write the constraints, then solve.
• Each limit on a resource is a $\le$ inequality. Add $x \ge 0$ and $y \ge 0$.
• The best plan is at a corner. Often it uses all of the scarce resources.
• Answer in words, with units. Check the plan against every limit.

References

• NCERT, Mathematics Textbook for Class XII, Part II, Chapter 12, Linear Programming (manufacturing problems).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A table needs $2$ hours of cutting and a chair needs $3$. There are $24$ hours of cutting. Write the cutting rule.

2. A table earns $50$ pounds and a chair earns $40$ pounds. Write the profit when $x$ tables and $y$ chairs are made.

3. In the worked example, what is the profit of the plan $(7, 1)$? Is the plan allowed?

4. Why is the best plan often at the meeting point of two limits?

5. The worked example gives $x = 6.4$ in another version. What should you ask before giving the answer?