# Linear Programming Sum

The Handy-Dandy Company makes three types of kitchen appliances ($A$, $B$ and $C$). To make each of these appliance types, just two inputs are required - labour and materials. Each unit of $A$ made requires $7$ hours of labour and $4$ kg of materials; for each unit of $B$ made the requirements are $3$ hours of labour and $4$ kg of materials, while for $C$ the unit requirements are $6$ hours of labour and $5$ kg of material. The company expects to make a profit of $40$€ for every unit of $A$ sold, while the profit per unit for $B$ and $C$ are $20$€ and $30$€ respectively. Given that the company has available to it $150$ hours of labour and $200$ kg of material each day, formulate this as a linear programming problem.

Formulation:

Step 1: Identify decision variables (what decisions can be made?)

For this production mix problem the company can decide on how many units of appliances $A$, $B$ and $C$ to make. Therefore, the decision variables are, say:

Let $x_1$ = number of units of appliance $A$ to make per day

Let $x_2$ = number of units of appliance $B$ to make per day

Let $x_3$ = number of units of appliance $C$ to make per day

Step 2: Identify restrictions or constraints

Evidently, $x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$.

Also, there are limitations (or constraints) on labour and materials:

Labour (per day): $7x_1 + 3x_2 + 6x_3 \leq 150$

Materials (per day): $4x_1 + 4x_2 + 5x_3 \leq 200$

Note: Here we have assumed linearity (scaling and additivity).

Step 3: Identify “best” criterion

It seems clear that profit should be maximised, that is the objective function is to maximise $40x_1 + 20x_2 + 30x_3$.

Note: Here also we have assumed linearity (scaling and additivity).

I'm having trouble understanding how they can up with the Maximise $40x_1 + 20x_2 + 30x_3$ as the final answer. Can someone explain how they go this. Thank You.

• "Maximise $40X_1 + 20X_2 + 30X_3$" is just a translation of "The company expects to make a profit of €40 for every unit of A sold, while the profit per unit for B and C are €20 and €30 respectively" and an assumption that the company wishes to maximise profits. This is not the final answer but the objective function, as you also have to to apply the labour and materials constraints – Henry Aug 10 '16 at 14:31
• Oh ok thanks. So is that all I need to write when I'm doing this type of question or do I actually need to figure out how many of each appliance A,B,C I need to make to maximise profit? Thanks – Tom Aug 10 '16 at 14:35
• The question says "formulate this as a linear programming problem" and the model answer is indeed a formulation, though it does not solve the problem. All the steps are part of the formulation – Henry Aug 10 '16 at 14:37
• If they gave, as an answer, that the formulation of the LP was "Maximize $40x_1+20x_2+30x_3$" (without mentioning constraints) then the solution provided was incomplete. An LP has an objective function (which is what they presented as the formulation) and a set of constraints. – TravisJ Aug 10 '16 at 15:56