# A small firm produces two types of wooden lampstands

A small firm produces two types of wooden lampstands: rounded and angular. Both types require two hand-crafted processes: cutting and smoothing. Rounded lampstands require 1 hour of cutting and 3 hours of smoothing whereas angular lampstands require 2 hours of cutting but only 1 hour of smoothing. The firm has 400 man-hours of cutting available each week and 300 man-hours of smoothing. The firm calculates that they can make 3 Euro profit on each rounded lampstand and 4 Euro profit on each angular lampstand.

• (4a) The problem is to maximise profit. Formulate this as a linear programming problem, giving the three steps and state any assumptions made.

• (4b) Solve the LP problem in (4a) graphically. Hence, state your recommendation for the number of each type of lampstands the firm should produce in order to maximise weekly profit. Give the total profit per week that would be expected given your solution.

I'm stuck on part b. How do I solve it graphically? I've never done that before. Hope you can help. Thanks.

• There are only two unknowns (two types of the product). Draw on the plane the constraint set (polyhedral set) and several level sets of the objective function (lines). Move the level set against the gradient to find the last touch with the constraint set (normally a corner). – A.Γ. Aug 12 '16 at 23:56
• Can you show me a graph of how it should look because it's hard to understand from your instructions. – Tom Aug 13 '16 at 0:03
• You may look here, for example. There are many other examples on Youtube too. – A.Γ. Aug 13 '16 at 0:31

For part b: the feasible allocations of resources are bound by:

$A=$ number of angled lampstands $R =$ number of rounded lampstands

$A \ge 0\\ R \ge 0$

$2A + 1 R \le 400$ (use of cutting resources)

$A + 3 R \le 300$ (use of smoothing resoures)

Graph this region, the most profitable will be at one of the corners.

• Is it not 2A+1R≤4002A+1R≤400 (use of cutting resources) A+2R≤300A+3R≤300 (use of smoothing resoures) – Tom Aug 13 '16 at 0:00
• I have that part I just don't know how to draw it graphically – Tom Aug 13 '16 at 0:01
• Graphing a system of inequalities is something you should have learned in 8th grade. Put $A$ on the $x$ axis, and $R$ on the $y$ axis. Plot the line $2A + R = 400.$ Then shade the the area below it and to the right of it. Do the same for the line $A + 3R = 0.$ Your other two lines are the axes themselves. This leaves a 4 sided shape that describes your feasible allocations. – Doug M Aug 15 '16 at 16:20