Question 1206298: A box company makes small and large wooden boxes. Small boxes require 0,8 square metres of wood, while large ones require 1,4 square metres. All boxes require 0,5 hours of labour, regardless of size. Wood is limited to 42 square metres, and only 24 hours of labour are available. Due to warehouse space limitations, no more than 20 large boxes can be made each day. Also, demand by customers for small boxes is limited to a maximum of 30 boxes. Each small box yields a profit of R42,00 and each large box earns only R14,00.
Formulate a linear programming model for the company as follows;
(1) Clearly identify the two decision variables in respect of each product to be produced.
Let x1 = _____________________________________________________
Let x2 = _____________________________________________________
PAGE 4
(2) Write down the objective function that maximizes total profit.
Maximize z = ________________________________________________
(3) Formulate the constraints in respect of the two resources used in producing the bowls and mugs, warehouse space limitation, and demand limitation.
______________________________________________ m2 of wood
______________________________________________ labour hours
______________________________________________ warehouse
______________________________________________ demand
(4) Indicate that all constraints are non-negative.
x1 ≥ 0 or x1, x2 ≥ 0
x2 ≥ 0
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of small boxes.
y = number of large boxes.
constraint inequalities are:
y <= 20
x <= 30
.8x + 1.4y <= 42
.5x + .5y <= 24
x >= 0]
y >= 0
objective function:
profit = 42x + 14y
using the decmos.com calculator, you would graph the opposite of the inequalities.
the feasible region is the area on the graph that is not shaded.
you would evaluate the objective function at each corner point to find the maximum profit.
since partial boxes are not allowed, you would round to the next lowest integer.
the profit at each corner point is shown below.
(0,20) = 840
(17.5,20) = (17,20) = 994
(30,12.857) = (30,12) = 1428
(30,0) = 1260
the maximum profit is at (20,12) where profit is equal to 42 * 30 + 12 * 14 = 1428.
all constraints are satisfied, such as:
wood constraint at .8 * 30 + 1.4 * 12 = 40.8 <= 42 is satisfied.
labor constraint at .5 * 30 + .5 * 12 = 21 <= 24 is satisfied.
0 <= x <= 30 is satisfied.
0 <= y <= 20 is satisfied.
here's what the graph looks like:
Answer by ikleyn(52786)  (Show Source):
You can put this solution on YOUR website! .
This minimax problem is very special.
It is special, because it can be easily solved MENTALLY, based on common sense ONLY,
without using heavy artillery technique of linear programming.
Indeed, after reading the post, it should be clear that the winning strategy
is the most aggressive strategy making SO MANY small boxes as possible,
until the restrictions allow do it; and when the restriction on small boxes
is reached, then to switch making large boxes, until the restrictions on
large boxes is reached.
It is so, because the profit of each small box is greater than the profit of each
large box (R42 against R14), while each small box requires less amount of
material than each large box (0.8 sq. m against 1.4 sq. m).
So, it is OBVIOUS that to produce small boxes is more profitable.
The other restrictions (total wood and total working time) do not give a preference to any
sort of production, and the limitation on the numbers of small boxes and large boxes (like
the warehouse space limitation and the demand) are in favor of small boxes.
So, following to this idea, as many of small boxes should be produced, making 30 small boxes.
(limited by the demand).
It will require 0.8*30 = 24 sq.m of wood, leaving 42-24 = 18 sq.m of wood for large boxes
and will require 0.5*30 = 15 hours of labor, leaving 24-15 = 9 hours for large boxes.
Now we determine the number of large boxes. The limitations are
18/1.4 = 12.86 for wood;
9/0.5 = 19 from labor hours;
20 from the warehouse space limitations.
So, for large boxes the number is 12.
The maximum profit is 30*42 + 12*14 = 1428 monetary units.
ANSWER. The optimal strategy is to make 30 small boxes and 12 large boxes,
making the maximum possible profit of R1428.
Solved.
From my point of view, this solution is much more educative than applying
heavy artillery of the linear programming method without necessity.
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