Page 597 - Sensors and Control Systems in Manufacturing

550
Ele v e n
Cha p te r

x ≤ 300 No more than 300 drill hours are available for product 2,
2
and 1 h per unit is required.
x + x ≤ 500 No more than 500 polishing hours are available for
1 2
both products, with 1 hour per unit required in both cases.
Solving the problem. The problem for the values given previously
is as follows:
Many solutions would satisfy all three constraints, but there is only
one maximizing solution. The possible solutions are shown graphically
in Fig. 11.5. The axes represent the number of product units; the remain-
ing solid lines, the production characteristics. Any point in the shaded
area will satisfy the constraints, but the maximizing solution will be at
one of the four corners of this area—point A, B, C, or D.
This point is the exact one which can be found by drawing a line that
represents the profit function 2x + 5x (the lower dotted line in Fig. 11.5).
1 2
When this line is shifted upward, always parallel to its original position,
it finally reaches the point where it touches the shaded solution space at
only one point, B. This point, where x = 200 and x = 300, is the solution
1 2
that gives the highest possible total contribution within the limited
resource constraints. The solution gives the following results:
Total contribution $ 1900 (200 × $2 + 300 × $5)
Lathe time used 200 h (200 × 1 h)
Drill time used 300 h (300 × 1 h)
Polishing time used 500 h (200 × 1 h + 300 × 1 h)

FIGURE 11.5 Profi t function.