Group of 45 persons. x super taxis and y mini taxis are used.
capacity of a super taxi = 5 and of a mini taxi = 3
Total number of passengers: 5 x + 3 y ≥ 45 --- (1)
a) Total number of taxis : N = 12 => x + y ≤ 12
b) y ≥ 4
c) see diagram attached.
d) see diagram attached.
The region above the line B is representation of 5 x + 3 y >= 45
The region below the line C (red) is representing x + y <= 12
The region to the right of vertical line D represents y >= 4
shading of unwanted regions.
cost of using a super taxi : $20 and of a mini taxi = $10.
Total cost of using x super taxis and y mini taxis = Cost = $ (20 x + 10 y)
To find the cheapest cost, we need to calculate the Cost on the
boundaries of the unshaded- white region in the graph. The region of
interest is the interior of triangle PQR. We evaluate this function
Cost at P, Q and R:
At P: x = 6.6 y = 4 Cost = $ (20 * 6.6 + 10 * 4) = $ 172
At Q : x = 8 y = 4 Cost = $ 160 + 40 = $ 160
AT R : x = 4.5 y = 7.5 Cost = $ 90 + 75 = $ 165
Minimum cost is at Q , when 8 super taxis and 4 mini taxis are used.
we draw a line on graph representing the cost. It is the green
line. 20 x + 10 y = K = constant. We move this line parallel to
itself. The point it touches the triangle PQR is the lowest cost or the
highest cost. We find that.
f) x + y = 11
case, draw a line (black thin on the graph) x + y = 11. It meets the
earlier lines y >= 4 and 5x + 3y >= 45 at point S : x = 7
and y = 4 and also at T (x = 6 and y = 5)
Cost at S = 20 x + 10 y = $ 140 + 40 = $ 180
Cost at T : = 20 x + 10 y = $ 120 + 50 = $ 170.
In this case the minimum cost point is at P. But as x = 6.6, a
fraction, we make x = 7. So we go to point S with x = 7 and y = 4.
Cost is $ 180.
The minimum cost will be at point T with $ 170. with x = 6 and y = 5.
Revenue from passengers through charges = $ 30 x + 16 y
Revenue at point S: x = 7 , y = 4: = $ 210 + 64 = $ 274
Revenue at point T : x = 6 , y = 5 : = $ 180 + 80 = $ 260
Revenue at point S is more.
Profit = Revenue - Cost :
At point S: $ 274 - $180 = $ 94
at point T : $ 260 - $ 170 = $ 90
at point S, the company makes more profit. That is with 11 taxis being
used, choose 7 super taxis and 4 mini taxis to give maximum profit.