Southern Sporting Company

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Chapter 3, Problem 8
Information from problem 7.
Formulate LPM to determine the number of basketballs and footballs to produce in order to maximize profit. X1 -- # of basketballs
X2 -- # of footballs
Maximize Z = 12x1 + 16x2
Subject to:
3x1 + 2x2 ≤ 500
4x1 + 5x2 ≤ 800
X1, x2 ≥ 0
Transform this model to standard form.
Maximize Z = 12x1 + 16x2 + 0s1 +0s2
3x1 + 2x2 + s1 = 500
4x1 +5x2 + x2 = 800
X1, x2, s1, s2 ≥ 0
a). Identify the amount of unused resources (slack) at each of the graphical points. 3x1 + 2x2= 5004x1 + 5x2 = 800
X1 = 0, x2 = 250 X1 = 0, x2 = 160
X2 = 0, x1 = 500/3X2 = 0, x1 = 200
Maximize Z = 12x1 + 16x2
At point A (0, 0)Z = 0
At point B (500/3, 0) Z = 12(500/3) + 16(0) Z = 2000
At point D (0, 160) Z = 12(0) + 16(160)Z = 2560
At point C (900/7, 400/7) Z = 12(900/7) +16(400/7) Z = 2457

Point D (0, 160) is an optimal solution. X1 = 0, x2 = 160, Z = 2560 Unused resources:
At point D (0, 160)
3x1 + 2x2 + s1 = 5004x1 +5x2 + s2 = 800
3(0) + 2(160) + s1 = 5004(0) + 5(160) +s2 = 800
S1 = 500 -320s2 = 800 - 800
S1 = 180s2 = 0

At point C (900/7, 400/7)
3(900/7) +2(400/7) + s1 = 5004(900/7) + 5(400/7) + s2 = 800 2700/7 +800/7 + s1 = 5003600/7 +2000/7 + s2 = 800
S1 = 3500/7 -500s2 = 800 – 5600/7
S1 = 0s2 = 0

At point B (500/3, 0)
3(500/3) + 2(0) + s1 = 5004(500/3) +5(0) +s2 = 800
500 + s1 = 5002000/3 + s2 = 800
S1 = 0s2 = 400/3

At point A (0, 0)
3(0) +2(0) + s1 = 5004(0) +5(0) +s2 = 800
S1 = 500s2 = 800

b). What would be effect on the optimal solution if the profit for a basketball changed from $12 to $13? Z = 12x1 + 16x2 becomes Z = 13x1 + 16x2
At point D (0, 160) Z = 2560
At point B (500/3, 0)Z = 2166.67
At point C (900/7, 400/7) Z = 2585.71
Point C would become an optimal solution if profit for basketballs changed from $12 to $13.

What would be the effect if the profit for a football changed from $16 to $15?...
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