#### Solution:

We will use linear programming method. Let’s use an example here. For the initial problem:

Balance: \$73,845

Suspension
Traction: +30
Aerodynamics: +50
Cost: \$30,000

Sport Stabilizer
Traction: +50
Cost: \$25,000

Transmissions
Power: +20
Traction: +30
Cost: \$35,000

Jet Kit
Power: +50
Cost: \$40,000

Nitrous Kit
Power: +100
Cost: \$80,000

Tires
Traction: +30
Cost: \$18,000

Throttle Body
Power: +25
Cost: \$17,500

Brakes
Traction: +20
Cost: \$12,000

Aerodynamics: +50
Cost: \$10,000

Exhaust
Power: +30
Cost: \$20,000

Air Filters
Power: +20
Cost: \$15,000

### Steps:

1. Assign a variable for each upgrade and find cumulative gain for each. Cumulative gain is addition of gains in each category (Power, Traction and Aerodynamics).
(gain x1000)

 variable upgrade gain a Suspension 80 b Sport Stabilizer 50 c Transmissions 50 d Jet Kit 50 e Nitrous Kit 100 f Tires 30 g Throttle Body 25 h Brakes 20 i Stealth Radiator Cover 50 j Exhaust 30 k Air Filters 20
2. Write down objective function as summation of each upgrade multiplied by its gain.
p = 80a+50b+50c+50d+100e+30f+25g+20h+50i+30j+20k
3. Write down balance constraint as summation of each upgrade multiplied by its cost.
30a+25b+35c+40d+80e+18f+17.5g+12h+10i+20j+15k <= 73
4. Write down supply constraints. Only one upgrade is available for each of them.
a <= 1, b <= 1, c <= 1, d <= 1, e <= 1, f <= 1, g <= 1, h <= 1, i <= 1, j <= 1, k <= 1
5. Go to an online linear programming solver site, ex. http://www.zweigmedia.com/RealWorld/simplex.html
6. Type your problem into problem box.
Maximize p = 80a+50b+50c+50d+100e+30f+25g+20h+50i+30j+20k subject to
30a+25b+35c+40d+80e+18f+17.5g+12h+10i+20j+15k <= 73
a <= 1
b <= 1
c <= 1
d <= 1
e <= 1
f <= 1
g <= 1
h <= 1
i <= 1
j <= 1
k <= 1
7. Click “Solve”. Optimal solution appears below.
p = 193.333; a = 1, b = 1, c = 0, d = 0, e = 0, f = 0.444444, g = 0, h = 0, i = 1, j = 0, k = 0
f (0.444444) is smaller than 1, so there is not enough balance left after buying a,b and i and it will not be bought.

That’s all. Good luck.

1. 1 Dmitry

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2. 2 a

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3. 3 a

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4. 4 a

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5. 5 a

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1. 1 ht wps breaker