Section One
Problem Formulation
1. Linear Programming Problem (LPP) Formulation:
It is the process of formulating the case study (text) into a linear programing Model.
 It’s consists of:
a. Decision Variables: what the decision maker can alter to attain his goals (X1, X2 …).
b. Objective Function: the Goals (MAX, MIN).
c. Constraints: the Resources and logic Limitations (<=, =, >=).
2. LPP Types:
a. Product Mix
b. Diet Problem.
c. Blend Problem.
d. Crop Mix Problem.
e. Transportation / Assignment Problem.
f. Portfolio Selection.
Examples:
1. Diet Problem:
 A student is trying to decide on lowest cost diet that provides sufficient amount of
protein, with two choices: steak: 2 units of protein pound, $3 pound; peanut butter: 1 unit
of protein pound, $2 pound in proper diet, need 4 units protein day. Formulate this LPP to
help this student to decide the number of pounds to be taken with as minimum cost as
possible.
Solution:
1. DVs:
x = # of pounds peanut butter/day in the diet.
y = # of pounds steak/day in the diet.
2. Objective Function:
minimize z = 2x + 3y (total cost)
3. Constraints:
x + 2y ≥ 4
x ≥ 0, y ≥ 0
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2. Product Mix:
 Goldilocks needs to find at least 12 lbs of gold and at least 18 lbs of silver to pay the
monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day
that Goldilocks spends in mine 1, she finds 2 lbs of gold and 2 lbs of silver. Each day that
Goldilocks spends in mine 2, she finds 1 lb of gold and 3 lbs of silver. Formulate an LP to
help Goldilocks meet her requirements while spending as little time as possible in the
mines.
Solution:
1. DVs:
𝑥1 = no. of time units (days) spent in mine 1.
𝑥2 = no. of time units (days) spent in mine 2.
2. Objective Function:
minimize z = 𝑥1 + 𝑥2
3. Constraints:
2𝑥1 + 𝑥2 ≥ 12
(Gold)
2𝑥1 + 3𝑥2 ≥ 18
(Silver)
𝑥1 , 𝑥2 ≥ 0
3. Blend Problem:
 A company needs to blend iron ore from 4 mines in order to produce tire tread. To assure
proper quality, minimum requirements of aluminum, boron, and carbon must be present
in the final blend. In particular, there must be 5, 100, & 30 pounds, respectively, in each
ton of ore. These elements exist at different levels in the mines. In addition, the cost of
mining differs for each mine. Given the data in the following table; State how much should
be mined from each mine in order to attain a proper blend per ton at minimum cost?
Mine
Aluminum
Boron
Carbon
Cost
1
10
90
45
2
3
150
25
3
8
75
20
4
2
175
37
800$
400$
600$
500$
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Solution:
1. DV:
𝑥𝑖 : 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑜𝑛 𝑡𝑜 𝑏𝑒 𝑚𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝑚𝑖𝑛𝑒 − 𝑖, where i = 1,2,3,4
2. Objective Fun:
min 𝑧 = 800𝑥1 + 400𝑥2 + 600𝑥3 + 500𝑥4
3. Constraints
10𝑥1 + 3𝑥2 + 8𝑥3 + 2𝑥4 ≥ 5
90𝑥1 + 150𝑥2 + 75𝑥3 + 175𝑥4 ≥ 100
45𝑥1 + 25𝑥2 + 20𝑥3 + 37𝑥4 ≥ 30
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 1
𝑥𝑖 ≥ 0
Good Luck 
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