Tarmac Chemical Corporation produces a special chemical compound called Chemix t

Question

Tarmac Chemical Corporation produces a special chemical compound called Chemix that is used in high school chemistry classes. Every 100mg (milligrams) of this compound must contain at least 20mg of sulfur, at least 30mg iron oxide, and at least 30mg but no more than 45mg of potassium. Tarmac’s marketing department has estimated that it will need at least 60kg of this compound to meet expected demand in the next school year. Tarmac can buy three compounds to mix together to produce Chemix. The makeup of these compounds is below in milligrams.


Compound 1: 20mg Sulfur, 60mg Iron Oxide, 20mg Potassium.
Compound 2: 40mg Sulfur, 30mg Iron Oxide, 30mg Potassium.
Compound 3: 10mg Sulfur, 40mg Iron Oxide, 50mg Potassium.

Compounds 1,2 and 3 cost $5.00, $5.25, and $5.50 per kg, respectively. Tarmac wants to find the least costly way of producing Chemix to meet the expected demand.

a. Formulate the problem as a linear program (i.e. define the variables, and write down the objective function and all constraints in algebraic form).

Explanation / Answer

Let xi be the amount of chemical i in compound 1 in mg,   i belongs to {Sulphur, Iron Oxide and Potassiium}

Let yi be the amount of chemical i in compound 2 in mg, i belongs to {Sulphur, Iron Oxide and Potassium}

Let zi be the amount of chemical i in compound 3 in mg, i belongs to {Sulphur, Iron Oxide and Potassium}

Let C1,C2,C3 be the anount of compund 1,2 and 3

Let C be the amount of Chemix made from compound 1, 2 and 3

Our objective is to minimize cost
Min 5*(xS + xI + xP) + 5.25*(yS + yI + yP) + 5.5*(zS + zI + zP)


subject to
Constraint 1: Minimum content of Sulphur, Iron Oxide and Potassium

So, (xS + yS + zS) >= (20/100)*C...............(1)

       (xI + yI + zI) /100 >= (30/100)*C ..............(2)

      (30/100)*C <=(xP + yP + zP ) <=(45/100)*C .............(3)

      C1 = xS + xI + xP; xS == (20/100)*C1; xI == (60/100)*C1; xP == (20/100)*C1

Similarly constraints for compounds 2 and 3.

       C == (C1+C2+C3) ....................(4)

Constraint: Demand

C >= 60*10^6; .......................(5)

Constraint3: Non-negativity
xS, xI, xP >=0 .....................(6)
yS, yI, yP >=0 .....................(7)
zS, zI, zP >=0 .....................(8)

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