Problem 14-01 (Algorithmic)

The RMC Corporation blends three raw materials to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 1/3 ton of material 1 and 2/3 ton of material 3. A ton of solvent base is a mixture of 1/3 ton of material 1, 1/2 ton of material 2, and 1/6 ton of material 3. RMCs production is constrained by a limited availability of the three raw materials. For the current production period, RMC has the following quantities of each raw material: material 1, 10 tons; material 2, 8 tons; material 3, 15 tons. Management wants to achieve the following P₁priority level goals:

Goal 1: Produce at least 21 tons of fuel additive.

Goal 2: Produce at least 18 tons of solvent base.

Assume there are no other goals.

1. Is it possible for management to achieve both P₁ level goals given the constraints on the amounts of each material available?

   	Amount needed to
   Raw Material	Achieve Both P1 Goals
   1
   2
   3

   (Yes, it is possible. No, it is impossible )
2. Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume that both P₁ priority level goals are equally important to management. If you don't need the variable in the model, enter "0". If you need a negative number, enter minus sign with it. If necessary, enter the numbers as a common fraction.

   Let
   	x1 = the number of tons of fuel additive produced
   	x2 = the number of tons of solvent base produced
   	d1+ = the amount by which the number of tons of fuel additive produced exceeds the target value of 21 tons
   	d1- = the amount by which the number of tons of fuel additive produced is less than the target of 21 tons
   	d2+ = the amount by which the number of tons of solvent base produced exceeds the target value of 18 tons
   	d2- = the amount by which the number of tons of solvent base is less than the target value of 18 tons

   Min	d1-	+	d2-
   s.t.
   	13 x1	+	13 x2							Material 1
   	x1	+	12 x2							Material 2
   	23 x1	+	16 x2							Material 3
   	x1	+	x2	+	d1+	+	d1-	=		Goal 1
   	x1	+	x2	+	d2+	+	d2-	=		Goal 2
   x1, x2, d1+, d1-, d2+, d2- 0

3. Choose the correct graph that solves the model formulated in part b.
4. d. If goal 1 is twice as important as goal 2, choose the correct graph for the optimal product mix?

(i) (ii) 90 90 Material 3 Material 3 80 80 70+ 70 60. 60 50 50 (0. 30) 40+ 40 Material 1 30 Material 1 (14, 16) 30 20+ 20+ Material 2 Material 2 Goal 2 Goal 2 10 10 x1 x1 10 50 30 20 30 40 10 20 Tons of Fuel Additive 40 60 Tons of Fuel Add itive (ii (iv) 90 90 x2 x2 Material 3 80 80 Material 3 70 70 60 60 50 50. 40 40 Material 1 Material 1 30 30 20 20+ Material 2 Material 2 Goal 2 Goal 2 10 10 (30, 0) (20, 10) x1 10 20 30 40 50 10 20 30 40 50 Tons of Fuel Additive Tons of Fuel Additive Tons of Solvent Base Goal 1 fons of Solvent Base (ii) (i) 90 90 80- Material 3 80- Material 3 70- 70 60 60 60+ (0, 30) 60 40 Material 1 30 Material 1 (14, 16) 30 20+ 20+ Material 2 Material 2 Goal 2 Goal 2 10 10 x1 x1 10 30 40 10 30 20 50 20 40 50 Tons of Fuel Additive Tons of Fuel Additive Tons of Solvent Base Goal1 Tons of Solvent Base Goal 1 (iii (iv) 90 90 x2 x2 Material 3 Material 3 80- 80 70 70 60- 60- 50 40 40 Material 1 Material 1 30 30 20- 20 Material 2 Material 2 Goal 2 Goal. 10- 10 (30, 0) x1 (20, 10) K1 10 30 10 20 40 50 20 30 A0 50 Tons of Fuel Additive Tons of Fuel Additive Tons of Solvent Base Goal 1 8 g Goal 1