Question

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 1/4 ton of material 1 and 3/4 ton of material 3. A ton of solvent base is a mixture of 1/2 ton of material 1, 1/8 ton of material 2, and 3/8 ton of material 3. RMC's 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, 15 tons; material 2, 3 tons; material 3, 30 tons. Management wants to achieve the following P1 priority level goals: Goal 1: Produce at least 32 tons of fuel additive. Goal 2: Produce at least 16 tons of solvent base. Assume there are no other goals. a. Is it possible for management to achieve both P, level goals given the constraints on the amounts of each material available? Amount needed to Raw Material Achieve Both P1 Goals 1 16 2 2 3 30 No, it is impossible b. 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 dit = the amount by which the number of tons of fuel additive produced exceeds the target value of 32 tons di = the amount by which the number of tons of fuel additive produced is less than the target of 32 tons dz? = the amount by which the number of tons of solvent base produced exceeds the target value of 16 tons da = the amount by which the number of tons of solvent base is less than the target value of 16 tons
everything is correct I just need the answer to part D on which graph is right.

Answer 1

Sol:

The Given data is tabulated as follows:

Resources	Requirement of each resource per ton	Available
	Fuel Additive	Solvent Base
RM 1 (in ton)	2/5	1/2	20
RM 2 (in ton)	0	1/5	5
RM 3 (in ton)	3/5	3/10	21

a.

	Amount needed to produce x1 = 30 and x2 = 15	Available amount	Comment
Raw Material
1	(2/5)(30) + (1/2)(15) = 12 + 7.5= 19.5	20	Requirement <Available
2	(0)(30) + (1/5)(15) = 3	5	Requirement <Available
3	(3/5)(30) + (3/10)(15) = 18 +4.5 = 22.5	21	Requirement >Available

To satisfy both the goals, raw material 3 required is 22.5 units, but available is only 21 units, thus it is not possible to achieve both goals. Any of the goal will be underachieved.

b.

Min	__1__d1-	+	__1__d2-
s.t.
	__2/5__x1	+	_1/2__x2					≤	20	_____Material 1
	_0__x1	+	_1/5___x2					≤	5	______Material 2
	__3/5__x1	+	__3/10__x2					≤	21	_____Material 3
	__1__x1	+	__0__x2	+	__−1__d1+	+	__1__d1-	=	30	______Goal 1
	__0__x1	+	__1___x2	+	___−1__d2+	+	__1__d2-	=	15	_______Goal 2

c.

Graph	Extreme point	d1-	d2-	Objective function Value Z = d1- + d2-
i	(18.75, 25)	30 – 17.75 = 11.25	0	11.25
ii	(30, 10)	30-30 = 0	15 – 10 = 5	5
iii	(27.5, 15)	30-27.5 = 2.5	15-15 = 0	2.5
iv	(25, 20)	5	0	5

At the point (27.5, 15) the objective function value is minimum.

The objective function is minimum at point (27.5, 15), graph (iii) is correct graph, optimal product mix is:

fuel additive = 27.5 tons

solvent base = 15 tons

Correct Option: Graph (iii)

d.

If the goal 1 is twice important a goal 2, new objective function is Min: P₁d₁^- + P₂d₂^-

P₁ = first priority to achieve at least 30 units of x₁

P₂ = second priority to achieve at least 15 units of x₂

First identify point at which d₁^-is minimum, from four points the d₁^-is minimum at (30, 10).

Graph	Extreme point	d1-	d2-	Objective function Value Z = d1- + d2-
i	(18.75, 25)	30 – 17.75 = 11.25	0	11.25
ii	(30, 10)	30-30 = 0	15 – 10 = 5	5
iii	(27.5, 15)	30-27.5 = 2.5	15-15 = 0	2.5
iv	(25, 20)	5	0	5

Thus, optimal product mix is:

fuel additive = 30 tons

solvent base = 10 tons

Correct Option: Graph (ii)

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