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 1/4 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 P 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? Explain. Amount needed to Raw Material Achieve Both P Goals 1 2 3 Since there are only 15 tons of Material 1 available, it is G > b. Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume that both P 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 d = 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 d+ = the amount by which the number of tons of solvent base produced exceeds the target value of 16 tons d2 = the amount by which the number of tons of solvent base is less than the target value of 16 tons Min d + d s.t. + Material 1 X1 + Material 2 Material 3 d+ Goal 1 + d2+ + Goal 2 X1, X2, d+, di, d+, d 0 X1 X1 X1 X1 + + + X2 X2 X2 X2 X2 d d VI VI VI || || 2. c. Choose the correct graph that solves the model formulated in part b. (0) (1) Goal I 80 Material 3 70+ 60+ 50+ 40- 30-Material I 20- 10- 0 10 80- Material 3 70+ 60- 790 20 30 40 Goal I Material 2 Goal 2 (33, 13) 50 60 21 (iv) 790 80- 70+ 60+ 50+ 40+ 30 Material 1 20- 10- (0) Material 3 (28, 16) 20 10 80- Material 3 70+ 60- 30 Goal 1 40 Goal I Material 2 Goal 2 + 50 60 21 780 80- Material 3 70+ 60+ 50+ 40+ (12, 24) 30-Material 1 20 10- 0 10 20 30 Goal 1 40 Material 2 50 Goal 2 60 21 (iv) 80- 70+ 60+ 50+ 40+ 30 Material 1 20 10- 0 Material 3 10 (32, 14) + 20 30 Goal 1 40 Material 2 50 Goal 2 60 21 d. If goal 1 is twice as important as goal 2, choose the correct graph for the optimal product mix? (i) (ii) 20 Goal 1 80- 70+ 60+ 50+ 40- 30 Material 1 20 10- 0 10 Material 3 20 30 40 Material 2 Goal 2 (33, 13) 50 60 21 490 80- 70+ 60+ 50+ 40+ 30- 20 10- 0 Material 3 Material 1 10 (28, 16) 20 30 Goal I 40 Material 2 Goal 2 + 50 60 21 (1) 790 80- 70+ 60- 50- 40+ 30 Material 1 20- 10- 0 10 Material 3 (12,24) 20 30 Goal I 40 Material 2 50 Goal 2 60 21 (iv) 200 80 70+ 60+ 50- 40+ 30 Material 1 20- 10- 00) Material 3 10 (32, 14) 20 30 Goal I 40 Material 2 + 50 Goal 2 60 201