Solution Key
Average=21.58 Range 15-25
BOSTON COLLEGE
GRADUATE SCHOOL OF MANAGEMENT
Operations and Strategic Management Department
MD716 Modeling and Decision Analysis
Quiz 1
Spring 2000
RMC, Inc. is a firm that produces a variety of chemical-based products. In a particular
production process, three raw materials are used to produce two products: a fuel additive
and a solvent base. The fuel additive is sold to oil companies and is used in the
production of gasoline and related fuels. The solvent base is sold to a variety of chemical
firms and is used in both home and industrial cleaning products. The three raw materials
are blended to form the fuel additive and solvent base. A ton of solvent base is a mixture
of .4 tons of material 1 and .6 tons of material 3. A ton of solvent base is a mixture of .5
tons of material 1, .2 tons of material 2 and .3 tons of material 3.
RMC’s production is constrained by a limited availability of the three raw materials. For
the current period, RMC has available 20 tons of material 1, 5 tons of material 2, and 21
tons of material 3. Because of spoilage and the nature of the production process, any
materials not used for current production are useless and must be discarded.
The accounting department has reviewed the relevant costs and prices for both products
and calculated a profit contribution of $40 for every ton of fuel additive produced and
$30 for every ton of solvent based produced. RMC’s management, after an analysis of
potential demand, has concluded that the prices established will ensure the sale of all the
fuel additive and solvent base produced. RMC would like to determine how many tons
of each product to produce in order to maximize total profit contribution.
The mathematical model for this problem is as follows:
Maximize
40 F  30S
Subject to :
.4 F  .5S  20
0.F  .2 S  5
.6 F  .3S  21
F, S  0
Where F=tons of fuel additive produced and S=tons of solvent base produced.
Attached are the following materials for this problem: a graph of the model constraints, a
spreadsheet model and Solver inputs, a Solver answer report, and a Solver sensitivity
report. Use this information to answer the following questions.
1. Clearly identify on the graph all of the product mixes that RMC has the capability to
produce (i.e. the feasible region).
: 0.6 F + 0.3 S = 21.0
S
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
: 0.0 F + 0.2 S = 5.0
: 0.4 F + 0.5 S = 20.0
Payoff: 40.0 F + 30.0 S = 1600.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Optimal Decisions(F,S): (25.0, 20.0)
: 0.4F + 0.5S <= 20.0
: 0.0F + 0.2S <= 5.0
: 0.6F + 0.3S <= 21.0
F
2. Suppose that RMC is uncertain about the contribution to profit of the fuel additive
and solvent base. That is, suppose that the objective function is linear but its
coefficients have not been specified. Which of the following solutions could possibly
be an optimal solution? Explain
a. Tons of fuel additive, F=20, tons of solvent base, S=20
This point cannot be optimal because it is on the interior of the feasible region
b. Tons of fuel additive, F=20, tons of solvent base, S=25
This point cannot be optimal because it is not feasible.
c. Tons of fuel additive, F=18.75, tons of solvent base, S=25
This point can be optimal because it is at a corner point of the feasible region.
3. Given the objective function, Maximize 40F+30S, what is the optimal solution to the
problem? How many tons of fuel additive should RMC produce? How many tons of
solvent base should they produce? How much will this product mix contribute to
profit? How many tons of material 1 are they using? How many tons of material 2
are they using? How many tons of material 3 are they using?
From the answer report:
Production of fuel additive=25 tons
Production of solvent base=20 tons
Total contribution is $1600.00
20 tons of material 1 are used
4 tons of material 2 are used
21 tons of material 3 are used
4. Suppose that upon further consideration, management believes that there is demand
for at most 30 tons of fuel additive. How would this change the mathematical model?
Would the total contribution to profit increase, decrease, or stay the same? Why
F  30
This is a non-binding constraint. The optimal production of fuel additive is 25 tons.
Therefore, the total contribution will not change.
5. Suppose that management is able to obtain an additional 5 tons of material 3. What
impact will this have on total contribution to profit? Explain. Will production of the
fuel additive increase or decrease? Why? Will production of the solvent base
increase or decrease? Why? (Base your answer on the original model presented in
the attachments.)
.The
increase of 5 tons is within the allowable increase for material 3. The shadow price
for material 3 is $44.44. Therefore, the total contribution to profit will increase by
5*44.44=$222.22.
Increasing the right hand side of constraint 3 causes a parallel shift outward of the
constraint line. This causes the optimal corner to move to the right and down (see the
attached graph. Thus, the production of fuel additive will increase and the production of
solvent base will decrease.
: 0.6 F + 0.3 S = 26.0
S
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
: 0.0 F + 0.2 S = 5.0
: 0.4 F + 0.5 S = 20.0
Payoff: 40.0 F + 30.0 S = 1822.2
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
Optimal Decisions(F,S): (38.9, 8.9)
: 0.4F + 0.5S <= 20.0
: 0.0F + 0.2S <= 5.0
: 0.6F + 0.3S <= 26.0
F