Midterm Questions

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University of Toronto
Department of Mechanical & Industrial Engineering
MIE165S - Modelling Integrated Systems
Midterm
Thursday, Feb. 26, 1998, 3:15pm - 4:45pm, Woodsworth 111

Notes:
1. Attempt all 4 questions. All questions are of equal value.
2. If doubt exists as to the interpretation of any question, you are urged to submit with your answer a clear statement of any assumption you have made.

Question 1 (15 marks total):

In assignment #3, you researched the year 2000 problem. State two essential actions required to solve the problem in each of the (i)analysis, and (ii)synthesis stages. (Provide a brief explanation of action.)

Question 2 (15 marks total):

SteelCo is required by the environment standards to reduce its annual emission of three types of pollutants, particulates, sulphur oxides and hydrocarbons by the amounts shown in Table 1.

Pollutant	Required reduction in annual emission (tonnes)
Particulates	30
Sulphur oxides	80
Hydrocarbons	60

Table 1. Clean Air Standards for SteelCo

Three abatement methods can be used to achieve the required reduction: (1) increasing the heights of the smoke stacks, (2) using filtering devices in the smoke stacks, and (3) using better fuels for the furnaces. Each method has a maximum technological capacity on how much emission of the pollutants it can reduce simultaneously, as shown (in tonnes per year) in Table 2, and any fraction of its full reduction capacity can be used independent of the other methods.

	Taller smoke stacks	Filters	Better fuels
Particulates	12	25	17
Sulphur oxides	35	18	56
Hydrocarbons	37	28	29

Table 2. Full Emission Reduction Capacities of the Abatement Methods

The total annual cost (in thousands of dollars) for using each abatement method at its full reduction capacity has been estimated and given in Table 3.

Method	Cost
Taller smoke stacks	8
Filters	7
Better fuels	11

Table 3. Annual Cost for Using the Abatement Methods at Their Full Reduction Capacities

The cost of fractional use of each method is proportional to its fractional capacity used.

Formulate a linear programming model for SteelCo to determine what fraction of the full capacity of each abatement method should be used to minimize the total cost.

Question 3 (15 marks total):

Solve the following LP problem using the graphical method.

Maximize        Z = 3X1+ 2X2
subject to
                -X1 + 3X2 <= 12
                 X1 +  X2 <= 8
                2X1 -  X2 <= 10
                  X1, X2 >= 0.

You must submit your graph, and state clearly the optimal solution of (X1, X2) and the optimal objective value.

Question 4 (15 marks total):

The following payoff matrix indicates the expected profit, in thousands, for four marketing strategies (namely M1, M2, M3, M4) and four potential levels of sales (namely S1, S2, S3, S4). An expert has assigned the probabilities of 0.2, 0.3, 0.4, and 0.1 to the potential levels of sales of S1, S2, S3, and S4, respectively. The payoff matrix is as follows:

	S1(0.2)	S2(0.3)	S3(0.4)	S4(0.1)
M1	-20	140	90	220
M2	-30	80	70	110
M3	40	100	200	120
M4	60	110	100	200

In each of the following cases, justify your answer by showing all calculations.

(a)(3 marks) State the best alternative by using the most probable future criterion.

(b)(3 marks) State the best alternative by using the expected value criterion.

(c)(3 marks) Suppose that the probabilities are not available. State the best alternative by using the Laplace criterion.

(d)(3 marks) Suppose that the probabilities are not available. State the best alternatives by using the maximin criterion.

(e)(3 marks) Suppose that the probabilities are not available. State the best alternative by using the Hurwicz criterion with alpha = 0.4.