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QUESTION 1: Linear Programming and Sensitivity Analysis (22 points)

Hanna Joy LTD. produces four products: W, X, Y and Z, using three resources, namely wood, metal and labour. The first three constraints in the algebraic formulation and SOLVER output below are resource availability constraints. Hanna Joy LTD. must produce a total of at least 205 units of the first two products and proper use of his machinery requires that the amount of Y produced be at exactly 1/3 of the amount of Z produced. The tables below give the revenue per unit for the products, and the cost per unit for the resources.

Product Revenue per unit Resources Cost per unit

W $82 Wood (meters) $3

X $65 Metal (meters) $5

Y $131 Labour (hours) $4

Z $63

Here is a correct algebraic formulation of the problem.

Let W, X, Y and Z denote the number units of products W, X, Y and Z to produce.

Maximize 45W + 26X + 53Y + 34Z

subject to: 3W + 10X + 2Y + 5Z ≤ 2900

4W + X + 12Y + 2Z ≤ 400

2W + X + 3Y + Z ≤ 300

W + X ≥ 205

Y - 0.33Z = 0

W, X, Y and Z ≥ 0

The SOLVER solution is shown below.

W X Y Z

0.00 286.05 2.33 6.98

45 26 53 34 7797.67

Wood (meters) 3 10 2 5 2900.00 <= 2900

Metal (meters) 4 1 12 2 327.91 <= 400

Labour (hours) 2 1 3 1 300.00 <= 300

First two products 1 1 0 0 286.05 >= 205

Y and Z requirement 0 0 1 -0.33 0.00 = 0

Instructions for parts (a) to (c) below: In these three parts, the problem above is altered by adding additional features to the problem description. In each part, you are to provide a correct linear programming formulation of the new problem. You do not necessarily have to rewrite the whole formulation, as long as you can clearly and unambiguously indicate what should be done. The three parts are independent of each other (i.e. when doing one of the parts, ignore all of the others).

(a) The price per meter of metal has risen to $6. (Hint: rewrite the objective function equation). (3 points)

(b) A total of at most $10,000 is available for acquisition of the wood material. (Hint: add an additional constraint). (3 points)

(c) Hanna Joy LTD. decided that the amount produced of product Y must constitute at least 30% of the total number of units produced. (Hint: add an additional constraint). (2 points)

The following correct output for this problem in its original form is provided below as an Excel Solver output. You will need this output to answer questions (d) through (g) each part of which is to be considered independently of all others.

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