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# Tutorial 6 - (d Is it necessary to increase the inspection...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 6 1. Solve the following linear program by the big-M method min 3 x 1 + 2 x 2 + 3 x 3 s.t. 2 x 1 + x 2 + x 3 2 3 x 1 + 4 x 2 + 2 x 3 8 x 1 , x 2 , x 3 0 . 2. A sports factory manufactures three types of footballs which require operations in three diﬀerent departments. The production times and maximum production availabilities are shown below: Production Time Type Sewing A 12 min 15 min 3 min B 10 min 15 min 4 min C 8 min 12 min 2 min Time Available 300 hours 200 hours 100 hours Current orders indicate that at least 1000 type A footballs should be manufactured. The proﬁts for each type A, B, and C football are respectively \$3, \$5, and \$4. (a) Formulate a linear programming problem to maximize proﬁt. (b) Can you obtain a feasible solution? Why? (c) Suppose the sewing time is increased to 300 hours and inspection and packaging time is increased to 150 hours by using overtime. What is the optimal solution?

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Unformatted text preview: (d) Is it necessary to increase the inspection and packaging time to reach the maximum proﬁt? 3. Consider the problem max x 1 + 2 x 2 + 3 x 3 s.t. x 1 + 2 x 2 + 3 x 3 ≤ 10 x 1 + x 2 ≤ 5 x 1 ≤ 1 x 1 ,x 2 ,x 3 ≥ Find all optimal basic feasible solutions and write a general expression for all (basic and nonbasic) optimal solutions. 1 4. Consider the following problem: max 3 x 1 + x 2 s.t. x 1 + 2 x 2 ≤ 5 x 1 + x 2-x 3 ≤ 2 7 x 1 + 3 x 2-5 x 3 ≤ 20 x 1 ,x 2 ,x 3 ≥ (a) In which direction is the solution space unbounded? (b) Without further computations, can one conclude that the problem has no bounded optimal solution? If not, ﬁnd the optimal solution. (c) Is the optimal solution degenerate? 5. Solve the following minimization problem: min 4 x 1 + 4 x 2 + x 3 s.t. x 1 + x 2 + x 3 ≤ 2 2 x 1 + x 2-3 x 3 ≥ 5 x 1 ,x 2 ,x 3 ≥ . 2...
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Tutorial 6 - (d Is it necessary to increase the inspection...

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