Tutorial 5 - earned from each litre of drink is shown...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 5 1. Consider a minimization linear programming problem in standard form. Let x be a basic feasible solution associated with the basis B . Prove the following: (a) If the reduced cost of every nonbasic variable is positive, then x is the unique optimal solution. (b) If x is the unique optimal solution and is nondegenerate, then the reduced cost of every nonbasic variable is positive. 2. The following is a simplex tableau of a minimization LP problem. Basic x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 Solution ¯ c 0 - 5 0 4 - 1 - 10 0 0 620 x 8 0 3 0 - 2 - 3 - 1 5 1 12 x 3 0 2 1 3 1 0 3 0 6 x 1 1 - 1 0 0 6 - 4 0 0 0 Determine the leaving variable and the resulting increase or decrease in the cost if the entering variable is (a) x 2 , (b) x 4 , (c) x 5 , (d) x 6 , (e) x 7 . 3. Fatimah’s drink stall sells three types of drink daily. Each type of drink is made from milk and rose syrup. Sixty litres of milk and twenty five litres of rose syrup are available daily. The composition of each type of drink per litre and the profit
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Unformatted text preview: earned from each litre of drink is shown below: Drink Amount of milk Amount of rose syrup Profit Type (litre) (litre) ($) 1 0.7 0.3 0.7 2 0.8 0.2 0.5 3 0.9 0.1 0.4 Past experience indicates that the total amount of drink of Type 1 and Type 2 sold is not more than the total amount of drink of Type 3 sold. Formulate an LP problem that will maximize profit and find the optimal solution using the simplex method. 1 4. Consider the following LP problem (P): max 3 x 1 + 2 x 2 + 3 x 3 s.t. 2 x 1 + x 2 + x 3 = 2-x 1-4 x 2-2 x 3 ≤ -6 x i ≥ ,i = 1 , 2 , 3 . (a) Add artificial variables where necessary, and write down the auxiliary problem for Phase 1 of the 2-phase method. (b) Solve the auxiliary problem in (a) and determine a basic feasible solution for the given LP problem (P). (c) Proceed to Phase 2 to solve the given LP problem (P). 2...
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Tutorial 5 - earned from each litre of drink is shown...

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