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# Linear Programming Using the M-technique

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a) Describe the purpose of using the 2-phase method or the M-technique and highlight the pluses and minuses of these two approaches.

b) Let the linear programming problem (S) be defined as follows:

Minimise Z = 6x1 + 3x2 + 4x3

Such that:

x1 + 6x2 + x3 = 10
2x1 + 3x2 + x3 = 15
x1,x2,x3 >=0

i) Use the M-technique to solve (S) by expressing the new objective function dearly.
ii) Construct the first tableau of phase 1 of the 2-phase method but do not continue to solve the problem.
iii) Produce the dual of (S), say (SD).
iv) Write down the complementary slackness conditions (CSC) for this problem and using the solution found in part (i) apply the CSC to solve the dual problem (SD).
v) Comment on your results found in (i) and (iv) and state the theorem which supports your observation.
(vi) What would the new dual problem be if the variable in (S) x1<=0 is changed to being unrestricted in sign and the second equality 2x1 + 3x2 + x3 = 15 is changed to inequality of the form 2x1 + 3x2 + x3 <= 15 instead?