Of the three options, production of 10 units of M and 5 units of Nwould cost the least, viz., Rs. 1050. So, this is the optimum production mix.4.5 USING SIMPLEXMETHOD TO SOLVE LPPLPP can be solved through another method called simplex method.It was mentioned earlier that graphical method is effective only when 2 variablesare there. If there are more number of variables, different method is called for.Simplex method is the other method.

107Simplex method is an interactive (repeat) procedure which eithersolves a LPP in a finite number of steps or gives an indication that there is anunbounded solution to the given LPP. Here we use some terms likeslackvariable or slack resource, artificial variable or resource basic feasiblesolution, improvement index, optimal column, intersectional elements, etc.These will be explained as we take up solutions to LPPs.4.5.1 Maximization through simplex methodThe simplex method, the iterative method, is now adopted to solvea maximization problem.Illustration: 7We may now take the illustration 1 we discussed earlier foradopting the simplex method. You may recollect that the:Objective function: Maximize 10a + 12bSubject to: 2a + b <60- (1)3a + 4b <120- (2)a, b >0Solution:To adopt the simple x method, the inequalities (1) and (2) have tobe converted into equalities by introducing slack variables. You may note that in2a + b <60, the LHS is less than the RHS.To equate the LHS to RHS, on the lower side we add few quantity,which we call as slack resource. This may be referred to by S1, S2, ... Sn. So theinequality (1) is converted into an equality as follows: 2a + b + S1= 60. The S1simply tells the unutilized material 1. If all 60 units of I are used S1will be zero.If some units are left S1will be a positive quantity. Whether S1is zero or somepositive quantity, the equation 2a + b + S1= 60 will hold good. Similarly theinequality 3a + 4b <120 is converted into an equation, 3a + 4b + S2= 120, S2being the slack variable, whose value would be zero or more than zero. In otherwords, the non-negativity condition would cover both real and slack variables.While products A and B would give us profits, unutilized material I, i.e., the

108slack resource S1and utilized material II, i.e., the slack resource S2would notgive any. So, the return from S1and S2is zero, each.In the simplex method the slack variables are also incorporated inthe objective function. So, the objective function is:Maximize P = 10a + 12b + 0S1+ 0S2Subject to: 2a + b + S1= 60- (1)3a + 4b + S2= 120- (2)a, b, S1, S2> 0Equations (1) and (2) are modified in such a way that both containsthe other slack variable also. The coefficient of the `other slack variable' wouldbe put as zero. This means `other slack variable' is included only to satisfy somestructural requirements, but no value is attached.So we get, 2a + b + S1+ 0S2= 603a + 4b + 0S1+ S2= 120Now, we can proceed with the solution process. Below, initialsolution table is presented. Tableau-1 gives the same.

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