Maximization Case Of Simplex Method
1. In the first column, programme column, are the problem variables or basis variables that are included in the solution or the company is producing at the initial stage. These are S1, S2 and S3, which are known as basic variables.
2. The second column, labeled as Profit per unit in Rupees shows the profit co-efficient of the basic variables i.e., Cb. In this column we enter the profit co-efficient of the variables in the program. In table 1, we have S1, S2 and S3 as the basic variables having Rs.0.00 as the profit and the same is entered in the programme.
3. In the quantity column, that is 3rd column, the values of the basic variables in the programme or solution i.e., quantities of the units currently being produced are entered. In this table, S1, S2 and S3 are being produced and the units being produced (available idle time) is entered i.e., 2500, 2000 and 500 respectively for S1, S2 and S3. The variables those are not present in this column are known as non-basic variables. The values of non-basis variables are zero; this is shown at the top of the table (solution row).
4. In any programme, the profit contribution, resulting from manufacturing the quantities of basic variables in the quantity column is the sum of product of quantity column element and the profit column element. In the present table the total profit is Z = 2500 × 0 2000 × 0 500 × 0 = Rs. 0.00.
5. The elements under column of non-basic variables, i.e., a and b (or the main body of the matrix) are interpreted to mean physical ratio of distribution if the programme consists of only slack variables as the basic variables. Physical ratio of distribution means, at this stage, if company manufactures one unit of ‘a’ then 10 units of S1, 5 units of S2 and 1 unit of S3
will be reduced or will go out or to be scarified. By sacrificing the basic variables, the company will lose the profit to an extent the sum of product of quantity column element and the profit column element. At the same time it earns a profit to an extent of product of profit co-efficient of incoming variable and the number in the quantity column against the just entered (in coming) variable.
6. Coming to the entries in the identity matrix, the elements under the variables, S1, S2 and S3 are unit vectors, hence we apply the principle of physical ratio of distribution, and one unit of S1 replaces one unit of S1 and so on. Ultimately the profit is zero only. In fact while doing successive modifications in the programme towards getting optimal; solution, finally the unit matrix transfers to the main body. This method is very much similar with G.J. method (Gauss Jordan) method in matrices, where we solve simultaneous equations by writing in the form of matrix. The only difference is that in G.J method, the values of variables may be negative, positive or zero. But in Simplex method as there is non-negativity constrain
7. Cj at the top of the columns of all the variables represent the coefficients of the respective variables I the objective functiont, the negative values for variables are not accepted.
8. The number in the Zj row under each variable gives the total gross amount of outgoing profit when we consider the exchange between one unit of column, variable and the basic variables.
9. The number in the net evaluation row, Cj – Zj row gives the net effect of exchange between one unit of each variable and basic variables. This they are zeros under columns of S1, S2 and S3. A point of interest to note here is the net evaluation element of any basis variable (or problem variable) is ZERO only. Suppose variable ‘a’ becomes basis variable, the entry in net evaluation row under ‘a’ is zero and so on. Generally the entry in net evaluation row is known as OPPORTUNITY COST. Opportunity cost means for not including a particular profitable variable in the programme, the manufacturer has to lose the amount equivalent to the profit contribution of the variable. In the present problem the net evaluation under the variable ‘a’ is Rs. 23 per unit and that of ‘b’ is Rs, 32 per unit. Hence the if the company does not manufacture ‘a’ at this stage it has to face a penalty of Rs. 23/– for every unit of ‘a’ for not manufacturing and the same of product variable ‘b’ is Rs. 32/–. Hence the opportunity cost of product ‘b’ is higher than that of ‘a’, hence ‘b’ will be the incoming variable. In general, select the variable, which is having higher opportunity cost as the incoming variable (or select the variable, which is having highest positive number in the net evaluation row.
In this problem, variable ‘b’ is having higher opportunity cost; hence it is the incoming variable. This should be marked by an arrow (↑)at the bottom of the column and enclose the elements of the column in a rectangle this column is known as KEY COLUMN. The elements of the key column show the substitution ratios, i.e., how many units of slack variable goes
out when the variable enters the programme. Divide the capacity column elements by key column numbers to get REPLACEMENT RATIO COLUMN ELEMENTS, which show that how much of variable ‘b’ can be
manufactured in each department, without violating the given constraints. Select the lowest replacement ratio and mark a tick (√) at the end of the row, which indicates OUT GOING VARIABLE. Enclose the elements of this column in a rectangle, which indicates KEY ROW, indicating outgoing variable