Question

A company owns two flour mills (A and B) which have different production capacities for HIGH, MEDIUM and LOW grade flour. This company has entered contract supply flour to a firm every week with 12, 8, and 24 quintals of HIGH, MEDIUM and LOW grade respectively. It costs the Co. $1000 and $800 per day to run mill A and mill B respectively. 

On a day, mill A produces 6, 2, and 4 quintals of HIGH, MEDIUM and LOW grade flour respectively. Mill B produces 2, 2 and 12 quintals of HIGH, MEDIUM and LOW grade flour respectively.

Required tasks:

•Formulate the LP model; how many days per week should each mill be operated in order to meet the contract order most economically standardize? Interpret the result; determine the surplus amount; determine the optimal value using simplex method. 

Answer 1

Minimize: $1000x_1+800x_2$

Subject to: $6x_1+2x_2\geq8$

$2x_1+4x_2\geq12$

$4x_1+12x_2\geq24$

$x_1\geq0, x_2\geq0$

Convert the minimization problem into its dual. Transposing matrix of coefficients:

Maximize: $8y_1+12y_2+24y_3$

Object to:

$6y_1+2y_2+4y_3\leq1000$

$2y_1+4y_2+12y_3\leq800$

$y_1\geq0, y_2\geq0, y_3\geq0$

1st iteration:

$Z_{min}=-24$ for $y_3$ - pivot column

$b_i/s_i$ in pivot column is minimal for $s_2$ ($800/12<1000/4$ ), so, pivot row is row of $s_2$

Pivot value is intersection of pivot column and pivot row.

Pivot value$=12$

Update table. The new coefficients of the tableau are calculated as follows:

In the pivot row each new value is calculated as: New value = Previous value / Pivot

In the other rows each new value is calculated as:

New value = Previous value - (Previous value in pivot column * New value in pivot row)

2nd iteration:

The pivot column is column of $y_1$ , the pivot row is row of $s_1$ , pivot value is $16/3$

3rd iteration:

The pivot column is column of $y_2$ , the pivot row is row of $s_2$ , pivot value is $5/16$

4th iteration:

Solution: $x_1=2/5,x_2=14/5, Z_{min}=2640$

The minimal cost per day of running mills is $2640. The company operates mills A and B for 0.4 days and 2.8 days per week respectively.