Suppose we have 2 machines and 3 products. We have 5 pieces of our first product, 4 pieces of our second product and 3 pieces of our third product. In our T39 machine, the first product is processed in 4.63 minutes, the second product in 2.5 minutes, and the third product in 9.88 minutes. In our T77 machine, the first product is processed in 4.68 minutes, the second product in 1 minute and the third product in 5 minutes.

Each product can be divided into parts of at least 100 units. A maximum of 500 requests can be made for each product. The partition of integer option of our 5 products is 6 pieces. Since there are 6 options for the 5 incoming products, there are 6 options for 5. For the first product, we will make 1partitions. We will partition from the product number i.

Therefore, there are (1+1)*(2+1)*(3+1)*(4+1)*(5+1) options.

Tmax= Total operation time of the plant

Ti,max= Total operation time of the machine i

Tj= Total production time of the item j

Ti,j = Operation time of the item j on the machine i

ci,j= Fraction of the production of item j on the machine i

Equality constraints: sum ci,j Ti,j= Tj

Inequality constraints :

Cost: Total production time T: minimize total production time of all items subject to equality and inequality constraints; decision variables are ci,j s.

This is a linear programming problem. If the ci,js are integers ( actually 0 or 1) this means all of item I is produced on the same machine, then it is integer programming

How can we write the minimum cost function according to all this information?

PRODUCT F4390162 (5 piece) F3365536 (4 piece) F3341182 (3 piece) MACHNE 1 T39 4,63 2,5 9,88 MACHINE 2 T77 4,68 1 5 PRODUCT F4390162 (5 piece) F3365536 (4 piece) F3341182 (3 piece) MACHNE 1 T39 4,63 2,5 9,88 MACHINE 2 T77 4,68 1 5