Production Mix

Problem definition

Chappie Ltd. has 3 production lines (A, B, and C) to make 4 different types of metal sheets of width 0.4, 05, 0.6, and 0.7 mm. The company needs to meet the monthly demand for each kind of product (in Tons) expressed in the next table:

Sheet 0.4 mm	Sheet 0.5 mm	Sheet 0.6 mm	Sheet 0.7 mm
500	1200	1500	300

The production lines do not have the same overall efficiency for all types of sheets. Furthermore, due to design constraints, not all the lines are capable of manufacturing all types of sheets. The table below indicates the manufacturing costs in € of the different products in the different lines: a dash “-“ means that it is not possible to manufacture a product in a line:

Line	Sheet 0.4 mm	Sheet 0.5 mm	Sheet 0.6 mm	Sheet 0.7 mm
A	60	50	50	45
B	80	70	75	70
C		60	60

The different lines have the following capacities (in hours) in the planning period:

A	B	C
500	480	370

And the following table represents the manufacturing time (hours) per ton of product required for each type of metal sheet in the different lines:

Line	Sheet 0.4 mm	Sheet 0.5 mm	Sheet 0.6 mm	Sheet 0.7 mm
A	0.4	0.3	0.3	0.25
B	0.8	0.6	0.7	0.6
C		0.5	0.4

Formulate a linear programming problem to obtain the monthly production plan (quantity of metal sheet in tons of each type of metal sheet to be manufactured in each production line)

indexes

i: Manufacturing lines (A,B,C)

j: Product types Sheet (0.4, 0.5, 0.6, 0.7) mm

Decision Variables

\(x_{ij}\) = Quantity of product \(j\) to produce in line \(i\)

**Objective function Minimize cost

\(\min z = \sum_i\sum_j c_{ij}*x_{ij}\)

where \(c_{ij}\) is the cost of manufacturing product \(j\) in line \(i\) as expressed in the second table.

Constraints Capacity

\(\sum_{j}a_{ij}·x_{ij} <= b_{i}, \forall i\)

Where \(a_{ij}\) represent the manufacturing time required to manufacture product \(j\) in line \(i\) as expressed in the last table and b_{i} represents the capacity of line i for the planning period

Demand

\(\sum_{i}{x_{ij}}>= d_{j}, \forall j\)