A manufacturing facility is dealing with a challenging production scheduling problem. The objective is to determine the most efficient production sequence for a set of complex manufacturing tasks, considering various constraints, resource availability, and production costs. You are tasked with optimising the production scheduling in a manufacturing plant to minimise production time and costs while meeting all operational constraints. This is a critical challenge, as it directly impacts the efficiency and competitiveness of the manufacturing process.

The Branch and Bound method is a powerful optimisation algorithm used to tackle complex decision$-$making problems in the manufacturing process. This technique systematically explores a wide range of potential solutions to identify the most optimal course of action. Solve the problem in Part A Question $2$ using the Branch and Bound method.

Subject to:

$2$x$1+3$x$2<24$

$5$x$1+3$x$2<30$

x$1,$ x$2>0$

The optimal linear programming tableau is shown in Table $1.$

Table $1/$Jadual $1$

Basic variable Coefficient of Solution

Z X$1$ X$2$ S$1$ S$2$ b

Z $1002/911/9382/3$

X$20105/9-2/962/3$

X$1001-1/31/32$