Toy R Us uses three operations to assemble three types of soft toys $-$ Peppa Pig, Hello Kitty

and Minnie Mouse. The daily available times for the three operations are $430,460,$ and $420$

mins, respectively, and the revenues per unit of toy $($Peppa Pig, Hello Kitty and Minnie

Mouse$)$ are $$3,$ $$2,$ and $$5,$ respectively. The assembly times per Peppa Pig at the three

operations are $1,3,$ and $1$ mins, respectively. The corresponding times per Hello Kitty and

per Minnie Mouse are $(2,0,4)$ and $(1,2,0)$ mins $($a zero time indicates that the operation is

not used$).$

$($a$)$ Letting x$1,$ x$2,$ and x$3$ represent the daily number of units assembled of Peppa Pig, Hello

Kitty and Minnie Mouse, respectively, construct the associated LP model.

$($b$)$ Using s$1,$ s$2$ and s$3$ as the slack variables for the constraints of operations $1,2,$ and $3,$

respectively, the initial simplex tableau is

x$1$ x$2$ x$3$ s$1$ s$2$ s$3$ RHS

z $-3-2-50000$

s$1121100430$

s$2302010460$

s$3140001420$

and the optimum tableau is

x$1$ x$2$ x$3$ s$1$ s$2$ s$3$ RHS

z $4001201350$

x$21$

$4101$

$21$

$40100$

x$33$

$20101$

$20230$

s$3200-21120$

How would you recommend the company manufacture the toys to maximise daily

revenue?

$($c$)$ Suppose Toy R Us wants to change the capacities of the three operations according to

$[$Operation $1,$ Operation $2,$ Operation $3]=[460,500,400].$

Use post$-$optimal analysis to determine the optimum solution.

$($d$)$ Suppose the fourth operation has the following specifications: The maximum production

rate based on $480$ minutes a day is $120$ units of Peppa Pig, $480$ units of Hello Kitty, or

$240$ units of Minnie Mouse. Determine the optimal solution, assuming that the daily

capacity is limited to $570$ minutes.

$($e$)$ Investigate the optimality of the Toy R Us solution for the following objective functions.

If necessary, use post$-$optimal analysis to determine the new optimum.

z $=3$x$1+6$x$2+$ x$3$