Recall the Par, Inc., problem introduced in Chapter $2.$ The linear program for this problem is

Max $10$x$119$x$2$

s$.$t$.$

$7/10$x$111$x$2$ # $630$ Cutting and dyeing time

$1/2$x$115/6$x$2$ # $600$ Sewing time

$1$x$112/3$x$2$ # $708$ Finishing time

$1/10$x$111/4$x$2$ # $135$ Inspection and packaging time

x$1,$ x$2$ $ $0$

where

x $15$

x$25$

number of standard bags produced

number of deluxe bags produced

The final simplex tableau is

x$1$ x$2$ s$1$ s$2$ s$3$ s$4$

Basis c B $1090000$

x$290130/160221/160252$

s$2000215/1615/320120$

x$11010220/16030/160540$

s$4000211/3209/64118$

z j $10970/160111/1607668$

c j $2$ zj $00270/1602111/160$

a$.$ Calculate the range of optimality for the profit contribution of the standard bag.

b$.$ Calculate the range of optimality for the profit contribution of the deluxe bag.

c$.$ If the profit contribution per deluxe bag drops to $$7$ per unit, how will the optimal solu$-$

tion be affected?

d$.$ What unit profit contribution would be necessary for the deluxe bag before Par, Inc.,

would consider changing its current production plan?

e$.$ If the profit contribution of the deluxe bags can be increased to $$15$ per unit, what is

the optimal production plan? State what you think will happen before you compute the

new optimal solution