Par, Inc., is a small manufacturer of golf equipment and supplies. Par's distributor believes a market exists for both a medium$-$priced golf bag, referred to as a standard

model, and a high$-$priced golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can make the bags at a competitive price, the

distributor will purchase all the bags that Par can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following

table, which shows the production time requirements for the four required manufacturing operations and the accounting department's estimate of the profit contribution

bag.

The director of manufacturing estimates that $630$ hours of cutting and dyeing time, $600$ hours of seving time, $708$ hours of finishing time, and $135$ hours of inspection a

packaging time will be available for the production of golf bags during the next three months. $($Let $S$ be the number of standard bags. Let $D$ be the number of deluxe bag

Suppose that Par's management encounters the following situations.

If each of these situations is encountered separately, what is the optimal solution and the total profit contribution?

$($a$)$ The accounting department revises its estimate of the profit contribution for the deluxe bag to $$18$ per bag.

The optimal solution occurs at $(S,D)=(,)$ with a profit of $

$($b$)$ A new low$-$cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $$20$ per bag. $($Assume that the profit

contribution of the deluxe bag is the original $59$ value.$)$

The optimal solution occurs at $(S,D)=(,)$ with a profit of

$($c$)$ New sewing equipment is available that would increase the seving operation capacity to $675$ hours. $($Assume that $10S+9D$ is the appropriate objective function.$)$

The optimal solution occurs at $(S,D)=(,)$ with a profit of $