$1.(30$p$)$ A local firm manufactures two types of children$$s toys: toy car and toy robot. The

projected demand over the next four months for the two toys are given in the table below.

Month, t Toy car demand, D$1$t Toy robot demand, D$2$t

$13,0004,000$

$22,0003,000$

$34,0002,000$

$45,0004,000$

Manufacturing is primarily an assembly operation, and capacity is governed by the

number of workers on the production line. On average, it requires one hour for a worker

to assemble one unit of toy car and two hours to assemble one unit of toy robot.

Assume that each worker works $20$ days per month, $8$ hours per day on regular time. If

needed, the workers can work overtime. A worker can work at most $20$ hours per month

on overtime. A worker$$s monthly salary for regular time work is $$3,000,$ and they are

paid an additional $$25$ for every hour they work on overtime. Currently, the workforce

consists of $50$ workers, and hiring and layoffs are possible at the beginning of each

month. The cost of hiring is $$3,000$ per worker and the cost of firing is $$5,000$ per

worker.

Currently, there are no toys in inventory. Holding cost is $$4$ per month for one unit of toy

car, and $$5$ per month for one unit of toy robot held at the end of each month. Stockouts

are allowed in a month, except for the last month. All the demand must be met by the end

of month $4.$ The cost of backordering one unit of toy car is $$6$ per month and one unit of

toy robot is $$8$ per month. The company aims to determine the optimal production plan

that has the minimum cost.

a$)(20$p$)$ Formulate a linear programming model for this aggregate planning problem.

Clearly define the decision variables, objective function, the constraints of the model

and any notation you use.

b$)(5$p$)$ Assume now that the company wants to know the cost of using a CHASE

strategy. Do the necessary modifications to the model in part a$)$ to evaluate the Chase

strategy.

c$)(5$p$)$ Assume now that each hour of idle time per worker $($i$.$e$.$ an hour of unused labor

capacity$)$ costs $$10$ to the company. Make the necessary modifications to the model in

part a$)$ to incorporate the idle time cost into the model.