please answer question number $1$ Note that it is very important for you to use the following rule in calculating the "Quarterly Inventory Carrying Cost":

Inventory carrying costs should be based on ending quarterly inventory in excess of safety stock.

The quarterly carrying cost is $$0\mathrm{.}25$ per case, or $$250$ per $1,000$ cases.

The Situation

You are the operations manager for a manufacturing plant

that produces pudding food products. One of your impor$-$

tant responsibilities is to prepare an aggregate plan for the

plant. This plan is an important input into the annual budget

process. The plan provides information on production rates,

manufacturing labor requirements, and projected finished

goods inventory levels for the next year.

You make those little boxes of pudding mix on packag$-$

ing lines in your plant. A packaging line has a number of

machines that are linked by conveyors. At the start of the

line, the pudding is mixed; it is then placed in small pack$-$

ets. These packets are inserted into the small pudding boxes,

which are collected and placed in cases that hold $48$ boxes of

pudding. Finally, $160$ cases are collected and put on a pallet.

The pallets are staged in a shipping area from which they are

sent to four distribution centers. Over the years, the technol$-$

ogy of the packaging lines has improved so that all the dif$-$

ferent flavors can be made in relatively small batches with

no setup time to switch between flavors. The plant has $15$ of

these lines, but currently only $10$ are being used. Six employ$-$

ees are required to run each line.

The demand for this product fluctuates from month to

month. In addition, there is a seasonal component, with peak

sales before Thanksgiving, Christmas, and Easter each year.

To complicate matters, at the end of the first quarter of each

year the marketing group runs a promotion in which special

deals are made for large purchases. Business is going well,

and the company has been experiencing a general increase

in sales.

The plant sends product to four large distribution ware$-$

houses strategically located in the United States. Trucks move

product daily. The amounts shipped are based on maintain$-$

ing target inventory levels at the warehouses. These targets

are calculated based on anticipated weeks of supply at each

warehouse. Current targets are set at two weeks of supply.

In the past, the company has had a policy of producing

very close to what it expects sales to be because of limited

capacity for storing finished goods. Production capacity has

been adequate to support this policy.

A sales forecast for next year has been prepared by the

marketing department. The forecast is based on quarterly

sales quotas, which are used to set up an incentive programfor the salespeople. Sales are mainly to the large U$.$S$.$ retail

grocers. The pudding is shipped to the grocers from the distri$-$

bution warehouses based on orders taken by the salespeople.

Your immediate task is to prepare an aggregate plan for

the coming year. The technical and economic factors that

must be considered in this plan are shown next.

Technical and Economic Information

The plant runs $5$ days each week, and currently is run$-$

ning $10$ lines with no overtime. Each line requires six

people to run. For planning purposes, the lines are run

for $7\mathrm{.}5$ hours each normal shift. Employees, though,

are paid for $8$ hours' work. It is possible to run up to

$2$ hours of overtime each day, but it must be scheduled

for a week at a time, and all the lines must run over$-$

time when it is scheduled. Workers are paid $$20\mathrm{.}00/$

hour during a regular shift and $$30\mathrm{.}00/$hour on over$-$

time. The standard production rate for each line is $450$

cases$/$hour$.$

The marketing forecast for demand is as follows:

Q$1-2,000$; Q$2-2,200$; Q$3-2,500$; Q$4-2,650$; and

Q$1($next year$)-2,200.$ These numbers are in $1,000-$

case units. Each number represents a $13-$week forecast.

Management has instructed manufacturing to main$-$

tain a two$-$week safety stock supply of pudding inven$-$

tory in the warehouses. The two$-$week supply should

be based on future expected sales. The following are

ending inventory target levels to comply with the

safety stock requirement for each quarter: Q$1-338$;

Q$2-385$; Q$3-408$; Q$4-338.$

Inventory carrying cost is estimated by accounting to

be $$1\mathrm{.}00$ per case per year. This means that if a case of

pudding is held in inventory for an entire year, the cost

to just carry that case in inventory is $$1\mathrm{.}00.$ If a case

is carried for only one week, the cost is $$\frac{1\mathrm{.}00}{52},$ or

$$0\mathrm{.}01923.$ The cost is proportional to the time carried

in inventory. There are $200,000$ cases in inventory at

the beginning of Q$1($this is $200$ cases in the $1,000-$

case units that the forecast is given in$).$

If a stock$-$out occurs, the item is backordered and

shipped at a later date. The cost