The chief operations officer $($COO$)$ believes the forecast is not good.

Overestimating demand can cause a loss of $$4500$ per product for product A$,$ $$3500$

for product B and $$6500$ for product C$.$ These are the costs of not selling the products

in time. The products are perishable and have limited shelf life. Therefore, once a

product expires, there is a cost. On the other hand, underestimation can cost the

company $$3500$ per unit of product A$,$ $$2500$ per unit of product B and $ $4500$ per

unit of product C$.$ Essentially, these are the profits made from selling each unit of

product. Assume that the COO hires your team $($or you$)$ to get some help.

Help the COO in the following ways $($these are the questions that you should

answer$)$:

$1.$ Use an appropriate method of forecasting and convince him that your method

is better than that of the analyst$$s method. You can use the following

information:

a$.$ If you use moving average, use a $3-$period moving average with weights

$0\mathrm{.}25,0\mathrm{.}6,0\mathrm{.}15$ for the periods $($t$-1),($t$-2)$ and $($t$-3)$ respectively.

b$.$ If you decide to use exponential smoothing, use $\backslash$alpha $=0\mathrm{.}4.$

c$.$ If you use trend adjusted exponential smoothing, use $\backslash$alpha $=0\mathrm{.}3,$ and $\backslash$beta $=$

$0\mathrm{.}4.$

Note: Do not consider any other method apart from the above three

mentioned.

$2.$ The COO tells you that they consider average loss $($combined from

overestimation and underestimation of forecast$)$ as the average of the two

losses. For example, if underestimation costs $$1000$ per item and

overestimation costs $$1200$ per item, the average loss is

$($$$1000+$$$1200)/2=$$$1100.$ Using this information, estimate approximately on

an average how much money you can save yearly for the company as

compared to the analyst$$s forecast. This estimate should be based on historical

data and not the future.

Hint: For part $2,$ think about what can give you average underestimation and

overestimation.

$3.$ The COO also wants to know the optimal product mix quantities for each

product for the first four months of the coming year. This can vary from month

to month, but as of now the COO is happy to have just the answer for January

$2024$ and this will be the solution for every month till April. Note that in the

coming year the company must adhere to the following constraints regarding

their budget, supplier raw materials and demand.

a$.$ Demand constraints: Produce according to your forecast.

b$.$ Budget: Production costs are $$1200$ per product for product A$,$ $$1500$

for product B and $$1400$ for product C$.$ Total budget is for production

is $$4000000.$

c$.$ The three products need raw materials X$,$ Y and Z$.$ the following table

gives how much raw material is need $($in lbs$)$ for each unit of a product.

Maximum available quantity of X$,$ Yand Z is $12000$ lbs each.