ELKO, a manufacturer of lighting fixtures, has been outstanding in developing new products, but

the new CEO has observed some potential opportunities in optimizing its distribution system.

Current Distribution System

The company has $100$ SKUs in its product line, all manufactured in its own plants located in the

Cleveland area. For demand management purposes, the contiguous United States was divided

into $5$ regions as shown below. Each region was served by a separate distribution center $($DC$)$

owned by ELKO. Customers $($all business: retailers, contractors, offices, etc.$)$ placed orders with

their DC$,$ which tried to supply them from inventory. Each DC ordered replenishments from the

company$$s plants, using a periodic review policy with a reorder interval of six days and a CSL of

$95\%($use z $=1\mathrm{.}645).$ The plants, in turn, scheduled make$-$to$-$order production based on DC

orders $($i$.$e$.,$ no finished$-$goods inventory at the plants$).$

ELKO used a third$-$party trucking company for all transportation. On average, shipments sent

from the plants to the DCs cost $$0\mathrm{.}08$ per item, whereas shipments from the DC to the customers

cost $$0\mathrm{.}15$ per item. Two days were necessary between the time a DC placed an order with a

plant and the time the order was put on a truck at a plant. Further, transportation lead time of

the trucking company for shipments from the plants to the DCs was normally distributed: $3$ days

on average, with $1-$day standard deviation.

A detailed study of the product line had shown that there were three types of products in terms

of the volume of sales: High, Medium, and Low. For a single representative product in each type

$($H$,$ M$,$ and L$)$ daily demand forecast is shown in the following table. Of the $100$ products that

ELKO sold, $10$ were of type High, $20$ of type Medium, and $70$ of type Low. The holding cost

incurred was $$0\mathrm{.}05,$ $$0\mathrm{.}10,$ and $$0\mathrm{.}15$ per item per day $($respectively$,$ for types H$,$ M and L$)$

whether the item was in transit or in storage. Assume that the product ownership is transferred

to the customers once it leaves the DCs$.$