$($ref$.$: Carlos F$.$ Daganzo: Logistics Systems Analysis, $4$th Ed$.$ Springer, $2004)3\mathrm{.}6$ Freight is to be exported from a region of variable width, lying on

one side of a transportation artery $($e$.$g$.,$ a highway or railroad line$)$

that is one thousand distance units $($Du$)$ long, $L=1,000Du\text{'}$s$.$ From

the origins freight can only be carried perpendicularly to the artery,

unless of course it moves on the artery itself. Freight must flow out

of this region through a system of terminals $($e$.$g$.,$ ports$)$ that are to

be located on this artery. The cost of travel within the region is one

monetary unit $()$ per weight unit $(Wu)$ per $Du.$ The cost of travel

beyond the terminals is not considered as part of our study. The

freight transportation needs per unit time $(Tu)$ are expressed as a

transportation demand density $W\frac{u}{D{u}^{**}Tu}$ which depends on the

position along the artery expressed in Du's, $x.$ The demand density,

$w(x),$ is expressed as follows:

Tun$=1,2c_T=80,000n^{**}{z}^{**}{n}^{**}{z}^{**}{c}_T{c}_T=80,000CA{c}_{T}0n=1,2,3,$dots,$10c_T{z}^{**}{c}_{T}w(x)=5if0x300$

$=1if300$

Then:

$(i)$ Determine the optimal location and the total access cost per $Tu$

when $we$ locate $n=1,2$ and $3$ terminals. $(Use$ the procedure $de-$

veloped $in$ Sec. $3\mathrm{.}4),$

$(ii)$ Show which locations you would use $if$ only the attached $50$ loca$-$

tions are feasible. Calculate for each case the percent change $inac-$

cess cost,

$(iii)If$ each terminal costs $c_T=80,000\text{'}s$ per $Tuto$ operate $(includ-$

ing any relevant amortized fixed costs$),$ determine whether the $op-$

timum number $of$ terminals that should $be$ operated, $n^{**},is1,2,or$

Calculate $as$ well the total regional cost per unit time, including

both transportation and terminal operations, $z^{**}.Do$ the calculation

with and without the location constraints described $in(ii),$

$(iv)$ Determine $n^{**}$ and $z^{**}$ for arbitrary $c_T$ using the continuum $ap-$

proximation $(CA)$ method. Compare the results for $c_T=80,000$

with those for part $(iii).$ Discuss how the $CA$ results might differ

from the true optimum $as{c}_{T}0,$ with and without the $50$ location

restriction,

$(v)$ Extra credit: Find the exact optimal solution $to$ part $(ii)$ using $dy-$

namic programming. Solve for $n=1,2,3,$dots,$10.$ Then determine

the ranges $of{c}_T$ for which the optimum number $of$ terminals $is1,$

$2,$ etc. Calculate and plot the resulting $z^{**}as$ a function $of{c}_T$ and

compare the result with your findings $in$ parts $(iii)$ and $(iv).$

Data for Problem $3\mathrm{.}6$

Coordinates $of50$ possible locations