Supply Chain Network Planning for Humanitarian Operations During Seasonal Disasters Page: 25
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Proof: Differentiation of expression 4 with respect to c1 yields ac, = x1 ; c2 yields
TC= (y - x); p yieldsTC ( - y)ped; and v yieldsTC = fY 19ed. Since all
-c (y - l;pyields -Oyp- =fy ( -Y)q1odl v
these derivatives are positive, The optimal cost is increasing in c1 , 2, p and v.
Since x1 is non-negative, the optimal cost is strictly increasing in c1 . The optimal order
size is independent of c1 because the optimal order size expression, y* = - 1 (p-c), does not
include c1. Since c1 is the lowest among all the purchasing costs (i. e., c1, c2, and p), purchasing
products at the first instance results in cost savings. However, before placing any first instance
orders, relief agencies need to evaluate other constraints such as financial, warehousing, and
other relevant constraints in conjunction with the cost analysis.
..... c2= 19 ;y*= 187.54
3600 - c2=17.5; y*= 1193.19
c2= 16 y*= 198.33
3400 - c2=145;y*=213.36
- c2= 13 ; y*=2x8.61
3200 ''" " .."
. ......... . ...-'
3000 . 00
120 140 160 180 200 220
C unlative order size in units
Figure 3-3 Impact of Second Instance Cost c2 on Optimal Order Size and Cost
The partial derivative of equation 4 with respect to c2 yields (y - x1). This quantity
(y - x1) is positive if x1 is strictly less than y. Therefore, the total cost is increasing in c2.
Figure 3-3 shows the impact of c2 on total cost. Differentiation of equation 4 with respect to p
yields =c ( - Since an
yiels -fy ( - y)qo~9d . Snethis derivative is positive, increase in spot market price
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Ponnaiyan, Subramaniam. Supply Chain Network Planning for Humanitarian Operations During Seasonal Disasters, dissertation, May 2013; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc271880/m1/33/: accessed June 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .