ECONOMICS OF TELKl'lK )\K KKLA^ A I'l'LK ' A TIOXS 227 



The average luimher of parts stored is one-half this, or 



where jV = rate of output of luachiue per year. 



The peiuUty per part due to lot-size and administration was given in 



(5), and when added to its base eost Ci gives the total value of one part as 



C^ + ^+'l. (8) 



n n 



The penalty per part, due to storage, equation (6), is then the product 

 of k/n times (7) times (8), where k = interest charges on stored parts 

 expressed as a ratio. Then the storage cost penalty, Cs , is 



C, = ^(c, + ^f+-i)«(i-»), (9) 



n \ n n / 2C\ N / 



1.23 Total Cost Penalty 



The entire penalty resulting from lot size charges and inventory 

 charges may now be stated as a function of annual demand by summing 

 equations (5) and (9). Writing q for (1 — n/N), and rearranging terms, 

 the total cost penalty in terms of annual demand is 



Equation (10) is the basic relation betw^een costs and annual demand, 

 and is of the general shape shown in graph (b) of Fig. 1. 



It is of greatest interest to determine conditions where this curve has 

 its lowest ^'alue, which occurs when dC/di = 0. The result is 



which gi\'es the lot-frequency, fo , for which total cost penalty is a 

 minimum. When fo is substituted for ( m equation (10), the resulting 

 optimum cost, as a function of annual demand, Co , is given: 



^^ ^ / 2kqS{C, + Ajn) _^ kqS ^ A 



Equations (11) and (12) furnish the means for deciding liow to plan 

 manufacture under differing conditions of annual demand, and how 

 much the product is penalized even after plaiming is carried out as 



