ECONOMIC LIQUID CONTAINERS-A NEW MODE ! DONALD C . DAUS
U m l economic lot-size equations do not a p p b to liquids in tanks or to solids in bins. A new equation and an example are presented for limiting and general cases.
Using this equation,
economic order quantities are signijcantb smaller than predicted by previous methods
A
'
pplying economic lot size equations to liquid inventories results in lot sizes (and tank sizes) larger than optimum. Classic inventory equations balance the calculated interest on the cost of the inventory against the cost per order (purchasing, receiving, and analysis) to maintain this inventory. The simplest equation (Model 1 ) is:
which is developed by Churchman and Rio and others (7, 3). A n additional cost for storage space is added hy Whitin (4, assuming this cost to be proportional to the order size:
An equation for the optimum size when a new container is needed is developed here, assuming that the order quantities are large enough to avoid further price breaks. If a process requires R gallons at constant rate in time T, then (C, C,) is the cost of holding 1 gallon in inventory for a unit of time. C, is the cost of ordering, analyzing, and receiving one order of liquid, Q gallons,
+
consumed in time interval t,.' t, = T number of orders in time T.
Q
R
- where -
=
Q If no cushion is kept, Q - is R
2
the average inventory. The first component of the inventory cost is .the interest calculated on the investment in the average inventory. The second component is the straight line depreciation on the storage containers, plus taxes, interest on the tank cost, and insurance. C, = F(Q). For one order period :
TEC
- + C,R ij-+ F(Q)
CiTQ 2
(Continued on next juge) T.E.C.. NEW MOM1 TANKAGf
Price breaks are also considered by Whitin (4). These equations can be applied to bulk and liquid storage. A major factor, cost of the storage tanks, is not included. The effect of ignoring this is to determine too large an order quantity, requiring larger storage containers and greater capital investment. Bulk storage problems fall into two general classesLe., where no suitable containersexist, requiring procurement of new units; and where existing containers are available. However, only the first class is discussed here. If the available containers have no alternate uses, depreciation should not be charged; Model 1 may be properly applied. The available containers may not be adequate; therefore, a decision must be made whether to use those available or to buy new. Cost of the old tanks can be plotted on a graph of annual cost us. order quantity, giving a discontinuous curve. Graphical addition to the order cost component will determine the minima.
COST
ORmR COST OmER QUANTITY
V O L 5 4 NO. 7 JULY 1 9 6 2
31
9
EXAMPLE
Considered an actual process requiring 120,000 gallons per year where = $30 and = $0.02/gallon-year. A reference 10,000-gallon tanks costs $16,000 in exotic
c,
c1
Qo
( 1 ) Q,
material or $4000 in steel. Interest and depreciation on the tank total 12%. a, the safety factor, i s 1.25.
= $1260 f
(2) Q,
TEc
= $200
=
= 19,000 gallons
(plus $3300 depreciation and interest on a 25,000-gallon tank = $3980/ yr.)
-
I Qi
+
$540 $360 iplus $100 storage cost = $1OOO/yr.)
These equations yield ideal order sizes. Adjustment of the results t o realistic delivery quantities and t o standard tank sizes must be made, generally t o larger sizes. The error from ignoring the tank costs is significant.
+ $480
By the Newton-Raphson method,
-
= 19,000 initial point
1.2 ( 1 60001 10.121 ( 1.251 190001.6 - (2130) 120,000)
119,0001
= 19,000 -
Q,
+
2!19000)
0.02 (10,000)0~~ 1.2 16,000) (0.12) (1.251
f 1.6
0.02 ! 10,0001 n.6
1
Q1
(2)
TEC
+
By Equation 10
(4) Williams (5) presented the “six-tenths rule” for the relation for the cost of a tank and one larger or smaller of similar type and material. = C’(d
+ i) a
(3 o.6
=
KQo.6
(5)
Where a is a factor of safety to allow extra capacity for delivery delays; C’, current installed cost for a tank of capacity Qe of similar design and materials. Differentiating and substituting:
Q.‘K __ Q“4
Let
C,R Q2 Q =
CiT 2
(6)
X2.5 (7)
This is general. Simplifying assumptions may be made. AUTHOR Donald G. Daus is Process Engineer, Glandular
Products Manufacturing Division, Eli Lilly @ Co. 32
0.02
19,000°,6
Continuing, Q:! = 4800, QB= 4105, and = 4070. Three iterations provide the solution even though the starting point, Q, was far from the solution.
(Continued from page 37)
Differentiating with respect to Q and setting equal to zero,
F(Q)
1
= 8300
$480 (plus $830 for depreciation and interest on tank = $1510/yr.)
= $200
$870 (plus $40 storage costs =
= 9900 gallons
TEC
0.02
ill
0.6 116,000) 10.121 1.25
0.626
$2170/yr.)
= ~ 2 ~ 1 2 0 , 0 0030 ~
Q,
1
(301 ( 120,0001 11 0,0001 0.6
= 4100 gallons
TEC
By M o d e l I (Equation 11, Qo
[
=
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
As c, + 0
x=-
Qo =
[-
1.2C’ (i
+ d)a and 4(X = 0)’s
C1TQ,’.‘
1.2C’ (i
+
d)aI2J and 4(Q = 0)’s C1TQ,0.6
(8) (9)
The imaginary Q, has no meaning; the trivial solutions (Q = 0) advise to order very frequently. This applies only where bulk material is received on a standing order where no analysis is required. As
c1+ 0
For common solvents valued at 50 cents per gallon, C1 can be safely treated as zero. As the value of the inventory approaches the value of the storage tank multiplied by the ratio of tank depreciation and interest to inventory interest and carrying charges, the assumption that C1 is insignificant becomes invalid. Since inventory is a short-term investment, the interest used in the inventory- carrying charge may be significantly smaller than the interest expected for the tank investment. The general case must then be treated. Rearranging Equation 6, Q2
+
AQl.6
+B = 0
(11)
,
If there is no clear-cut policy for a, it may be inferred by observing plant operations, dividing lot size into t d . size. If a is consistent between materials, apolicy is provided for an initial’ assumption. Other values of a should be substituted to show its effect on costs.
where
A = 1.2C’(i
+ d)a
CiT Q.O.6 and
B = - 2C LR CIT
NOMENCLATURE
factorofaafety; Q . 0 = tankmtofrrfcrcncemt-ff, S CI cmt of holding O Q mil ~ in storage m e time unit, f / y r . C. = cost per order or lot, I C, = cmm per gallon ofeontaina capacity d = depreciation, t a m , and iosurancc on container i = interatwpectcd; companypolicy Q = order quantity, gal. Q. = capacity of refumce container, gal. Q. = eeanomic order quantity, gal. R = conrumption of material in time T S = space charges, $/unit-year T = time; can be omitled without d u c b g ‘ t h e validity of the equationa, but is raained to d u c e the probability of an erroneous substitution TEC = total e x p a d X = substitute variable D
A solution can be reached using the Newton-Raphson (2)method of iteration, where Yi+t
.
Qt
Substituting: Qi+t
*
=
=
Qg-
Q?
f(QJ - f7 (43
++A1.6AQ,aS Q P +B
24,
(‘2)
(l3,
Either Equation 1 or 10 may be used to provide the initial estimate for Q. Equation 10 and 1.1are nwdeLp for the economic order quantity and container siz&for bulk materials requiring tanks or bins. The M.riaMcs.am easily measured with the possible exception of a. Choosing a value for a is a .matterof particular portance. Factors as lead..*, reliability of supplier and shipper and potential espamionmust be considered. Specifying a is ultimately a l l ~ ~ n a e ent-decisi pm
c‘
= = =
BACKGROUND LITERATURE (1) Churdunan, C. W., A&&, R. L., Amoff, E. L., “Intmduction to Opffations Research,” Wiley, New York, 1957. (2) Nielsm, Kaj L.,“Methods in Numerical Analysis,” pp. 172-: MacMillan, New York,1956. (3) E o , J., S h a r e , A. F., C h .Eng. &om. 53,175-205 (1957). (4) Whitin, T.M.,“Theory of Inventory Managcmmt,” Rinu ton University Rem, Prinoetan, N. J., 1953. ) Williams, R., C h .Eng. 54, No. 12,124-5 (194
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