2.5 Stochastic Effects
2.5.2 Stochastic Effects Using Two Shipping Modes
the overall demand can be met in the long run. Although inventories at the origin would then tend to grow with time, every once in a while (every many reorders, presumably) the production process could be interrupted for a while to allow the demand to catch up with the cumulative number of items produced. The frequency of these stoppages would depend on pro-duction and inventory cost considerations.
A simple strategy would stop production whenever the inventory at the origin (after a shipment) reaches a critical value, v1 , and would resume it (also after a shipment) when the inventory dips below another value, v2; see Fig. 2.10. The maximum inventory is therefore: v1 + v , and the aver-age inventory: 1/2(v1 + v2 + v). The cost of production should be a func-tion only of D'p and the duration of the on and off periods. The on and off periods, however, only depend on v1 , v2 and on the statistical properties of the smooth curve tangent to the crests of the orders sent curve (see Fig.
2.10). On a scale large compared with v , this curve shares the statistical properties of the demand curve which do not depend on v. Therefore, the optimal production decisions (i.e., the choices of v1, v2 , and D'p) do not depend on v.
As before, the inventory (maximum and average) can be decomposed into a portion that is proportional to v (represented by the shaded area in Fig. 2.10) and independent of the production strategy, and a remaining portion which is influenced by the production scheme and is independent of v . Thus, the extra production and inventory costs arising at both the origin and the destination due to the unpredictability of demand are largely independent of v . They can be ignored when determining the optimal shipment size.
The foregoing discussion is not an exception; stochastic effects can be captured within the scope of a deterministic EOQ model in other situations as well. Problem 2.6 discusses the use of a private vehicle fleet, and the following subsection considers an operation where two different transpor-tation modes are used.
deci-Stochastic Effects 43
sion. For the following discussion it is assumed that the expedite mode is so fast that its lead time can be ignored.
Fig. 2.10 Inventory effect of production and transportation decisions
Most of the time the expedite mode lies in wait, and the system operates as if the primary mode was the only mode (see Fig. 2.9). The trigger point v0 , however, does not have to be chosen as conservatively as before, because when a stock-out is imminent enough items can be sent by the premium mode to avoid it.
The analysis is simple. If, as is commonly the case, the time between re-orders is large compared with the primary mode's lead time (i.e., so that when the trigger point, v0 , is reached there aren't any unfilled orders) then the probability that some items have to be expedited in the time between ordering and receiving a lot (of size v ) does not depend on v . It is a de-creasing function of v0 , approaching zero when (v0 - E (PQ))2 >>
var(PQ).
The exact form of the expected amount expedited per regular shipment will depend on the strategy used for choosing the expedited lot sizes. (Al-though these could be fixed, if possible they should be chosen just large
enough to meet demand until the regular order arrives). In any case, the expected amount expedited per regular shipment will also be a decreasing function of v0 , f(v0) . Assuming that the cost per item expedited is a con-stant, ce , we find that the expected expediting cost per regular shipment is:
cef(v0) . The moving cost per item is as a result:
v +c .
f + v
v f c -c +
= c v
f + v
v f c + v c +
= c item per
cost moving
v v
e f e
v f
0 0 0
¸ 0
¹
¨ ·
©
§
The maximum inventory still occurs when PQ is as small as possible, and remains: v + v0 - PQ ; the total cost per item is thus:
v + c
v + v - PQ
/D .
f + v
v f c -c + +c c
=
cost/item v f e v h 0 c
0 0
For a given v0 , if we think of the expected amount shipped by both modes with every regular shipment, v1 = [v + f(v0)] as the "lot size," the equation is still of the EOQ form (2.8a), where the fixed moving cost has been in-creased to include the expected cost of expediting, (ce - cv)f(v0):
> @
^ `
. D / v c v +
v f c -c + +c
D / v f -PQ -v c + c
= cost/item
h v
e f
h v
c c c
c
0
0 0
Unlike in the previous case, though, the trigger point v0 should not be cho-sen independently of v. If v is large so that shipments are infrequent, expe-diting a significant amount of freight with the average shipment only in-creases the moving costs marginally. But if v is small, the penalty for expediting is paid more often; it may be more efficient to increase v0.
Suggested Exercises 45
Suggested Exercises
2.1 Prove that if, as depicted in Fig. 2.1, the production and consump-tion rates are constant and shipments always carry all the producconsump-tion that has accumulated prior to their departure, then the maximum ac-cumulation at the destination equals the maximum acac-cumulation at the origin.
2.2 Use multiple regression analysis to validate Eq. (2.5c) using a recent book of rate tables.
2.3 Prove that the graph of Fig. 2.3 depicts a subadditive function.
2.4 Prove that the following functions are subadditive: (i) any positive, increasing and concave function defined for x > 0 , (ii) the sum of subadditive functions, and (iii) G(x) = min{f(x)+F(x-z): z [0,x]} if f and F are subadditive.
2.5 If the optimum shipment size obtained with the construction of Fig.
5.8, v* , is in the interval ((n-1) v1max , nv1max) , evaluate the differ-ence c(nv1max) - c(v*) and show that it cannot exceed (5n)-1(cf/v1max).
2.6 Assume that a firm operating its own vehicle fleet uses the trigger point strategy of Section 2.5 and Fig. 2.9, with trigger point v0 and shipment size v . A dispatched vehicle becomes available after a cy-cle time Tr that varies with every shipment; Tr can be viewed as a random variable (independent of all the others) with mean tr and standard deviation ır . If the demand has the stochastic properties discussed in the text, prove that the fleet size needed to ensure that the firm does not run out of vehicles is of the form:
where constant 3. (Hint: the fleet size should be as large as the maximum number of requests that are likely to be received during a vehicle cycle).
Comment: Multiplied by an appropriate constant, the second term of this expression is the contribution to transportation cost caused by uncertainty. Notice that it is proportional to (1/v) and independent of v0. Therefore, the optimal v is still given by the solution of Eq.
>
Dı /v+
DȖt /v
@
,/v + t
D
c
rconstant c
r 2c
r 2 1/2(2.8a) if the fixed transportation cost is duly modified. A modifica-tion to the transportamodifica-tion coefficient of Eq. (2.8a) can also be found if, in order to lessen the need for a larger fleet, we allow the inven-tory buffer at the destination to be increased in anticipation of (rare) vehicle shortages.
Glossary of Symbols 47
Glossary of Symbols
A: EOQ formula constant, A = ch/D', B: EOQ formula constant, B = cf,
cd: Transportation cost per vehicle-mile ($/vehicle-distance),
c'd: Marginal transportation cost per item, per distance unit ($/item-distance),
ce: Cost per item expedited, when using two shipping modes,
cf: Fixed transportation cost for a shipment, independent of size ($/shipment),
c'f: Fixed handling cost of moving a pallet,
c"f: Fixed motion cost (transportation + handling) per shipment, ch: Holding cost per item, per unit time, ch = cr + ci,
ci: Waiting cost per item, per unit time ($/item-time), cr: Rent cost per item, per unit time ($/item-time),
cs: Fixed transportation cost of stopping for a shipment (part of cf in-dependent of distance),
c's: Added transportation cost of carrying an extra item (part of cv in-dependent of distance) ($/item),
cv: Added transportation cost per extra item carried ($/item), c'v: Added handling cost per extra item handled,
c"v: Incremental motion cost per item moved (transportation + han-dling),
d: Distance traveled,
D': Demand rate (items/time), D'p: Production rate (items/time),
fh(): Handling cost function per shipment,
fm(): Motion cost function per shipment, fm = ft + fh, ft(): Transportation cost function per shipment,
Ȗ: Index of dispersion of the demand arrival process (items),
H: Generic headway between successive shipments (time), assumed to be the first in a sequence,
H1: Maximum interval between successive dispatches (time), Hi: Headway between the ith and the (i+1)th dispatch (time),
i: Annual discount rate for money ($/$-year), used in Sec. 2.1 only, n: Number of shipments,
ns: Number of stops,
ʌ: Value of one item ($/item), ʌ0: Production cost of one item,
ʌ1: Selling price for one item, PQ: Generic segment of a figure, pt: Costs incurred during year t, ı5: Standard deviation of the lead time, t: Time,
t5: Mean of the "lead time", T5,
T5 : Lead time (period between order placement and arrival), tm: Transportation time between origin and destination, v: Generic shipment size (items),
V: Total number of items shipped, v¯ : Average shipment size,
v': Average number of items shipped (regular plus expedited) per regular shipment,
v*: Optimal shipment size, v0: Inventory trigger point, vi: Size of the ith shipment, vmax: Capacity of a vehicle (items), v'max: Capacity of a pallet (items).
3 Optimization Methods: One-to-One Distribution
Readings for Chapter 3
Newell (1971) shows how to find an optimal sequence of headways for a transportation route serving a changing demand over time with a contin-uum approximation method that avoids "details." This problem is mathe-matically analogous to the problems with time dependent demand ad-dressed in this chapter, which are traditionally solved with dynamic programming. Section 3.3, is based on this reference. Daganzo (1987) shows that a continuous approximation of a function and its variables can be more accurate than the exact, detailed and discontinuous world repre-sentation they replace. This result is discussed in Section 3.2.