INVENTORY MODELS WITH TWO CONSECUTIVE DELIVERY MODES
3.6. An Example
Similarly, (3.45)-(3.46) can be written as
= inf \c{(y-q,) + Cf{z-y)-^E[H2(y-9iiilllvi))]
z>y
+ E[u2{z-gi{illlvi)Jl)]], (3.111)
= inf [ciiy-q2) + E[Hs{y-g2{iillv2))]}. (3.112)
y>q2
Let
p + h
Now we find the optimal order quantity for the fast order at period 2 when I^
is observed with J] = ^2- To this end, solving equation
j^ajmy-^^JlvM^. (3.113)
we get the solution
dy
y* = £a. ( / ? - - ) + 4 .
Hence, the optimal order quantity for the fast order when I2 is observed with I2 = i^j is as follows:
ea . (/? - i ) + 4 - q2, if ea • (/5 -~ ^) + 4 > ^2, ^ ^ otherwise.
In (3.114), the optimal order quantity for the fast order at period 2, / | , is a piecewise function of the observed information i^ and the inventory position g^2. Therefore, the value function V2(92?^2) ^^ ^"^^ ^ piecewise function of
^2. By (3.114), the manufacturer needs to make a fast order at period 2 only when the inventory position is lower than the ordering point—that is, / I > 0 if q2 < sa{P - 1/2) + i^', otherwise, / I = 0. Further, in view of (3.108) and (3.109), 5^2(4^ ^2 ^ ^2) < 4 + ^^/2 ^.p.l when I2 is observed; therefore, no penalty cost arises if the quantity of the inventory position is sufficiently large—that is, ^^2 > 4 + ^<^/2. Thus, we present the value function in the
following cases:
V2fe,4)
( -h-il + h-q2, if 4 < 92 - f ,
if g 2 - f < 4 < 9 2 - £ a ( / ? - i ) ,
I cl-il-4-q2-\-Y, if 52 - £a(/3 - ^) < 4 '
(3.115) where 1^1(52), ^2(92) and y are defined as
yi{q2) = —^ r—92, sa
^2(92) =
25a (92) + - •92 + £a,
^ , . ( , _ i ) ^ M ! ± i ^
Based on ^2(92, ^2) given by (3.115), we find the optimal fast-order quantity fi and the optimal slow-order quantity s | at period 1. First, it follows from C({u) = octhat
Let
and
/r = 0.
Gi(z) = c?-z + E[V2(z,/l)],
ai = p + h, a2= p — cl.
It follows from (3.115) that if 0 < z < V2 — -+ea((3—), taking the derivative of Gi(z), then 52 > 0 and
(3.116)
if ^^2 - 2 + ^^(/^ - id <z <V2- -{I- e), then
dCifz) a i 2 1 r / ^ \ ^ 1
-j = 2-^ + 9 2\-'^Ci'^ -^ ^CiW - ^012) - ^Cil'^2\z Oil 2 1 r / r \ r. fi
+ r—2^2 + ^—2 L ^'^i + ^^v^ - h) - 2ea&2\v2
"^''i 2 "^ 2 2 ^ 4 "^ 8 "^8g' (3.117) if V2--{!-£) <z <V2 + - + £a{(5 - - ) , then
(3.118) dGi(z) /i + 4 /i + c^ £a2(^ + ci) 2cf + / i - c i
• = -z -V2 ^ - + ^—1^ ^;
az a a la\ I a /^ Ix ^ a ea ^
if ^^2 + 2 + ^^(^ ~ 2 ' ^ ^ ^ ' ^ ^ ' ^ 2 ' ^ T '
d G i ( 2 ; ) a i 2 , o;i ^ i 2
a i . , eai 4cf + 3/i ~ p a i
- 2 ; ^ [ 1 + ^ 1 " ^ - — + — 4 — &
and if z > i;2 + f + ^ , then
(3.119)
Gi(;^) = clz + h 6t. (3.120) This implies that
^ = 05 + /^.
az
To demonstrate the cost function graphically, we depict the cost function with three sets of parameters, where the parameters are for the base case, h — 0.1,p = 5.3,cf = 1.0,c^ = 2,a - 15,£ = 0.5,i;2 = 50; for the higher holding cost, h = 0.5; and for the higher penalty cost, p = 5.5. The graphs appear in Figure 3.2.
Now we discuss some insights into the relationship of the optimal order quantity and other parameters. By Theorem 3.1, we know that the cost function Gi {z) is convex in z. Therefore, it is sufficient to find the optimal order quantity from the first-order condition.
Expected total cost 74
72 70 68 66 64 62 60 58 56 54
1— 1 1 1 \ r~
odse cdse
Higher holding cost Nv ' ^. Smaller penalty cost
k" N. * * '
\ >v * ^ *
1 ^^ ^ V '
^ N N. * » ' '
^\ \>. ^..^ N. ^v^^^ ^^'^
1 \ 1 1 l _
-_
—
35 40 45 50 55
Slow-order quantity z at period 1
60 65
Figure 3.2. Sample cost curves with different cost parameters
LEMMA 3.1 Assume that {?)AQ5)-{?).\09) hold. Then we have (i) when c\ > 03, s^ = 0 is optimal;
(ii) when cf < c^, the optimal s* > t'2 — f + £a(/? — 5)-
Proof First we prove (i). By (3.116), we get 6Gi{z)/diZ > 0. Therefore, the optimal (minimum) value is achieved at the extreme point zero.
Next we establish (ii). The cost function is convex. In addition, dGi(^)/d2;
< 0; therefore, in the interval [0, V2 — a/2 + ea{(3 — 1/2)], the minimum lies
on the right boundary—that is, s\>V2 — a/2 + ea(/3 — 1/2). D Lemma 3.1 indicates that actions must be taken at period 1 when the unit slow-
order cost is less than the fast-order cost at period 2. From the point of view of just-in-time production, no material should be ordered at period 1. Similarly, the quick-response program rejects ordering materials at this time. In other words, just-in-time production and a quick-response program apply only for the cases where there is no per unit order cost difference between different supply sources. This observation is corroborated by other researchers. For example, Fisher, Hammond, Obermeyer, and Raman [5] observed that "to address the problem of inaccurate forecasts, many manufacturers have turned to one or
another popular production-scheduling system. But quick-response programs, just-in-time (JIT) inventory systems, manufacturing resource planning, and the like are simply not up to the task." Lemma 3.1 provides an analytical example of why manufacturers need a better system than a pure just-in-time approach.
Further, it is interesting to consider scenarios with capacity constraints where only one source of raw material is available and the lead time and price are constant. The just-in-time production strategy suggests that no decision on raw-material order quantity or production commitment be made until the latest stage. In many industrial settings, especially in the quick-response systems, the available capacity limits the production lead time. If the production lead time can be reduced, the demand forecast is more accurate at a later time.
Lemma 3.1 suggests that decisions on raw-material ordering and production commitments can be made earlier. With some earlier productions, precious capacities could be reserved at a later stage when the forecasts become more accurate. This coincides with the findings of Cohen and Mallik [3]. They argue that by holding excess capacity, a firm has an option to respond to uncertain events and may be able to take advantage of arbitrage opportunities.
We summarize results in this section into the following theorem.
T H E O R E M 3.8 Assume that (3.\05)-(3.109) hold. For any setting of parame- ters, we have
(i) if cf > c(, then s^ = 0;
(ii) if cf = C2, then s^ = x, for any x that satisfies 0 < x < f 2 — a/2 + ea{(5-l/2);
(iii) let
if V2 - a/2 + £a{(5 - 1/2) <x<V2-al2 + {so) 12, then s\ = x;
(iv) let aci
X = V2 r-^—I-
ci — h
4 + h 2(4 + h) ^ -1 2(c^ + h)ai
+
ea hp — cUh — a2)if V2 - a/2 + ea/2 < x < V2 + a/2 + ea((5 - 1/2), then s\ = x;
(v) let
a ea a 1 , „ ~ ifv2 + a/2 + ea[(5 - 1/2) <x<V2-\- a/2 + ea/2, then s\ = x;
(vi) else, s^ = V2 + a/2 + sa/2.
It can be proved that the optimal order quantity is a linear function with respect to either a or e. Figure 3.3 provides examples of the changes of cost
54 35
Expected total cost 74
72 70 68 66 64 62 60 58 56
[7 1 1 I 1 1
1 **» Base case
r~ *», v 1 rtn111r'ot^f' WYMTwrw/f^nnf^'nt i n TCWf^/^c^QtAincf hv N^ O l g l l U l L d l U U U p i U V C l I l C l l L 111 lUlCv^abLlllg
r x^^ *^, - ~ -" Marginal improvement in forecasting
u \ \ . "••"•••.,
\ N. ""^^ y
N ^ V **'% ' ' / ^
- ^"^ ^ v '^'^ '-'X
^^ >v " ' • * , ' ' ' ' X
\ \ v ^ ^ ^ /
\ ^^-^^ ^.^-^ y
^~-- - ^ ^
~ ^ V. ^ " ^
^ ^ -'"^
1 1 ^ ' ^ ^ n — ' ^ " ^ I \ - -
~
1
—1 40 45 50 55
Slow-order quantity z at period 1
60 65
Figure 3,3, Sample cost curves with different forecasting-improvement factors function with respect to the changes in s. The smaller e is, the less that the stage 1 order quantity will be. We find that, similar to the s, the larger variance requires a larger order quantity. We demonstrate this feature in Figure 3.4, where the larger the variance is, the higher the cost and the larger the order quantity are. On the other hand, it seems that the order quantity is more sensitive to the degree of improvement than the variance itself.
REMARK 3.8 If a is sufficiently small, for the case of cf < C2, si nearly optimal.
V2 IS
The analysis given above reveals the existence of optimal purchase policies with respect to cost parameters and demand information. These findings an- swer questions such as how well the manufacturer can forecast and what the optimal expenditure is. However, the manufacturer would be interested to know the value of information updates and, further, the opportunities for continuous improvement. In this section, we explore these managerial implications by marginal cost/benefit analysis.
It is reasonable to assume that the manufacturer does not have control over cost parameters in the short run. After knowing the optimal material-purchase policies, the manufacturer would look for other directions of further improve-
54
Expected total cost
74 72 70 68 66 64 62 60 58 56
1 ' ^
k L\
\ \,
L. \ k \ X '^
\ "^
* \ ^^
h \ \ \
\ \ \
*» \ '^
N^ X ^^^^
^ \
1 1
\ 1
- Base case
• Smaller variance Larger variance
^^^-—- — ' '^
— y
^'^Y " \
1 1
_ / / / / / / / /
/ — /
^•^ 1
-J
1 1 1 '35 40 45 50 55 60
Slow-order quantity z at period 1
65 70
Figure 3.4. Sample cost curves with different forecasting errors
ment, such as improving its demand forecast. Intuitively, improving either stage 1 and stage 2 forecasts results in the cost reduction. The marginal costs of information updates with respect to s and a provide indications on the value of information updates. Specifically, the marginal benefit of information updates can be expressed as follows. If V2 —1/2+ea{i3—1/2) < s^ < V2 — a/2-\-sa/2,
de da
(/t+cf)(/t-cf)+(4-cf)- 2{p+h)
v / 2 ( 4 - c f ) ^ / ^ V ^ 3 VpTTI s / i ' 2 3 y/^+h \ / S '
if V2 - a/2 + ea/2 <sl<V2 + a/2 + sa{/3 - 1/2),
(3.121)
dG*isl) de da
{h+cjfe , {p-c{){h+c\) I2{p+h)'^a
+
{h+c. / \ 3 ^ 2
2{p+h) {ci-c\){h+c\) WP+W^ + 2{h+ci)
(3.122)
and if t;2 + a/2 + ae((3 - 1/2) <s\<V2-\- a/2 + {ae)/2,
de dG\{sl)
da
h+c\
~ 2 _h+cl
~ 2
v/2 {h+clf/^ v ^ 3 y/p+h i/e v/2(/^+cf)^/^yg
3 y/p+h y/a
(3.123) Based on equations (3.121), (3.122), and (3.123), the manufacturer is able to determine the impact that one unit of demand forecast improvement in either stage 1 or stage 2 has on its cost structure and further to determine whether its effort in improving demand information is worthwhile. From these equations, when cl is large, improving the first-stage forecasts would yield a larger payoff;
similarly, when the difference of cf and C2 is small, improving the second- stage forecasts would be much beneficial. More important, the notion of equal principle (Samuelson and Nordhaus [11]) suggests that the company should put its last dollar to the place where a higher retum is expected. Equations (3.121), (3.122), and (3.123) provide formulas for calculating marginal retum with respect to demand-forecast improvement. Comparing dGl{sl)/da and dGl{sl)/de indicates the potential retum. For example, if dGl{sl)/da >
dGl(sl)/ds, the manufacturer should concentrate its effort to improve the demand forecasting at stage 1.