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Safety production plans in MRP-systems using transform methodology
international journal of
production
econOmlcs
ELSEVIER
Int. J. Production Economics 46M7 (1996) 297 309
i
,
Safety production plans in MRP-systems I using transform methodology Robert W. GrubbstriSm*, Anders Molinder Department of Production Economics, Link6ping Institute q["Technology, S-58l 83 Link@ing, Sweden
Abstract In recent papers relationships between input-output analysis, material requirements planning (MRP 1 and multiechelon production-inventory systems have been studied. In this paper previous findings are generalised with external demand as a random variable. Immediate rescheduling because of random disturbances makes an MRP-system behave in a nervous way. Therefore correctly dimensioned safety stocks are important for use as a buffer when creating feasible master schedules. In this paper we develop a basic method illustrated with some examples of how such safety master production plans can be determined in simple cases in terms of our previously developed theory using the Laplace transform. Keywords: MRP; Input-output; Multi-echelon systems; Safety production; Laplace transform
I. Introduction 1.1. Background
Material requirements planning ( M R P ) systems were originally designed for a deterministic environment. Often, however, d e m a n d for end-items or lead times in the system are uncertain. Therefore, detailed c o m p o n e n t plans are frequently extended with safety stocks in the M R P records. The safety
~Revised version of the paper presented at the Eighth International Working Seminar on Production Economics, lgls/Innsbruck, Austria, 21 25 February 1994. * Corresponding author.
stock is a buffer of stock which is used for protection against uncertainty in the gross requirements over lead time. Safety stocks could also be used for protection against unplanned usage of spare parts and c o m p o n e n t s which means that lower levels in the system also are affected by stochastic disturbances. By those means the inventory holding costs can be balanced against the service level of the end-items, cf. [1, pp. 511-514]. W h e n the master p r o d u c t i o n schedule is replanned, variability of d e m a n d and lead times may cause earlier p r o d u c t i o n runs, but safety stocks m a y serve to protect for some or most of these. In this paper some thoughts for setting accurate safety stocks are outlined in terms of our earlier developed theory (cf. [2, 3]). The theory presented below applies the stochastic extensions of the
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fundamental equations to the backlogging of external demand in simple cases. In this paper, external demand for end-items is assumed to take place as individual random events separated by independent stochastic time intervals, an example of which is a Poisson process. This has the consequence that inventory and backlogs will be stochastic in nature. Production decisions may be decided individually in terms of both batch sizes and their timing. These decisions are taken in the beginning of the process to be planned which lasts until a given horizon T is reached, and enables safetyproduction decisions to be incorporated simultaneously. As our objective function, we choose the traditional sum of expected average costs of set-ups, inventory holding and backlogs. We restrict our attention to a one-level and the most simple twolevel, serial system. We also note that the increase of the number of product levels from one to two increases the degree of complexity in our analytical expressions considerably. As the uncertainty in demand on the highest level of a multi-level system increases over time when looking into the future it amplifies the uncertainty in the current production schedules on the lower levels. Therefore, it is important to capture the effects of cumulative uncertainties in external demand on the highest level, despite the fact that there might well be time to reschedule the top level plan before it is realised. Hence, future cumulative uncertainties of external demand certainly have an effect on the safety production plan and need be taken into account before the parts explosion begins. A production plan which takes uncertainty into consideration must be more reliable to use over a longer time period (until the next rescheduling occurs) than a plan which does not.
1.2. Notation We use the abbreviations f(t) for the time integral (cumulative function) of any time function f(t) and f(s) or ~ { f ( t ) } for the Laplace trans-
form of f(t), where s is the complex frequency (cf. [4, 5]): o0
f(s)
= ~cF{f(t)} =
fe-~'f(t)dt.
(1.1)
0
The inverse transform is the time function having a given transform f(s) and is written as
f(t)
= ~ - 1 {f(s)}.
(1.2)
For the sake of treating nonnegativities, we use the representation I f ( t ) ] + M a x { f (t), 0} for any time function f(t) If(s)]+ Laplace transform of I f ( t ) ] +, i.e. the transform of a time function which is truncated and for treating probability distributions we use Pr(.) for the probability of its argument occurring and E(.) for the corresponding expectation operator. Further assuming the total number of items in the system to be N, we use: P~(t) production of item i at some time t (as a rate), i = 1,2 . . . . ,N; the Pg(t) for different values of i are collected into the column vector P(t) /5~(t) cumulative production of item i at some time t; the Pi(t) are collected into the column vector P(t) Bi(t) volume of backlogs (cumulative stockouts) of item i at some time t; these are collected into the vector B horizon, i.e. the total length of time periods covered by the plan In the special but normal case of batch production to be treated in the next section, we adopt the notation Qik order quantity of item i, i = 1, 2 . . . . , N, of the kth batch; these are the decision variables to be determined in advance of the process tik time at which the production of batch Qik is decided to take place (is completed) n~ number of batches of item i produced within the horizon
R. ~ Grubbstr6m, A. Molinder/Int. J. Production Economics 46 47 (1996) 297 309
ri
lead time for the production of item i from the time its production is ordered (demand for components are generated) until the time the order is completed Si total inventory of item i at some time Ri available inventory of item i at some time Hi i number of units of item i required for the production of one unit of item j Di external demand rate of item i at some time In the batch production case, the entity P~(t) is interpreted as follows. Since tgk, k = 1,2 . . . . . are the times at which batches Qik of item i are completed, then production P~(t) will be a sequence of Dirac impulses ~kQik~(t -- tik) and/~.(t) will be the time integral of this sequence (a staircase function). The Dirac impulse 6(t - t~k) is a generalised function having an integral of unity if it is integrated over an interval including t~k and an integral of zero otherwise. As abbreviations, we also use Pig = P~(tik), which is the cumulative production of item i at time t = t~k, when the kth batch has been completed. H denotes the input matrix with element H~i in position (i,j), H* = (I - H) -1 being the Leontief inverse of H, and S and R represent total and available inventory as N-dimensional vectors (as functions of time). D~ is the (stochastic) external demand of item i at some time. The Dj are arranged into a column vector D. ri is the constant lead time for the production (acquisition) of item i from the time its production is ordered until the time the order is completed. The lead times r~, r2 . . . . . Lv create internal demands for components these times ahead of the completion of the products they enter and correspond to translations backwards in time. The diagonal lead time matrix ~, having e ~', in its ith diagonal position captures these lags, and n = H'~ is then the generalized input matrix describing component requirements in volumes as well as the times in advance when they are needed.
299
accounts for all items in the system and is initial stock plus cumulative net production less external demand:
S=[~,S(O)+,I-H)P-/5)]+
7! °'+'' •
,,3,
+"
As seen from the point of view of total inventory, the internal demand is instantaneous. At the same time as a component is assembled into a higherlevel product, the stock of this item is reduced by the same amount. The net production as seen from the available inventory point of view is gross production P(s) less external demand and less internal demand at the time the components are reserved, i.e. H~P(s). Available inventory represents the physical existence of items at the time they might be needed physically and is given by 1
/~ = - [R(O) + (I - H~)P -/5]+ S
71
•
+"
,,4,
The physical necessity of available inventory being nonnegative accounts for constraints of inventory on lower levels in production. The first two terms of available inventory, R(O)/s + ( I - H'~)P, must always add up to some nonnegative vector (in the time domain) otherwise the corresponding production plan would be infeasible, i.e. internal demand is always met, If external demand cannot be met, we assume that this demand is backlogged and satisfied at the time available inventory is positive. Backlogs therefore only concern external demand and are given by /~ = _1 [/5 + B(0) - (I - H~)P] + S
1.3. Fundamental equations Due to the lead times, we make a distinction between total inventory S, available inventory R and allocated component stock ~ - R. Total inventory
=
3 + B(0) - (I - H~tP 1 S
+"
(1.sl
Eqs. (1.3)-(1.5) are referred to as the Jundamental
equations•
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Grubbstr6m, A. Molinder/Int. J. Production Economics 46-47 (1996) 297-309
2. One-level model
2.1. Stockouts and inventory We first consider the case of producing only one product in batches at times t~, t2 . . . . , t,. This implies that we do not need the first subscript i (the item index) and the remaining index will refer to the batch number and not the item number according to Xk = X~k for a typical variable. The horizon of the process is finite t = 7~. For convenience, we define t,+ a = 7~. The fundamental equations then contain 1 x 1-matrices and one-dimensional vectors and there is no need to distinguish between S and R since H ? vanishes. We also assume that S(0) = O. Demand is considered to be made up of individual events occurring at independent stochastic intervals. In this section, our developments are general, but in the numerical examples we confine our attention to a Poisson process with an average interval of 1/2 and therefore an average demand rate of 2. Our decision variables are the produced amounts, their timing and the number of batches to be produced within the horizon. The objective function is chosen as traditional average costs (for set-ups, for holding inventory per unit and time unit, and for backlogs per unit and time unit), although the numerical examples to follow should easily be amenable for optimisation adopting the annuity stream principle. With H ? = 0, the fundamental equations for inventory and stockouts become ~(s) = [/~(s) --/~(s)] +,
(2.1)
/3(s) = [/~(s) -/~(s)] +.
(2.2)
Expected inventory and expected stockouts are thus related by
E(~(s)) = P(s) -- E(B(s)) + E(~(s))
(2.3)
(2.1) and (2.2). The major problem is to solve an optimisation problem with (2.1) (or alternatively (2.2)) as a fundamental constraint. If this is done, then E(~(s)) (or alternatively E(~(s)) is easily determined. We choose to use cumulative production rather than batch quantities as decision variables, although they are easily translated into each other. For convenience we set to = 0, Qo = 0 and Po = 0, and the transform of cumulative production and of its time integral will be
=1~
Qke-m--e-~'t k=O ~ Qkl
s2
Qke-st" _ e-S¢ k=O
Qk .
(2.6)
k
From the definition of the Laplace transform (1.1), the value of the time integral of cumulative production at the horizon T is then
='im:Es
e k=l
k=O
(2.7) using, for instance, a series expansion in s. The transform of the expected cumulative demand and its integral is
and the integral of expected inventory and expected stockouts by
E(~(s)) = ~ (1 E(S(s)) = P(s) -- E(D(s)) + E(B(s))
(2.5)
- --s e
,
(2.8)
(2.4)
which is used below for determining the time averages concerned. Eq. (2.3) is a simple consequence of
2
E(B(s)) = U(1 - e - ~ )
2T e_St
sT
(2.9)
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and the value of the time integral at the horizon is E(/5(~))
lim[ 2 -- _2T _ e-sT ] ~ o ~ (1 - e - ~ ) s
•
CumulativeProduction and ExpectedStockouts
)~-p: -
(2.10)
2
: 2lL . . . . .
If cumulative production has a given integer level Pk at some time, the transform of the probability of a certain stockout j/> 0 at that time will be [6, 7]
B(tl
,
{Pr(B(t) = j)}
-_ ~ S I P r ( ~ ( t ) = P~ +j)~,
t~
;~ t~ ;
tk,:
tk.3
T
t
~ S IPr(/)(t) ~< Pk)], Fig. 1. Expected backlog as a function of time (striped level).
= f/'i+Pql-- - - f ) / s ,
j > 0 j = o
t ( i -f)ls,
(PR given),
(2.11)
where f(s) is the transform of the probability density of the interarrival time between two demand events. The transform of the expected stockouts for a given production Pk will then be = .i =
o
(1 -f)i's
jf
Summing over all batches n produced over the production interval T we have
i
s(l - - f ) "
(2.12)
In the case of the Poisson process applied in the next section, we have f ( t ) = 2e - ~ and therefore f = 2/(2 + s). In [6], it is shown that a primitive function for the expected cumulative stockouts in that case can be written as E(B(t)) -
)~t2 2
Pt + P ( P + 1) 22
P~2 (~/)i+ 1 -
2P
j=O
~-v~-
e-X'[ 22
1)i~('(2t)Jj! P(P +
.=)
P~3 (,~t)j+ 21 +
j=O
J!
,
(2.13)
where P = Pk is the given production level. In our transform representation the expected cumulative stockouts for a general independent interarrival time distribution can be obtained integrating (2.12), " 1 /Pk + 1 E(B) - s2 (1 -- f ) '
and the contribution from each production interval k to the integral as
(2.14)
'
which may be evaluated using (2.13) in the Poisson case. The expected stockouts during the production intervals are depicted in Fig. 1 as the striped levels above cumulative production. We may note that stockouts increase progressively in the beginning of each production interval k and they will approach E(/)) - P for larger values of t. An analysis of the asymptotic behaviour of stockouts for a planned production process is given in [8], where it is shown that this behaviour for a large class of probability distributions follows essentially the square root of time.
2.2. Optimisation
The classical inventory problem attempts to find a balance between the economic consequences of holding inventories, of production and of stockouts. The main decision variable is the batch
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R.W. Grubbstr6m, A. Molinder/Int. J. Production Economics 46-47 (1996) 297-309
quantity Q, but an additional production decision can be the amount of a safety stock. We are now in a position to define our objective function in terms of the integrals obtained above. Let h denote the inventory holding cost per unit and time unit, b the backlog cost per unit and time unit and K the cost for each set-up. O u r objective function to minimise then becomes Kn
h
C = --:- + - : [ 5 ¢ - 1{E T T
(~)}] ,=~-
b
+ ~ [L/~-' {E (/~)}], = f,
(2.17)
where E(S(T)) = f i ( T ) - E(5(IB)) + E(/~(]B)) ~. k=,
=
Pk(tk+,
- -
2~ 2 tk) - - ~ + E(B(T)).
(2.18) F o r a given value of n, the decision variables may be interpreted as the tk and /~k, k = 1, 2 . . . . . n. The first-order optimisation conditions are the following: OC
h
h
77=77
ji,:,
= o,
(2.19) OC
h
--=--~(tk+i--tu)--
h ; b ((~_ i ,~i'k +1~~
. - - - Sr - - .l".~' : ,x.I (<.~-1 C JJ
[
~ O,
s2
J/t=,. (2.20)
where the second set of conditions is to be interpreted as a set of difference (rather than differential) conditions, since the Pk are only allowed to be varied in integral steps using the interpretation
=fP*(s) ( f ( s ) - 1).
(2.21)
As seen from these conditions, the optimum solution, given n, must depend on the ratio (h + b)/h = 1 + h/b only. Furthermore, we must require that tk + ! > t k.
2.3. N u m e r i c a l solution p r o c e d u r e and results
Since the optimisation problem cannot be solved analytically because of the complexity of the expression for the expected stockouts in (2.15), we have solved the minimisation problem with a numerical iteration procedure according to the steps 1. The finite fixed horizon 7~ is first divided into (n + 1) equal time intervals as a trial solution. 2. For a given sequence tl, t2 . . . . , t , , optimal integer values of PR, k = 1, 2 . . . . . n, are calculated each in turn using an interval-halving procedure until Eq. (2.20) is satisfied as close to zero as possible. 3. F o r a given sequence, Pk, k = 1, 2 . . . . , n, the corresponding optimal time intervals tk are calculated using an interval-halving procedure until (2.19) is satisfied. 4. The iterations in each of these two steps are stopped when an indicator for each equation reaches a small value. Steps 2 and 3 are then repeated in sequence until there is no change in any of the two solution sequences. For each n = 1, 2 . . . . , steps 1 4 are carried out and average costs also including set-up costs are computed until the minimum sum of these are found, constituting the overall optimum of n, and the Pk and tk, k -- 1, 2, ... ,n. We have not investigated conditions for the convergence of this suggested procedure, but all numerical results indicate that no problems are encountered as long as reasonable values of the parameters are chosen. The problem was solved numerically for six different parameter combinations regarding set-up costs, inventory holding costs and backorder costs. The parameter values as well as the numerical optimisation results are shown in Table 1. The expected average demand rate was chosen to unity at the same time as the ratios between the backorder cost and the inventory holding cost were varied.
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303
Table 1 Results of the safety production plan calculations in Section 2.3 using six different parameter combinations, ,~ = 1, n* = optimal n u m b e r of batches (*)
h=
k
1
h=50
K = 100
n* = 6
Pk
tk
Qk
SSk
26 47 65 83 99 115
0.5 21.3 39.2 55.1 70.9 85.4
26 21 18 18 16 16
0
1 2 3 4 5 6
1,*)
h=
k
1
h=
k
- 0.5 4.7 7.8 9.9 12.1 13.6 15.0
P~
~
Q~
ss~ - 9.5
0
19 40 62 87
9.5 29.5 51.1 74.6
19
- 10.5
21 22 25
- 11.1 - 12.6 - 13.0
t7=50
b= 1
K = 100
n* = 5
k
Pk
t~
Qk
SS~
0 1 2 3 4 5
12 26 40 53 69
16.4 32.9 49.2 64.2 81.4
12 14 14 13 16
---
16.4 20.9 23.2 24.2 28.4 -- 31.0
In Fig. 2, a case with a high given backlog cost is shown (Table 1, Case (*)). We note that the distance between the cumulative production level and the expected cumulative demand increases with time. Hence, the distance at the end of each step represents the safety stock (SSk in the figure). As the uncertainty in demand increases over time a compensation takes place in the optimal safety stock. During the iterations, it was notable that the convergence adjustments towards optimum during the
n* = 9
P~
tk
O~
SS~
17 33 47 60 72 83 94 106 117
0.3 13.2 26.4 38.3 49.5 59.8 69.7 79.8 90.4
17 16 14 13 12
~ 3 4 5 6 7 8 9
n*=4
2 3 4
K =50
l
K = 100
1
h=50
0
b=l
0
1
h=
k
I
b=
1
O.3 3.8 1~.6 8.7 1(/.5 12.1 13.3 14.2 15.6 17.0
11 11 12 II K =50
n* = 5
s%
&
t~
Q~
1
17
2 3 4 5
35 53 72 90
8.5 26.1 44.1 62.6 81.1
17 18 18 19 18
h=50
b= 1
K =50
n*--:6
k
Pk
tk
Qk
SSk
0 1 2 3 4 5 6
8 22 33 46 58 72
14.3 28.6 41.8 56.3 70.2 85.3
10 12 II 13 12 14
- ~.5 9.1 9.6 9.1 .- 9.1 10.0
- 14.3 18.6 19.8 - 23.3 - 24.2 27.3 - 28.0
iteration were particularly intensive in the first few intervals compared to the initial trial solution. In Fig. 3, a case with an equal ratio between backorder cost and inventory holding cost is shown (Table 1, Case (**)). We note that the optimal solution shows a symmetric alternating backorder and inventory situation and that the plan both begins and ends with a backorder situation. In this case we found that the convergence adjustment towards optimum during the iterations was
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Cumulative production/demand 2=1 K=I00 b=50
,, ,
~ " "
H * = 6
•
Slope=2
/ / / / / -
-
/ / / / /
/
I fi
-%----+-
I
I
t2
h
t~
h
t
T
It is notable for all six cases that the time intervals between the set-ups and their mean value differ at most in the beginning and at the end of the production horizon• When the backlog cost parameter is high, cumulative production increases faster than cumulative demand, resulting in an increase in safety stocks over time. The correlation between the optimal time intervals and the corresponding batch quantities is very high in such cases, indicating that the time intervals essentially increase in a proportional way. In the converse case with a high inventory holding cost parameter, the safety stock becomes increasingly negative, as expected. However, no obvious pattern in the relationship between batch quantities and intervals has been found in such cases and the correlation coefficients computed are low.
Fig. 2. Safety production with high backlog costs•
3. Two-level, serial model
Cumulative2:lproduction/demand
K=100 b=l h=l n*=4
3.1. Introduction
r _ ~ i ~
~
We now turn our attention to the two-level serial assembly system. Backlogging is not permitted on the second level and we assume no external demand on this level. Notations will now have two indices, the first being the item number (in this case coinciding with the level number) and the second the batch index. In terms of our previous developed model the input matrix is
..'" Slope=2
r--
i
/'"
.=i0001 A
t2
ts
T
Fig. 3. Safety production with equal backlog and storage costs.
Production on the top and lower levels take place in batches of sizes Qlt, Q2t at times tl~, l = 1,2 . . . . , n~, and t2z, / = 1, 2 . . . . , he. Cumulative production on the lower level will be /52(s)=![f_~ a Q2te-S'~' -
particularly intensive halfway through the production horizon. Apparently, the safety stock then was negative throughout. The possibility of negative safety stocks may be questioned by definition. However, we feel no reason for confusion by defining safety stocks to be positive as well as negative.
e-sT"l~= Q2t1.
(3.2)
The intermediate stock on the lower level (inventory of item 2) is g2(s) =-sl / Qzle-St~"~ ~1 - k~Q~ke-S"kl= 1 '
(3.3)
R.W. Grubbstr6m, A. Molinder/lnt. J. Production Economics 46 47 (1996) 297 309
where it is assumed that initial stock is zero. The available inventory on this level will be
assuming initial available inventory also to be zero and that cumulative inventory on the lower level ends at the same value as that of the upper level. Both S2(s) and 1~2(S ) are differences between staircase functions and it will always pay to keep the area between these two functions as small as possible for given values of nl and n2. We can therefore make the following remark. Whatever staircase the upper-level cumulative production follows, at the optimum there can never be a gap between the lower-level cumulative function and the highest of the steps of the upper function within a particular lower-level production interval. If there were such a gap, the lower-level production level could be lowered reducing inventory holding costs with no other effects. Also, whatever staircase the upper-level cumulative production follows, at the optimum there can never be a gap in time between the next replenishment of the lower level and the next replenishment of the upper level, because if this were the case, then the lower-level replenishment could be postponed reducing holding costs with no other effects. Therefore in the two-level serial case, we note:
In the optimal solution, the lower-level replenishments ahvays must take place at the same time as some upper-level replenishment and at that time (immediately prior to the replenishment in question) both cumulative levels coincide. Geometrically, this means that the two staircases fit into each other at the "inner" corners when lower-level replenishments take place (indicated with circles in Fig. 4). This also implies that the optimal number of batches on the lower level cannot be higher than on the upper level, as illustrated in Fig. 4. The deterministic lead time r~ is of no consequence in this respect, since it only translates the staircase uniformly backwards in time. Hence, the set of times (t~k--Z~), on the one hand, and the set of cumulative levels P~k, on the other, in the optimum will contain the set of times
305
..... lower level upper level
t. I -
t p'-
Fig. 4. A two-level serial system with the left staircase showing P2~ and the right showing Plk. "Inner corners" are marked with circles.
t2t and the set of levels/521 as subsets, respectively. For handling the different combination opportunities, we introduce the two index functions k(l) and l(k), where k(l) is the batch index of the upper-level cumulative production determining the lower level of interval 1, i.e. P21 = Plk(l), and l(k) the batch index of the lower level as a function of the index of each upper-level interval (the kth interval). Therefore, we always also have P21(k)=/5).k(l(k)) and t z l t + l ) ÷ z"1 ~ t l ( g ( t ) + l ) . The function I(k), in general, has no unique inverse, whereas k(l) does. The functions k(I) and l(k) have the properties 1 <~k(l) < k(I + 1),
k(n2) = n , .
l(k+l)-l<~l(k)<~l(k+l)<~k+l,
l(n~)--n2. (3.5)
The time average of inventory on the lower level (which is deterministic) may now be written as
S2(T) - lim (/~(s) -/~(s))
= ~
--
1 T
P21(t2(l+ 1) -- 12t) +
k-1 ~" " l k ( t l
/31.~Zl
(k + l ) -- t l k ) ]
(P2uk) -- Plk)(tl (k~-1~-- tlk) + /51.,z'l k
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R.W. Grubbstr6m, A. Molinder/Int. .Z Production Economics 46-47 (1996) 297-309
=~ T
k 1
('Olk~l~k))-- Plk)(tl~k+ 1) -- tlk)
+ h2
m
-~ JillntZ1] .
(3.6)
If k(1) is given as a function, meaning that the relation between the two staircases Plk and P2t is fixed, then attaching a unit holding cost h 2 to S2(T)/T, we can increment the objective function C with this quadratic expression of the same decision variables as in the one-level case of (2.17). To ensure the plan to be feasible, we need t21 >~ 0, i.e. h 1 - zl >I 0, which is captured by including a Lagrangian term q(rl - t11), where q is a nonnegative multiplier.
C=-~I [Klnl + K2n2 -- hi ~-~2 + (hi + b l ) E ( B ( 7 ~ ) ) ] + ~(T1
--
tll)
+ 7~ Lk=l X (/l(k+
1)
~
k(l(m))=k
tik) + h:Pi,,zi ].
(3.7)
The derivatives ~C/OPlk and ~C/~tlk now contain further terms depending on h2 and on the structure l(k), but their essential form remains the samel
0C I 0txk - ~ [(hl - hz)(Pltk-1) -- Pak)
Jr h2(Plk(l(k-1))
--
ff lk(l(k)))]
(s
_{qO
(if k = l ) = (if k > l)
0,
-
-1{ --7-1)
, : ,,, )
(1 --f)
J),-,,. (3.8)
+
(tl(m+ 1 ) - tlm) I J
0
(if k < nl)
h2z--A
(if k = hi) ~< 0
and we now also have the complementary condition: r/(rl - tll) = 0.
(3.10)
We may note that the one and only term depending on the lead time which occurs in the derivative with respect to the final cumulative production level Pl,,. This is the effect from the influence of external demand when hedging against backlog effects. Had no final backlogs been allowed, this term would have been lacking, since the final cumulative production would then have been forced to equal total cumulative demand. Let us take a look at the optimal function k(1) which prescribes the geometry of the lower-level staircase, once the upper-level staircase is given. The top level has steps at tit, t12 . . . . ,tt,,. The lower level can only have steps at some or all of these points. If the lower level has a step then it must depart from the same value as that of the top level. Hence, whatever P 2 ( / + l ) , /~l can either remain or increase to P2t~+ i). Hence, for each interval there are exactly two possibilities, see Fig. 5. Furthermore, tlx = t21 + T1 and at t = t,, the two levels Pl,, and P2,2 must coincide. Hence the number of possible lower-level functions are 2 ~"'-1~ This is illustrated in Fig. 5 for n~ = 4 together with the corresponding functions l(k) and k(l(k)). One might also view this set of combinations as whether an "inner" top staircase corner is included in the lower staircase or not, the leftmost always being included. As regards the optimal number of batches on the lower level, if given the top-level structure, a lowerlevel increase with one batch is only justified if the inventory holding cost increase is less than the additional set-up cost K2: h2(P2(/+ 1) -- P2t)(t2(l + l) -- tel) <
+ T [(hi--h2)(tltk+i)--t,k)
(3.9)
K 2.
(3.11)
However, the optimal top structure certainly depends on its lower-level counterpart.
307
R.W. Grubbstr6m, A. Molinder/lnt. J. Production Economics 46-47 (1996) 297 J09
l
F
following reasonable set of values:
T k(ICk))
'i .
°
°
•
k
•
k
j.-
F f
~ l(k) ° .
FFU 4
k
F
T l(k)., k f
T t(k)., k F
t
>
FF
h2 : 0.5, b 1 = 50, K1 =
100,
K2 = 300, rl = 20, T = 100.
~ l(k). . . k f
2 = 1, hi = 1,
p-
T t(k).
For each value of nl, the 2 "1 l) different structures have been examined and evaluated. The values of different partial terms of the objective function in the suboptimum, given each value of nl, are shown in Table 2. The overall optimum is obtained for n l = 4. In each solution apart from the first the optimal number of set-ups on the lower level is shown to be two (Column 2 in the table). For nl = 4 we demonstrate the effects of the different lower-level structures (eight options) in Table 3, adopting the same enumeration as in Fig. 5. In Table 4 we have included the planning data for the solution of the optimal structure (No. 2) in Table 3). For this structure l(k)= 1, 1, 2, 2. The safety stock on level 2 SS2,k~ is defined as cumulative production less expected cumulative internal demand at the end of the interval. We may note that the optimal safety stock on the upper level SSlk behaves irregularly, in particular when it falls back at the end of the final interval.
k 4. Conclusions jr-
Vj5 V4 f
l l(k)°
k
lk(t(k)), k
j.-
lowerlevel......... upperlevel-Fig. 5. Possiblecombinations of staircase functions in a twolevel systemwith n~ = 4. The eightcasesare found by choosing among all possiblenondecreasingfunctionsl(k).
3.2. A two-level numerical example For purposes of illustrating our foregoing discussion we include the following example. The solution procedure is a slight modification of the one described in Section 2. Parameters are given the
The objective of this paper has been to apply the previously developed extensions of the fundamental equations in the most simplest cases with the aim of computing optimal safety stock values. Consequences of stockouts have been investigated for a general class of probability distributions for demand together with conditions for optimality. Optimal solutions for safety production plans have been determined numerically using the assumption of Poisson-distributed demand in a onelevel case and the corresponding analysis has also been sketched in the most simple two-level case. We note that for the case with a high cost ratio between backorder cost and inventory holding cost, the distance between the cumulative production level and the expected cumulative demand increases with time. As the uncertainty in demand increases over time, our interpretation is that a compensation takes place in the form of an increasing safety
R. ~14 GrubbstrOm, A. Molinder/Int. J, Production Economics 46-47 (1996) 297-309
308
Table 2 Values of the objective function for different values of nl
nl
n2
bIE(B(T))
hiE(S1 (T))
h2E(Se(T))
Klnl
K2n2
CT
1 2 3 4 5 6 7
1 2 2 2 2 2 2
10 603.57 10417.90 10334.45 10 337.59 10 340.36 10 339.92 10 340.95
8320.00 7001.06 6588.77 6426.54 6303.80 6238.54 6192.33
1040.00 1080.00 1346.52 1438.02 1498.28 1524.75 1543.05
100 200 300 400 500 600 700
300 600 600 600 600 600 600
20363.57 19298.97 19237.71 19 226.18 19284.24 19354.21 19436.95
Table 3 Values of the objective function for different structures on the lower level for the optimal value of n~ No,
n2
blE(B(7~))
h,E(S1 ( 7 ~ ) )
h2E(S2(T))
K,n~
K2nz
CT
0 1 2 3 4 5 6 7
l 2 2 3 2 3 3 4
10 358.51 10 336.23 10337.59 I0 360.81 10 336.34 10360.71 10 358.01 10 306.31
6487.26 6573.40 6450.57 6454.80 6503.16 6435.54 6414.16 6426.54
2156.37 1387.29 1438.02 1179.13 1442.08 1194.79 1215.83 1120.00
400 400 400 400 400 400 400 400
300 600 600 900 600 900 900 1200
19 702.14 19 296.91 19 226.18 19 294.74 19 281.58 19 291.04 19288.01 19452.85
Table 4 Optimal production decisions in numerical example
k
ilk
Plk
Qlk
fllk+ 1~ -- tlk
0 1 2 3 4
20.0 39.3 60.8 76.9
49 69 92 110
49 20 23 18
20 19.3 21.5 16.1 23.1
stock. In a converse case with a high inventory holding cost parameter, the safety stock becomes more and more negative over time in the example. Furthermore, we note for the optimal solution in the two-level production case that the lower-level replenishments always must take place at the same time as some upper-level replenishment and at that time both cumulative levels must coincide. A number of issues remain to be investigated. In future research, the models will be extended to more general cases including multi-level models
SSlk - 20.0 9.7 8.2 15.1 10.0
P2ttk~
Q2tlk~
SS21{k) - 20.0
69
69
110
41
8.2 10.0
with stochastic external demand also on lower levels. More general interarrival distributions and more general product structures are but two trails for further progress, Also, it would be of interest to model practically oriented lot-sizing rules used in real-life MRP-systems and investigate consequences of augmenting these with heuristic safety stock procedures derived from considerations of the kind presented above. Simulation studies based on the theoretical principles underlying these models will also be undertaken in order to examine more general stochastic settings. A number of
R. H,i Grubbstr6m, A. Molinder/Int. J. Production Economics 46 47 (1996) 2 9 7 309
i n t e r e s t i n g e x t e n s i o n s h a v e a l r e a d y b e e n m a d e by B o g a t a j a n d H o r v a t [13, 14-1 o p e n i n g u p several n e w lines of i n v e s t i g a t i o n ,
Acknowledgements T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e the m a n y v a l u a b l e s u g g e s t i o n s m a d e by Dr. L a r s - E r i k A n dersson, Department of Mathematics, Link6ping I n s t i t u t e of T e c h n o l o g y , w h i l e d e v e l o p i n g t h e b a c k g r o u n d t h e o r y of this p a p e r .
References [I] Vollmann, T.E., Berry, W.L. and Whybark, D.C., 1988. Manufacturing Planning and Control Systems. Irwin, Homewood, IL. [2] Grubbstr6m, R.W. and Ovrin, P., 1992. Intertemporal generalization of the relationship between material requirements planning and input output analysis. Int. J. Prod. Econom., 26:311 318. [3] Grubbstr6m, R.W. and Molinder, A., 1994. Further theoretical considerations on the relationship between MRP. input-output analysis and multi-echelon inventory systems. Int. J. Prod. Econom., 35:299 311. [4] Churchill, R.V., 1958. Operational Mathematics, 2nd ed. McGraw-Hill, New York.
309
[5] Aseltine, J.A., 1958. Transform Method in Linear System Analysis. McGraw-Hill, New York. [6] Grubbstr6m, R.W., 1994a. Stochastic relationships of a multi-period inventory process with planned production using transform methodology. Research Report RR-126, Departmem of Production Economics, gink6ping Institute of Technology. [7] Grubbstr6m, R.W., 1994b. Stochastic properties of a production-inventory process with planned production using transform methodology. Int. J. Prod. Econom., forthcoming issue. [8] Andersson. L.E. and Grubbstr6m, R.W., 1994. Asymptotic behaviour of a stochastic multi-period inventory process with planned production. Working Paper WP-210. Department of Production Economics, Link/sping Institute of Technology. [9] Hax. A.C. and Candea, D., 1984. Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ. [10] Molinder, A., 1995. Material requirements planning employing input output analysis and Laplace transforms. PROFIL 14, Production Economic Research in kink6ping, kmk6ping. [11] Ross, S.M.. 1989. Introduction to Probability Models. 4th ed. Academic Press, San Diego. [12] Whybark, D.C. and Williams, J.G., 1976. Material requirements under uncertainty. Dec. Sci., 7:595 6{)6. [13] Bogataj, L. and Horvat, L., 1996. Stochastic considerations oi" GrubbstrSm Molinder model of MRP, input output and multi-echelon inventory systems. Int. J. Prod. Econom., 45:329 336. [14] Horvat, [_. and Bogataj, L., 1996, MRP, Input-Output Analysis and Multi-Echelon Inventory Systems with Exponentially Distributed External Demand. Proceedings of GLOCOSM Conference, Bangalore, India.
production
econOmlcs
ELSEVIER
Int. J. Production Economics 46M7 (1996) 297 309
i
,
Safety production plans in MRP-systems I using transform methodology Robert W. GrubbstriSm*, Anders Molinder Department of Production Economics, Link6ping Institute q["Technology, S-58l 83 Link@ing, Sweden
Abstract In recent papers relationships between input-output analysis, material requirements planning (MRP 1 and multiechelon production-inventory systems have been studied. In this paper previous findings are generalised with external demand as a random variable. Immediate rescheduling because of random disturbances makes an MRP-system behave in a nervous way. Therefore correctly dimensioned safety stocks are important for use as a buffer when creating feasible master schedules. In this paper we develop a basic method illustrated with some examples of how such safety master production plans can be determined in simple cases in terms of our previously developed theory using the Laplace transform. Keywords: MRP; Input-output; Multi-echelon systems; Safety production; Laplace transform
I. Introduction 1.1. Background
Material requirements planning ( M R P ) systems were originally designed for a deterministic environment. Often, however, d e m a n d for end-items or lead times in the system are uncertain. Therefore, detailed c o m p o n e n t plans are frequently extended with safety stocks in the M R P records. The safety
~Revised version of the paper presented at the Eighth International Working Seminar on Production Economics, lgls/Innsbruck, Austria, 21 25 February 1994. * Corresponding author.
stock is a buffer of stock which is used for protection against uncertainty in the gross requirements over lead time. Safety stocks could also be used for protection against unplanned usage of spare parts and c o m p o n e n t s which means that lower levels in the system also are affected by stochastic disturbances. By those means the inventory holding costs can be balanced against the service level of the end-items, cf. [1, pp. 511-514]. W h e n the master p r o d u c t i o n schedule is replanned, variability of d e m a n d and lead times may cause earlier p r o d u c t i o n runs, but safety stocks m a y serve to protect for some or most of these. In this paper some thoughts for setting accurate safety stocks are outlined in terms of our earlier developed theory (cf. [2, 3]). The theory presented below applies the stochastic extensions of the
0925-5273/96./$15.00 Copyright 4", 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 5 - 5 2 7 3 ( 9 5 ) 0 0 1 5 8 - 1
298
R.W. Grubbstr6m, A. Molinder/Int. J. Production Economics 46-47 (1996) 29~309
fundamental equations to the backlogging of external demand in simple cases. In this paper, external demand for end-items is assumed to take place as individual random events separated by independent stochastic time intervals, an example of which is a Poisson process. This has the consequence that inventory and backlogs will be stochastic in nature. Production decisions may be decided individually in terms of both batch sizes and their timing. These decisions are taken in the beginning of the process to be planned which lasts until a given horizon T is reached, and enables safetyproduction decisions to be incorporated simultaneously. As our objective function, we choose the traditional sum of expected average costs of set-ups, inventory holding and backlogs. We restrict our attention to a one-level and the most simple twolevel, serial system. We also note that the increase of the number of product levels from one to two increases the degree of complexity in our analytical expressions considerably. As the uncertainty in demand on the highest level of a multi-level system increases over time when looking into the future it amplifies the uncertainty in the current production schedules on the lower levels. Therefore, it is important to capture the effects of cumulative uncertainties in external demand on the highest level, despite the fact that there might well be time to reschedule the top level plan before it is realised. Hence, future cumulative uncertainties of external demand certainly have an effect on the safety production plan and need be taken into account before the parts explosion begins. A production plan which takes uncertainty into consideration must be more reliable to use over a longer time period (until the next rescheduling occurs) than a plan which does not.
1.2. Notation We use the abbreviations f(t) for the time integral (cumulative function) of any time function f(t) and f(s) or ~ { f ( t ) } for the Laplace trans-
form of f(t), where s is the complex frequency (cf. [4, 5]): o0
f(s)
= ~cF{f(t)} =
fe-~'f(t)dt.
(1.1)
0
The inverse transform is the time function having a given transform f(s) and is written as
f(t)
= ~ - 1 {f(s)}.
(1.2)
For the sake of treating nonnegativities, we use the representation I f ( t ) ] + M a x { f (t), 0} for any time function f(t) If(s)]+ Laplace transform of I f ( t ) ] +, i.e. the transform of a time function which is truncated and for treating probability distributions we use Pr(.) for the probability of its argument occurring and E(.) for the corresponding expectation operator. Further assuming the total number of items in the system to be N, we use: P~(t) production of item i at some time t (as a rate), i = 1,2 . . . . ,N; the Pg(t) for different values of i are collected into the column vector P(t) /5~(t) cumulative production of item i at some time t; the Pi(t) are collected into the column vector P(t) Bi(t) volume of backlogs (cumulative stockouts) of item i at some time t; these are collected into the vector B horizon, i.e. the total length of time periods covered by the plan In the special but normal case of batch production to be treated in the next section, we adopt the notation Qik order quantity of item i, i = 1, 2 . . . . , N, of the kth batch; these are the decision variables to be determined in advance of the process tik time at which the production of batch Qik is decided to take place (is completed) n~ number of batches of item i produced within the horizon
R. ~ Grubbstr6m, A. Molinder/Int. J. Production Economics 46 47 (1996) 297 309
ri
lead time for the production of item i from the time its production is ordered (demand for components are generated) until the time the order is completed Si total inventory of item i at some time Ri available inventory of item i at some time Hi i number of units of item i required for the production of one unit of item j Di external demand rate of item i at some time In the batch production case, the entity P~(t) is interpreted as follows. Since tgk, k = 1,2 . . . . . are the times at which batches Qik of item i are completed, then production P~(t) will be a sequence of Dirac impulses ~kQik~(t -- tik) and/~.(t) will be the time integral of this sequence (a staircase function). The Dirac impulse 6(t - t~k) is a generalised function having an integral of unity if it is integrated over an interval including t~k and an integral of zero otherwise. As abbreviations, we also use Pig = P~(tik), which is the cumulative production of item i at time t = t~k, when the kth batch has been completed. H denotes the input matrix with element H~i in position (i,j), H* = (I - H) -1 being the Leontief inverse of H, and S and R represent total and available inventory as N-dimensional vectors (as functions of time). D~ is the (stochastic) external demand of item i at some time. The Dj are arranged into a column vector D. ri is the constant lead time for the production (acquisition) of item i from the time its production is ordered until the time the order is completed. The lead times r~, r2 . . . . . Lv create internal demands for components these times ahead of the completion of the products they enter and correspond to translations backwards in time. The diagonal lead time matrix ~, having e ~', in its ith diagonal position captures these lags, and n = H'~ is then the generalized input matrix describing component requirements in volumes as well as the times in advance when they are needed.
299
accounts for all items in the system and is initial stock plus cumulative net production less external demand:
S=[~,S(O)+,I-H)P-/5)]+
7! °'+'' •
,,3,
+"
As seen from the point of view of total inventory, the internal demand is instantaneous. At the same time as a component is assembled into a higherlevel product, the stock of this item is reduced by the same amount. The net production as seen from the available inventory point of view is gross production P(s) less external demand and less internal demand at the time the components are reserved, i.e. H~P(s). Available inventory represents the physical existence of items at the time they might be needed physically and is given by 1
/~ = - [R(O) + (I - H~)P -/5]+ S
71
•
+"
,,4,
The physical necessity of available inventory being nonnegative accounts for constraints of inventory on lower levels in production. The first two terms of available inventory, R(O)/s + ( I - H'~)P, must always add up to some nonnegative vector (in the time domain) otherwise the corresponding production plan would be infeasible, i.e. internal demand is always met, If external demand cannot be met, we assume that this demand is backlogged and satisfied at the time available inventory is positive. Backlogs therefore only concern external demand and are given by /~ = _1 [/5 + B(0) - (I - H~)P] + S
1.3. Fundamental equations Due to the lead times, we make a distinction between total inventory S, available inventory R and allocated component stock ~ - R. Total inventory
=
3 + B(0) - (I - H~tP 1 S
+"
(1.sl
Eqs. (1.3)-(1.5) are referred to as the Jundamental
equations•
300
R. ~
Grubbstr6m, A. Molinder/Int. J. Production Economics 46-47 (1996) 297-309
2. One-level model
2.1. Stockouts and inventory We first consider the case of producing only one product in batches at times t~, t2 . . . . , t,. This implies that we do not need the first subscript i (the item index) and the remaining index will refer to the batch number and not the item number according to Xk = X~k for a typical variable. The horizon of the process is finite t = 7~. For convenience, we define t,+ a = 7~. The fundamental equations then contain 1 x 1-matrices and one-dimensional vectors and there is no need to distinguish between S and R since H ? vanishes. We also assume that S(0) = O. Demand is considered to be made up of individual events occurring at independent stochastic intervals. In this section, our developments are general, but in the numerical examples we confine our attention to a Poisson process with an average interval of 1/2 and therefore an average demand rate of 2. Our decision variables are the produced amounts, their timing and the number of batches to be produced within the horizon. The objective function is chosen as traditional average costs (for set-ups, for holding inventory per unit and time unit, and for backlogs per unit and time unit), although the numerical examples to follow should easily be amenable for optimisation adopting the annuity stream principle. With H ? = 0, the fundamental equations for inventory and stockouts become ~(s) = [/~(s) --/~(s)] +,
(2.1)
/3(s) = [/~(s) -/~(s)] +.
(2.2)
Expected inventory and expected stockouts are thus related by
E(~(s)) = P(s) -- E(B(s)) + E(~(s))
(2.3)
(2.1) and (2.2). The major problem is to solve an optimisation problem with (2.1) (or alternatively (2.2)) as a fundamental constraint. If this is done, then E(~(s)) (or alternatively E(~(s)) is easily determined. We choose to use cumulative production rather than batch quantities as decision variables, although they are easily translated into each other. For convenience we set to = 0, Qo = 0 and Po = 0, and the transform of cumulative production and of its time integral will be
=1~
Qke-m--e-~'t k=O ~ Qkl
s2
Qke-st" _ e-S¢ k=O
Qk .
(2.6)
k
From the definition of the Laplace transform (1.1), the value of the time integral of cumulative production at the horizon T is then
='im:Es
e k=l
k=O
(2.7) using, for instance, a series expansion in s. The transform of the expected cumulative demand and its integral is
and the integral of expected inventory and expected stockouts by
E(~(s)) = ~ (1 E(S(s)) = P(s) -- E(D(s)) + E(B(s))
(2.5)
- --s e
,
(2.8)
(2.4)
which is used below for determining the time averages concerned. Eq. (2.3) is a simple consequence of
2
E(B(s)) = U(1 - e - ~ )
2T e_St
sT
(2.9)
301
R. W, Grubbstr6m, A. Molinder / lnt. d. Production Economics 46 ~47 (1996) 297-309
and the value of the time integral at the horizon is E(/5(~))
lim[ 2 -- _2T _ e-sT ] ~ o ~ (1 - e - ~ ) s
•
CumulativeProduction and ExpectedStockouts
)~-p: -
(2.10)
2
: 2lL . . . . .
If cumulative production has a given integer level Pk at some time, the transform of the probability of a certain stockout j/> 0 at that time will be [6, 7]
B(tl
,
{Pr(B(t) = j)}
-_ ~ S I P r ( ~ ( t ) = P~ +j)~,
t~
;~ t~ ;
tk,:
tk.3
T
t
~ S IPr(/)(t) ~< Pk)], Fig. 1. Expected backlog as a function of time (striped level).
= f/'i+Pql-- - - f ) / s ,
j > 0 j = o
t ( i -f)ls,
(PR given),
(2.11)
where f(s) is the transform of the probability density of the interarrival time between two demand events. The transform of the expected stockouts for a given production Pk will then be = .i =
o
(1 -f)i's
jf
Summing over all batches n produced over the production interval T we have
i
s(l - - f ) "
(2.12)
In the case of the Poisson process applied in the next section, we have f ( t ) = 2e - ~ and therefore f = 2/(2 + s). In [6], it is shown that a primitive function for the expected cumulative stockouts in that case can be written as E(B(t)) -
)~t2 2
Pt + P ( P + 1) 22
P~2 (~/)i+ 1 -
2P
j=O
~-v~-
e-X'[ 22
1)i~('(2t)Jj! P(P +
.=)
P~3 (,~t)j+ 21 +
j=O
J!
,
(2.13)
where P = Pk is the given production level. In our transform representation the expected cumulative stockouts for a general independent interarrival time distribution can be obtained integrating (2.12), " 1 /Pk + 1 E(B) - s2 (1 -- f ) '
and the contribution from each production interval k to the integral as
(2.14)
'
which may be evaluated using (2.13) in the Poisson case. The expected stockouts during the production intervals are depicted in Fig. 1 as the striped levels above cumulative production. We may note that stockouts increase progressively in the beginning of each production interval k and they will approach E(/)) - P for larger values of t. An analysis of the asymptotic behaviour of stockouts for a planned production process is given in [8], where it is shown that this behaviour for a large class of probability distributions follows essentially the square root of time.
2.2. Optimisation
The classical inventory problem attempts to find a balance between the economic consequences of holding inventories, of production and of stockouts. The main decision variable is the batch
302
R.W. Grubbstr6m, A. Molinder/Int. J. Production Economics 46-47 (1996) 297-309
quantity Q, but an additional production decision can be the amount of a safety stock. We are now in a position to define our objective function in terms of the integrals obtained above. Let h denote the inventory holding cost per unit and time unit, b the backlog cost per unit and time unit and K the cost for each set-up. O u r objective function to minimise then becomes Kn
h
C = --:- + - : [ 5 ¢ - 1{E T T
(~)}] ,=~-
b
+ ~ [L/~-' {E (/~)}], = f,
(2.17)
where E(S(T)) = f i ( T ) - E(5(IB)) + E(/~(]B)) ~. k=,
=
Pk(tk+,
- -
2~ 2 tk) - - ~ + E(B(T)).
(2.18) F o r a given value of n, the decision variables may be interpreted as the tk and /~k, k = 1, 2 . . . . . n. The first-order optimisation conditions are the following: OC
h
h
77=77
ji,:,
= o,
(2.19) OC
h
--=--~(tk+i--tu)--
h ; b ((~_ i ,~i'k +1~~
. - - - Sr - - .l".~' : ,x.I (<.~-1 C JJ
[
~ O,
s2
J/t=,. (2.20)
where the second set of conditions is to be interpreted as a set of difference (rather than differential) conditions, since the Pk are only allowed to be varied in integral steps using the interpretation
=fP*(s) ( f ( s ) - 1).
(2.21)
As seen from these conditions, the optimum solution, given n, must depend on the ratio (h + b)/h = 1 + h/b only. Furthermore, we must require that tk + ! > t k.
2.3. N u m e r i c a l solution p r o c e d u r e and results
Since the optimisation problem cannot be solved analytically because of the complexity of the expression for the expected stockouts in (2.15), we have solved the minimisation problem with a numerical iteration procedure according to the steps 1. The finite fixed horizon 7~ is first divided into (n + 1) equal time intervals as a trial solution. 2. For a given sequence tl, t2 . . . . , t , , optimal integer values of PR, k = 1, 2 . . . . . n, are calculated each in turn using an interval-halving procedure until Eq. (2.20) is satisfied as close to zero as possible. 3. F o r a given sequence, Pk, k = 1, 2 . . . . , n, the corresponding optimal time intervals tk are calculated using an interval-halving procedure until (2.19) is satisfied. 4. The iterations in each of these two steps are stopped when an indicator for each equation reaches a small value. Steps 2 and 3 are then repeated in sequence until there is no change in any of the two solution sequences. For each n = 1, 2 . . . . , steps 1 4 are carried out and average costs also including set-up costs are computed until the minimum sum of these are found, constituting the overall optimum of n, and the Pk and tk, k -- 1, 2, ... ,n. We have not investigated conditions for the convergence of this suggested procedure, but all numerical results indicate that no problems are encountered as long as reasonable values of the parameters are chosen. The problem was solved numerically for six different parameter combinations regarding set-up costs, inventory holding costs and backorder costs. The parameter values as well as the numerical optimisation results are shown in Table 1. The expected average demand rate was chosen to unity at the same time as the ratios between the backorder cost and the inventory holding cost were varied.
R.W. GrubbstriCm, A. Molinder/Inr J. Production Economics 4t~ 47 (1996) 297 309
303
Table 1 Results of the safety production plan calculations in Section 2.3 using six different parameter combinations, ,~ = 1, n* = optimal n u m b e r of batches (*)
h=
k
1
h=50
K = 100
n* = 6
Pk
tk
Qk
SSk
26 47 65 83 99 115
0.5 21.3 39.2 55.1 70.9 85.4
26 21 18 18 16 16
0
1 2 3 4 5 6
1,*)
h=
k
1
h=
k
- 0.5 4.7 7.8 9.9 12.1 13.6 15.0
P~
~
Q~
ss~ - 9.5
0
19 40 62 87
9.5 29.5 51.1 74.6
19
- 10.5
21 22 25
- 11.1 - 12.6 - 13.0
t7=50
b= 1
K = 100
n* = 5
k
Pk
t~
Qk
SS~
0 1 2 3 4 5
12 26 40 53 69
16.4 32.9 49.2 64.2 81.4
12 14 14 13 16
---
16.4 20.9 23.2 24.2 28.4 -- 31.0
In Fig. 2, a case with a high given backlog cost is shown (Table 1, Case (*)). We note that the distance between the cumulative production level and the expected cumulative demand increases with time. Hence, the distance at the end of each step represents the safety stock (SSk in the figure). As the uncertainty in demand increases over time a compensation takes place in the optimal safety stock. During the iterations, it was notable that the convergence adjustments towards optimum during the
n* = 9
P~
tk
O~
SS~
17 33 47 60 72 83 94 106 117
0.3 13.2 26.4 38.3 49.5 59.8 69.7 79.8 90.4
17 16 14 13 12
~ 3 4 5 6 7 8 9
n*=4
2 3 4
K =50
l
K = 100
1
h=50
0
b=l
0
1
h=
k
I
b=
1
O.3 3.8 1~.6 8.7 1(/.5 12.1 13.3 14.2 15.6 17.0
11 11 12 II K =50
n* = 5
s%
&
t~
Q~
1
17
2 3 4 5
35 53 72 90
8.5 26.1 44.1 62.6 81.1
17 18 18 19 18
h=50
b= 1
K =50
n*--:6
k
Pk
tk
Qk
SSk
0 1 2 3 4 5 6
8 22 33 46 58 72
14.3 28.6 41.8 56.3 70.2 85.3
10 12 II 13 12 14
- ~.5 9.1 9.6 9.1 .- 9.1 10.0
- 14.3 18.6 19.8 - 23.3 - 24.2 27.3 - 28.0
iteration were particularly intensive in the first few intervals compared to the initial trial solution. In Fig. 3, a case with an equal ratio between backorder cost and inventory holding cost is shown (Table 1, Case (**)). We note that the optimal solution shows a symmetric alternating backorder and inventory situation and that the plan both begins and ends with a backorder situation. In this case we found that the convergence adjustment towards optimum during the iterations was
304
R.W. Grubbstrgm, A. Molinder/Int. J. Production Economics 46-47 (1996) 297-309
Cumulative production/demand 2=1 K=I00 b=50
,, ,
~ " "
H * = 6
•
Slope=2
/ / / / / -
-
/ / / / /
/
I fi
-%----+-
I
I
t2
h
t~
h
t
T
It is notable for all six cases that the time intervals between the set-ups and their mean value differ at most in the beginning and at the end of the production horizon• When the backlog cost parameter is high, cumulative production increases faster than cumulative demand, resulting in an increase in safety stocks over time. The correlation between the optimal time intervals and the corresponding batch quantities is very high in such cases, indicating that the time intervals essentially increase in a proportional way. In the converse case with a high inventory holding cost parameter, the safety stock becomes increasingly negative, as expected. However, no obvious pattern in the relationship between batch quantities and intervals has been found in such cases and the correlation coefficients computed are low.
Fig. 2. Safety production with high backlog costs•
3. Two-level, serial model
Cumulative2:lproduction/demand
K=100 b=l h=l n*=4
3.1. Introduction
r _ ~ i ~
~
We now turn our attention to the two-level serial assembly system. Backlogging is not permitted on the second level and we assume no external demand on this level. Notations will now have two indices, the first being the item number (in this case coinciding with the level number) and the second the batch index. In terms of our previous developed model the input matrix is
..'" Slope=2
r--
i
/'"
.=i0001 A
t2
ts
T
Fig. 3. Safety production with equal backlog and storage costs.
Production on the top and lower levels take place in batches of sizes Qlt, Q2t at times tl~, l = 1,2 . . . . , n~, and t2z, / = 1, 2 . . . . , he. Cumulative production on the lower level will be /52(s)=![f_~ a Q2te-S'~' -
particularly intensive halfway through the production horizon. Apparently, the safety stock then was negative throughout. The possibility of negative safety stocks may be questioned by definition. However, we feel no reason for confusion by defining safety stocks to be positive as well as negative.
e-sT"l~= Q2t1.
(3.2)
The intermediate stock on the lower level (inventory of item 2) is g2(s) =-sl / Qzle-St~"~ ~1 - k~Q~ke-S"kl= 1 '
(3.3)
R.W. Grubbstr6m, A. Molinder/lnt. J. Production Economics 46 47 (1996) 297 309
where it is assumed that initial stock is zero. The available inventory on this level will be
assuming initial available inventory also to be zero and that cumulative inventory on the lower level ends at the same value as that of the upper level. Both S2(s) and 1~2(S ) are differences between staircase functions and it will always pay to keep the area between these two functions as small as possible for given values of nl and n2. We can therefore make the following remark. Whatever staircase the upper-level cumulative production follows, at the optimum there can never be a gap between the lower-level cumulative function and the highest of the steps of the upper function within a particular lower-level production interval. If there were such a gap, the lower-level production level could be lowered reducing inventory holding costs with no other effects. Also, whatever staircase the upper-level cumulative production follows, at the optimum there can never be a gap in time between the next replenishment of the lower level and the next replenishment of the upper level, because if this were the case, then the lower-level replenishment could be postponed reducing holding costs with no other effects. Therefore in the two-level serial case, we note:
In the optimal solution, the lower-level replenishments ahvays must take place at the same time as some upper-level replenishment and at that time (immediately prior to the replenishment in question) both cumulative levels coincide. Geometrically, this means that the two staircases fit into each other at the "inner" corners when lower-level replenishments take place (indicated with circles in Fig. 4). This also implies that the optimal number of batches on the lower level cannot be higher than on the upper level, as illustrated in Fig. 4. The deterministic lead time r~ is of no consequence in this respect, since it only translates the staircase uniformly backwards in time. Hence, the set of times (t~k--Z~), on the one hand, and the set of cumulative levels P~k, on the other, in the optimum will contain the set of times
305
..... lower level upper level
t. I -
t p'-
Fig. 4. A two-level serial system with the left staircase showing P2~ and the right showing Plk. "Inner corners" are marked with circles.
t2t and the set of levels/521 as subsets, respectively. For handling the different combination opportunities, we introduce the two index functions k(l) and l(k), where k(l) is the batch index of the upper-level cumulative production determining the lower level of interval 1, i.e. P21 = Plk(l), and l(k) the batch index of the lower level as a function of the index of each upper-level interval (the kth interval). Therefore, we always also have P21(k)=/5).k(l(k)) and t z l t + l ) ÷ z"1 ~ t l ( g ( t ) + l ) . The function I(k), in general, has no unique inverse, whereas k(l) does. The functions k(I) and l(k) have the properties 1 <~k(l) < k(I + 1),
k(n2) = n , .
l(k+l)-l<~l(k)<~l(k+l)<~k+l,
l(n~)--n2. (3.5)
The time average of inventory on the lower level (which is deterministic) may now be written as
S2(T) - lim (/~(s) -/~(s))
= ~
--
1 T
P21(t2(l+ 1) -- 12t) +
k-1 ~" " l k ( t l
/31.~Zl
(k + l ) -- t l k ) ]
(P2uk) -- Plk)(tl (k~-1~-- tlk) + /51.,z'l k
306
R.W. Grubbstr6m, A. Molinder/Int. .Z Production Economics 46-47 (1996) 297-309
=~ T
k 1
('Olk~l~k))-- Plk)(tl~k+ 1) -- tlk)
+ h2
m
-~ JillntZ1] .
(3.6)
If k(1) is given as a function, meaning that the relation between the two staircases Plk and P2t is fixed, then attaching a unit holding cost h 2 to S2(T)/T, we can increment the objective function C with this quadratic expression of the same decision variables as in the one-level case of (2.17). To ensure the plan to be feasible, we need t21 >~ 0, i.e. h 1 - zl >I 0, which is captured by including a Lagrangian term q(rl - t11), where q is a nonnegative multiplier.
C=-~I [Klnl + K2n2 -- hi ~-~2 + (hi + b l ) E ( B ( 7 ~ ) ) ] + ~(T1
--
tll)
+ 7~ Lk=l X (/l(k+
1)
~
k(l(m))=k
tik) + h:Pi,,zi ].
(3.7)
The derivatives ~C/OPlk and ~C/~tlk now contain further terms depending on h2 and on the structure l(k), but their essential form remains the samel
0C I 0txk - ~ [(hl - hz)(Pltk-1) -- Pak)
Jr h2(Plk(l(k-1))
--
ff lk(l(k)))]
(s
_{qO
(if k = l ) = (if k > l)
0,
-
-1{ --7-1)
, : ,,, )
(1 --f)
J),-,,. (3.8)
+
(tl(m+ 1 ) - tlm) I J
0
(if k < nl)
h2z--A
(if k = hi) ~< 0
and we now also have the complementary condition: r/(rl - tll) = 0.
(3.10)
We may note that the one and only term depending on the lead time which occurs in the derivative with respect to the final cumulative production level Pl,,. This is the effect from the influence of external demand when hedging against backlog effects. Had no final backlogs been allowed, this term would have been lacking, since the final cumulative production would then have been forced to equal total cumulative demand. Let us take a look at the optimal function k(1) which prescribes the geometry of the lower-level staircase, once the upper-level staircase is given. The top level has steps at tit, t12 . . . . ,tt,,. The lower level can only have steps at some or all of these points. If the lower level has a step then it must depart from the same value as that of the top level. Hence, whatever P 2 ( / + l ) , /~l can either remain or increase to P2t~+ i). Hence, for each interval there are exactly two possibilities, see Fig. 5. Furthermore, tlx = t21 + T1 and at t = t,, the two levels Pl,, and P2,2 must coincide. Hence the number of possible lower-level functions are 2 ~"'-1~ This is illustrated in Fig. 5 for n~ = 4 together with the corresponding functions l(k) and k(l(k)). One might also view this set of combinations as whether an "inner" top staircase corner is included in the lower staircase or not, the leftmost always being included. As regards the optimal number of batches on the lower level, if given the top-level structure, a lowerlevel increase with one batch is only justified if the inventory holding cost increase is less than the additional set-up cost K2: h2(P2(/+ 1) -- P2t)(t2(l + l) -- tel) <
+ T [(hi--h2)(tltk+i)--t,k)
(3.9)
K 2.
(3.11)
However, the optimal top structure certainly depends on its lower-level counterpart.
307
R.W. Grubbstr6m, A. Molinder/lnt. J. Production Economics 46-47 (1996) 297 J09
l
F
following reasonable set of values:
T k(ICk))
'i .
°
°
•
k
•
k
j.-
F f
~ l(k) ° .
FFU 4
k
F
T l(k)., k f
T t(k)., k F
t
>
FF
h2 : 0.5, b 1 = 50, K1 =
100,
K2 = 300, rl = 20, T = 100.
~ l(k). . . k f
2 = 1, hi = 1,
p-
T t(k).
For each value of nl, the 2 "1 l) different structures have been examined and evaluated. The values of different partial terms of the objective function in the suboptimum, given each value of nl, are shown in Table 2. The overall optimum is obtained for n l = 4. In each solution apart from the first the optimal number of set-ups on the lower level is shown to be two (Column 2 in the table). For nl = 4 we demonstrate the effects of the different lower-level structures (eight options) in Table 3, adopting the same enumeration as in Fig. 5. In Table 4 we have included the planning data for the solution of the optimal structure (No. 2) in Table 3). For this structure l(k)= 1, 1, 2, 2. The safety stock on level 2 SS2,k~ is defined as cumulative production less expected cumulative internal demand at the end of the interval. We may note that the optimal safety stock on the upper level SSlk behaves irregularly, in particular when it falls back at the end of the final interval.
k 4. Conclusions jr-
Vj5 V4 f
l l(k)°
k
lk(t(k)), k
j.-
lowerlevel......... upperlevel-Fig. 5. Possiblecombinations of staircase functions in a twolevel systemwith n~ = 4. The eightcasesare found by choosing among all possiblenondecreasingfunctionsl(k).
3.2. A two-level numerical example For purposes of illustrating our foregoing discussion we include the following example. The solution procedure is a slight modification of the one described in Section 2. Parameters are given the
The objective of this paper has been to apply the previously developed extensions of the fundamental equations in the most simplest cases with the aim of computing optimal safety stock values. Consequences of stockouts have been investigated for a general class of probability distributions for demand together with conditions for optimality. Optimal solutions for safety production plans have been determined numerically using the assumption of Poisson-distributed demand in a onelevel case and the corresponding analysis has also been sketched in the most simple two-level case. We note that for the case with a high cost ratio between backorder cost and inventory holding cost, the distance between the cumulative production level and the expected cumulative demand increases with time. As the uncertainty in demand increases over time, our interpretation is that a compensation takes place in the form of an increasing safety
R. ~14 GrubbstrOm, A. Molinder/Int. J, Production Economics 46-47 (1996) 297-309
308
Table 2 Values of the objective function for different values of nl
nl
n2
bIE(B(T))
hiE(S1 (T))
h2E(Se(T))
Klnl
K2n2
CT
1 2 3 4 5 6 7
1 2 2 2 2 2 2
10 603.57 10417.90 10334.45 10 337.59 10 340.36 10 339.92 10 340.95
8320.00 7001.06 6588.77 6426.54 6303.80 6238.54 6192.33
1040.00 1080.00 1346.52 1438.02 1498.28 1524.75 1543.05
100 200 300 400 500 600 700
300 600 600 600 600 600 600
20363.57 19298.97 19237.71 19 226.18 19284.24 19354.21 19436.95
Table 3 Values of the objective function for different structures on the lower level for the optimal value of n~ No,
n2
blE(B(7~))
h,E(S1 ( 7 ~ ) )
h2E(S2(T))
K,n~
K2nz
CT
0 1 2 3 4 5 6 7
l 2 2 3 2 3 3 4
10 358.51 10 336.23 10337.59 I0 360.81 10 336.34 10360.71 10 358.01 10 306.31
6487.26 6573.40 6450.57 6454.80 6503.16 6435.54 6414.16 6426.54
2156.37 1387.29 1438.02 1179.13 1442.08 1194.79 1215.83 1120.00
400 400 400 400 400 400 400 400
300 600 600 900 600 900 900 1200
19 702.14 19 296.91 19 226.18 19 294.74 19 281.58 19 291.04 19288.01 19452.85
Table 4 Optimal production decisions in numerical example
k
ilk
Plk
Qlk
fllk+ 1~ -- tlk
0 1 2 3 4
20.0 39.3 60.8 76.9
49 69 92 110
49 20 23 18
20 19.3 21.5 16.1 23.1
stock. In a converse case with a high inventory holding cost parameter, the safety stock becomes more and more negative over time in the example. Furthermore, we note for the optimal solution in the two-level production case that the lower-level replenishments always must take place at the same time as some upper-level replenishment and at that time both cumulative levels must coincide. A number of issues remain to be investigated. In future research, the models will be extended to more general cases including multi-level models
SSlk - 20.0 9.7 8.2 15.1 10.0
P2ttk~
Q2tlk~
SS21{k) - 20.0
69
69
110
41
8.2 10.0
with stochastic external demand also on lower levels. More general interarrival distributions and more general product structures are but two trails for further progress, Also, it would be of interest to model practically oriented lot-sizing rules used in real-life MRP-systems and investigate consequences of augmenting these with heuristic safety stock procedures derived from considerations of the kind presented above. Simulation studies based on the theoretical principles underlying these models will also be undertaken in order to examine more general stochastic settings. A number of
R. H,i Grubbstr6m, A. Molinder/Int. J. Production Economics 46 47 (1996) 2 9 7 309
i n t e r e s t i n g e x t e n s i o n s h a v e a l r e a d y b e e n m a d e by B o g a t a j a n d H o r v a t [13, 14-1 o p e n i n g u p several n e w lines of i n v e s t i g a t i o n ,
Acknowledgements T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e the m a n y v a l u a b l e s u g g e s t i o n s m a d e by Dr. L a r s - E r i k A n dersson, Department of Mathematics, Link6ping I n s t i t u t e of T e c h n o l o g y , w h i l e d e v e l o p i n g t h e b a c k g r o u n d t h e o r y of this p a p e r .
References [I] Vollmann, T.E., Berry, W.L. and Whybark, D.C., 1988. Manufacturing Planning and Control Systems. Irwin, Homewood, IL. [2] Grubbstr6m, R.W. and Ovrin, P., 1992. Intertemporal generalization of the relationship between material requirements planning and input output analysis. Int. J. Prod. Econom., 26:311 318. [3] Grubbstr6m, R.W. and Molinder, A., 1994. Further theoretical considerations on the relationship between MRP. input-output analysis and multi-echelon inventory systems. Int. J. Prod. Econom., 35:299 311. [4] Churchill, R.V., 1958. Operational Mathematics, 2nd ed. McGraw-Hill, New York.
309
[5] Aseltine, J.A., 1958. Transform Method in Linear System Analysis. McGraw-Hill, New York. [6] Grubbstr6m, R.W., 1994a. Stochastic relationships of a multi-period inventory process with planned production using transform methodology. Research Report RR-126, Departmem of Production Economics, gink6ping Institute of Technology. [7] Grubbstr6m, R.W., 1994b. Stochastic properties of a production-inventory process with planned production using transform methodology. Int. J. Prod. Econom., forthcoming issue. [8] Andersson. L.E. and Grubbstr6m, R.W., 1994. Asymptotic behaviour of a stochastic multi-period inventory process with planned production. Working Paper WP-210. Department of Production Economics, Link/sping Institute of Technology. [9] Hax. A.C. and Candea, D., 1984. Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ. [10] Molinder, A., 1995. Material requirements planning employing input output analysis and Laplace transforms. PROFIL 14, Production Economic Research in kink6ping, kmk6ping. [11] Ross, S.M.. 1989. Introduction to Probability Models. 4th ed. Academic Press, San Diego. [12] Whybark, D.C. and Williams, J.G., 1976. Material requirements under uncertainty. Dec. Sci., 7:595 6{)6. [13] Bogataj, L. and Horvat, L., 1996. Stochastic considerations oi" GrubbstrSm Molinder model of MRP, input output and multi-echelon inventory systems. Int. J. Prod. Econom., 45:329 336. [14] Horvat, [_. and Bogataj, L., 1996, MRP, Input-Output Analysis and Multi-Echelon Inventory Systems with Exponentially Distributed External Demand. Proceedings of GLOCOSM Conference, Bangalore, India.