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Protective inventories and bottlenecks in production systems
European Journal of Operational Research 22 (1985) 319-328 North-Holland
319
Protective inventories and bottlenecks in production systems M . R . L A M B R E C H T , R. L U Y T E N a n d J. V A N D E R E E C K E N
Operations Management Group, Department of Applied Economics, K.U. L uven, Belgium Abstract: In this paper we study the allocation of safety s t o c ~ and safety t systems. We concentrate on the impact of demand uncertaivty on the presence of capacity constraints. We describe how this problem can be Process. The form of the optimal policies is discussed and illus'~rated witl
rues in multi-echelon production amount of safety stocks in the iewed as a Markovian Decision examples.
Keywords: Inventory, manufacturing industries, Markov processes, stocha tic processes
1. Introduction The purpose of this paper is to study the allocation of safety stocks and safety time in multiechelon production systems with limited capacity. Uncertainty does exist in several forms~variability of demand from the forecast, uncertainty in the supply from stage to stage due to the variability in the yields from each production batch and uncertainty in the lead times for both production and procurement. Protectic.n against the~e disruptions, is required and this results in buffer stocks a n d / o r safety times. Research on protective stocks in multi-stage systems has primarily been experimental, We refer the reader to the work of Banerjee [2], Whybark and Williams [20], Buechel [4], Carlson and Yano [5,6], Collier [91, De Bodt et al. [11], Meal [17], New [191 and Joensson, LundeU and Thorstenson [14]. The authors of this paper have shown iv, a recent paper [15] that the multi-echelon stochastic problem can be viewed as a Markovian Decision Problem. Because of the limited practical value of this approach heuristic solutions methods have been suggested based on the well-known results of Clark and Scarf [7,8]. The authors also have demonstrated that generalized (s, S) policies appear to
ReceivedApril t984; revisedOctober 1984
be optimal. The -esulting (s, S) policies are next simulated in ord:zr to determine the amount of safety ~tock and safety time. The simulation is necessa~ becaus~ it is impossible to determine the exact amount ot safety stock or time by only knowing the valu.~s of the policy variables. Safety stock i~ delermin,:d as the expected arr, ount of on hand inventory at the time an order or a lot arrives Safety tinge is defined as the expected time between the completion of a lot at a preceeding stage until any milts frc,m that lot are :ased in the production of a ot at the succeeding ,,rage. The computar,ional r~sults indicate that both safety time and safety stock can eaist at intermediate stages. The above observations will be illustrated by means of an exaraple in Section 2. The purpose of this paper is to extend the" previous anaIysis [15] for situations where capacity constraints are limiting the production output at certain facilities, In Section 2 the multi-echelon system is ir~troduced together with the underlying assumptions and a description is given of how this problem can be viewed as a Markovian Decision Process. In Section 3 we discuss the form of the optima! operating policies for the case in which there are capacity constraints at the end product level, tn particular we show that generalized (s~ S) polJcie~ appear to be optimal. The discussion is illustrated with examples.
0377-2217/85/$3.30 'co ~985, ElsevierSciencePub}isher~B.V.(North-Holland)
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M I ~ i.ambrecht et al. / Protective stocks and bottlenecks in production systems
1. M o d d fommb~tioa
A. Nomenclature In order to facilitate the understanding of the model and its resuits, we recall in this section some generally accepted detrmitions of inventory con,zepts. The problem considered is a single-item multilevel production/inventory system of the serial -ype. Multi-level, or multi-echelon, production/inventory systems have traditionally been subdivided in two types, assembly and arborescence sy~_,ems. An ~sembly system is a system in which each facility, or production stage, requires inputs from a number of immediate predecessor stages while supplying, m turn, at most one immediate successor stage. An arborescence structure, on the other hand, is a system in which each facility supplies a rmmber of immediate successor stages while it gets input from at most one predecessor stage. A serial system is a special c a ~ of both an assembly system and an arbore.w.ence system; every facility has at most one preceeding facility and one succeeding facility. 'Ive installation inventory at facility j, also called the physical inventory, is the number of units pi~ysically on-hand at facility j. The installation invent~,ry position at facility j, also called the economic inventory, is the ;nstallation invento,"y at facility j augmented with5 the quantity on order and decreased with the backorders. The quantity on order reflects replenishment orders which have been relezsed but not yet received whereas backorders reflect orders for the product at facility j which have been received but not yet delivered because of a stockout. The echelon inventory definitions are taken from Tinmmr, Monhemius and Bertrand [20]. They define the echelon inventory at facility j as the number of units in the system which have gone through the operations of facility j but have as yet not been sold. The echelon inventory position at facility j, also called the available echelon inventor3, is defined as the echelon inventory, at facility j plus the quantity on order at facility j minus the backorders for the finished items which contain the item under consideration. Another look at the echelon inventor, position is that it is the installation inventory at facility j plus the installation inventory position at all succeeding facilities.
The symbol liP) will be used to represen: the installation inventory position and the symbol E1P/ will be used to represent the echelon inventory position at facility j.
B. Methodology In this section, we introduce the multi-echelon production/inventory system and its underlying assumptions. We also describe how this system can be modelled as a Markovian Decision Process. Attention is restricted to the situation where the demand process is stationary and the planning horizon is infinite. The product structure we investigate, is depicted in Figure 1. It is a two echelon serial structure, echelon 2 is the component level and echelon 1 is the final assembly level. The model we present is based on the following assumptions: 1. A periodic review policy is followed; 2. The production and procurement lead times, which are measured in periods, are known constants and are independent of the batch size; 3. The demand distributions are known; demand is independent from period to period for each final product; 4. All excess demand is backordered; 5. Prcrduction capacity is limited at the final assembly level and the production of one batch is not allowed to last for more than one period. We also assume that the relevant costs consist of (1) production costs, at each stage, which are composed of a fixed setup cost per production run and a constant cost per unit produced; (2) holding costs which are charged proportional to the installation inventory at the end of the period; (3) linear shortage costs, which are charged proportional to the number of final products backordered at the end of each period.
Eche!on 2
(~) .f
Echelon I
O
.f Final demand Figure 1. A two echelon serial structure
~f.P,. Lambrecht et a L / Protective stocks and bottlenecks ir productto ~ systems
Note that the constant co~t per unit produced can be ignored in our analysis ~.s we assume that costs remain stationary over time and we have an infinite planning horizon. The multi-echelon stochastic production/inventory problem introduced above can be modelled as a Markov Decision Process. It is assumed that the system is observed at time t = 0, 1. . . . . T and classifted into one of a finite number of states i labeled 1, 2 . . . . . N. For every state i, a number of decision alternatives, k, are possible, labeled I, 2 , . . . , K ( i ) . (The number of possible policies might differ from state to state). Whenever the system is in state i and alternative k is chosen the system moves to state j with known ~ransition probability p k/ for all i, j = 1, 2 . . . . . N and k = 1, 2 . . . . . K(i). The cost structure is introdueexi through q~, the expected one-period cost incurred when the system is in state i and decision alternative k is chosen. In terms of the multi-echelon stochastic production planning problem the states i, i = I, 2. . . . . N, of the Markov Decision Process correspond to the echelon inventory position at each facility. Every state thus corresponds to specific inventory positions at each facility; these echelon inventory positions themselves can be positive, negative or zero. The total number of states equals the product of all possible echelon inventory positions at all facilities. The decision alternatives k, k -~ 1. . . . . Kfi), r e p r e ~ n t the production decisions at each facility. In case there are capacity constraints, some of these production decisions are impossible which implies that the number of decision alternatives for state i in the capaeitated case is lower than (or equal to) the number of decision alternatives in the uneapacitated case. The transition probabilities p~, depend on the current state, i, the decision made, k, and the demand distribution, which together with i and k, determines j. The proposed solution mmhod is based on the value iteration or successive approximation method developed by Howard [13]. The recursive relation is given by
I
E(T) = m n.
E j=l
p*,,V/T-1)
}
where V,(0) = 0, Yi. ~ ( T ) stands for the expected total (discounted) cost of a system starting in state i and evolving for T periods when an optimal
321
policy is followed, a denotes the discount factor and q~ and p~) are elements of the cost matrix Q and the transition matrix pk. For large values of T, a stationa:y cost structure and stationary transition probabilities, the system will reach its steady state. This means that the decision to be taken in a certain state i is unaffected by the time period in which the de,'ision has to be taken. After stabilisation, only a limited number of states will be possible; these are the recurrent states. C. A numerical example An example will illustrate the methodology mtroduced in Section 2.B. Consider a two-echelon serial production system, shown in Table 1. Demand is assumed to be independent from period-to-period, all excess demand is backordered. We further assume that the demar:d process is stationary, the costs are time independent and the planning horizon is infinite. In a first step, we restrict the number of states and decision alternatives considered. We therefore determine minimum and maximum levels for the echelon inventory position of each item and a maximum production quantity for each item. These figures are chosen such that no rea:,onabte de:ision ahernatives or states are excluded. For the above mentioned problem the following limits we~c imposed: gable I Two-ezhelon serial production aystem Setupcost
a ~
Echelon holding Lead t;rne coat (perio:l_
Facility 2 30
5
]
Facility 1
9
1
9
t v
Demard Shortage ¢o~t per unit per perie_,d = 150 Demand distribution Demand per period 0 i 2 3 4 MeRR = 2 Variance = 3
Probability 0.4 0 OA rj.2 0.3
322
~tt.R~ ~ t ' ~
el a£ / P r a ~ l i ~ JtockJ' and botdentd~ in production rj~tems
- M a x i m u m production at echelon I, d e n o t e d b y M 1 ,,- 10.
"l'~bl¢ 2 S¢:~p e o ~ at f~,ility I Holding ¢c~t at facility e0A × 3 - 0.1 × 1) 0 4 ) Shortage o ~ t (0.3 × 1) t lSO) Holding c,o~ls at facility 2 ~8-5) (5] Work in process holding cosls (5) (5)
9.00 18.20 45.00 15.00 25~0
To~.al
112.20
- Minimum eche:on inventor3" position at echelon 2=-2.
- Maximum echelon inventory position at echelon 2 = 20.
- Maximum production at 2, denoted b y M 2 = 15. - Minimum echelon inventory position at echelon 1=-6.
- Maximum echelon inventory position at echelon 1 =15,
The decision alternatives, k, are formed by e combination of the production at 1, Z1 {0, 1. . . . . M1}, and the production at 2, Z 2 ~ {0, 1 . . . . . M2}. The cost dements, q~, represent the set-up cost and invt~atory rehted costs. Suppose that the echelon inventory position at facility 2 is 11 units and at facility i we have an echelon inventory position of 3 and that there are no outstanding orders nor backorders (the installation inventory position at facility 2 thus is 8 units). Furthermore, assume that we decide to produce 5 units at facility 1 and nothing at facility 2. We then incur a setup cost at facility 1 and no setup cost at facility 2. We have holding costs at facility 1 whenever demand is less than three and incur shortage costs whenever demand is greater than three m:it_, Each unit at facility 1 leads to an inventory l*olding cost of 14, i.e. the summation of the echelc.w~ holding cost at facility 2 and the
Table 3 The recurrent states Slate t IEIP.~
EIP 1
Steady state
Decisiot k
probabilily
Z2
Z1
E~pected cost~ q~
6 7 7 8 8 8 9 9 10 10 10 10 10 11 11 11 11 12 12 12 12 14 14 14
4 4 5 4 5 6 5 6 2 3 4 6 8 3 4 5 8 4 5 6 8 6 7 8
0.7808 E - 0l 0.4932 E - Ol 0.5205 E--O1 0.2466 E - O I 0.3288 E - O I 0.2603 E - Ol 0.1644 E - O 1 0.1644 E - 0 1 0.2342 E - 02 0.3041E-01 0.9411 E - Ol 0.8219 E - 0 1 0.1041E+00 C.1562 E - 0 1 6.2027 E - 0 1 0.6274 E - O f 0.6575 E - 0 1 0.7808 E - 0 2 ,~.1014 E - 0 1 ¢~.3137 E - 0 1 ('.3288 E - 0 1 C.3123 E - O 1 ~.4055 E - 0 1 01255 E - O I
8 7 7 6 6 6 5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 3 2 4 3 2 3 2 6 5 4 2 0 5 4 3 0 4 3 2 0 2 1 o0
77 82 9t 87 96 105 101 110 180.2 1072 67 85 94 112.2 72 81 99 77 86 95 104 105 I14 I14
(1)
(2)
(3)
~4)
(5)
(6)
Gain = q5.449493, service = 0,974479 fdl rate = 0.983726.
Echelon ~to~k p ~ i t i o n after dexi.~i~m ;z! 2
at 1
14 14 14 14 14 14 14 14 10 10 10 10 10 11 il 11 11 12 12 12 12 I4 14 t4
6 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
(7)
(8)
M.R. Lambrecht et al. / P,otective stocks and bottlenecks in production systems
echelon holding cost at facility I. The installation inventor)' at 2 is 3 units, which is the difference between the beginning installation inventory at facility 2 and the production at facility 1 (there are no open orders at facility 2). The production decision at facility 1 leads to a work in process inventory of 5 units and each unit is charged at five monetary units. Adding all these figures we obtain an expected one period cost of 112.20 as is summarized in Table 2. The solution for the stated problem is given in Table 3. From Table 3 we learn that the production level is zero at the second facility when zhe echelon inventory position at that stage is 10 or more. The production quantity is positive if the echelon inventory position is less than 10. A closer look at the table shows that we always order up to a level of 14. Consequently, at facility 2 we have a pure (s, S) or m i n - m a x policy with s = 10 and S = 14. At stage 1 we also have an (s, S) policy with s = 8 and S = 8. The first three states in Table 3 seem to be in conflict with this policy. This, however, is not true, the (s, S) policy for the first state e.g. suggests we should produce 4 units but we only have 2 units available (the installation inventory at stage 2 is 2 units), so production is 2 units. Sometimes it is impossible to reach th.: order-up-to level because of the unavailability ~,f components at the preceeding facility. The optimal average cost per period is 95.45. The service level defined in term~ of percentage ~Jf periods without stockouts is 97.4~. "Ihe fill rate equals 98.3%. It is interesting to note that all policies are based on the echelon inventory position concept. As was already explained in Section 2.A, the echelon inventory position at facility i can be seen as the installation inventory position at that facility plus the installation inventory position at all succeeding facilities. As a consequence, the echelon holding costs reflect the value added cost s!ructure. The following statistics can be obtained from a simulation experiment:
323
Average order quantity at stage 1 : 2.97, Average order quantity at stage 2: 6.79, Fill rate : 98.3%, Safety stock at stage 1 : 2.62, Safety stock at stage 2 : 0.099, Safety time at stage 2 : 0.24 periods. We can now test, e.g., the impact of the cost structure on the amount of safety stock. In the present example the installation holding cost for facility 1 is 14 and for facility 2, the installation holding cost is 5. The value added ratio is 5/14 × 100 = 35.71%. The optimal policy was computed for tile following value added ratios presented in T~ble 4. The higher the ratio the more expensive the component. The results are given in Figure 2 The impact of the length of the lead time on safety stocks can also be examined. It is a wellknown result that the amount of safety st3ck is much higher when the lead times are longer (see
[15]). We now extend the anaysis to the case in which there is a capacity constraint at the end product level (stage !~. Assume that no more than 3 units can be produced during a specific time pericd and that the production of one batch cannot last for more than one period. The optimal policy is now given by Facilityt Facility 2
s=8, S-9 If the installation inventory position at facility 2 (IIP 2 ) i:~ greater than or equal six units, there is no production. If IIP 2 < 6, we have s=ll, S = m i n { E I P I ~9,15}.
Safety ~teck echelon 2
%
0.3
0.2
i
0." Table 4
F
\
Echelon holding costs Facility 2 Facility 1 Ratio
l 9 t0%
3 9 25%
5 9 35.71%
7 9 43.75%
9 9 50%
L
\
\
t
~0%
25% '>5% 43%
50%
Ratio
Figure 2. Buffer stock as a functicrt of the value added ratio
M. R Lam&tz~ et aL / Pr~rertive uocks and boultnecks in production systtms
324
At first sight, thi~ policy seems difficult t o understand. First of all we repeat that the policy paxameters (s, S) are to be interpreted in 'echelon inventory position' terms. The additional condition for facility 2, namely liP,. < 6, is in that respect not in contradiction with the (s, S) parameters. Simulating the above policy (the simulation experiment always covers 7000 periods) indicates that the average order quantity at facility 1 is 2.85 units and for facility 2 the average order quantity equals 5.'~6. This shows that the production quantity is at-aost 3 for facility 1 and 6 for facility 2. This remit can easily be explained by the fact that all facilities must be coordinated with the bottleneck, which in our case is facility 1. A more detailed analysis in section III will :eveal that the safety stock at echelon 2 is increased. In a multi-echelon production system, the planning depends on the bottleneck processes.
Table 6 Dmnand distributions for the experiment Demand dlqribution A
Deman,t d~tribution B
Demand per period
Probability
Demand per perk~
Probability
0 1 2 3 4 Mean = 2 Variance ~ 1
0 0.4 0.3 0.2 0.1
0 1 2 3 4 Mean -- 2 Variance = 3
0.4 0 0.1 0.2 0.3
Table 7 Lead time combinations used in the experiment Lead time combination l e a d time facility 1 Lead time facility 2 ! 2
t 2
1 1
Table 8 The eight test problems
3. Safety stocks and bottlenecks: Computational results
A. Dewription of the experimant In this section we analyze the results obtained fc)r eight lest problems covering two lead time structures, two demand distributions and two cost combinations. In all cases, a two echelon serial production system was examined. The cost structures we used in the experiment are given in Table 5. In both cases the shortage cost equals 150. Two demand distributions for :he final product are used in the experiment. The expected demand is thz same for both distributions; however, the variance is higher for the second distribution. The specifications are given in Table 6. The lead time combinations are given in Table 4. The resulting eight test problems are summarized in Table 8. For each problem, 3 experiments were con-
Problem number
Cost structure
Demand distribution
Lead time combination
1 2 3 4 5 6 7 g
2 2 1 1 2 2 1 t
A B A B A It A B
1 1 I l 2 2 2 2
ducted: a. The optimal policy is computed in case there are no capacity constraints. b. The optimal operating policies are computed for the same eight examples but with the restriction that production output at facility I (end product level) is limited to 3 units. c. In a third experiment we examine what would
Table Cost data for the experiment Facilit ¢
1 2
Cost structure 1
Cost atructure 2
Setup cost ( K )
Eche'on holding cost ( H )
Setup cost ( K )
Echelon holding cost ( H )
30 9
5 9
9 30
9 5
M.R. Lambrecht ~ af. / Protective stocks and bottlenecks in production system :
325
Table 9 Operating po|icies for each problem Problem Experiment a (no cap. constraints) Facili,t I
1
t6, 7)
Experiment b (cap. constraints)
~'adlity 2 Exp.cost/period Facility 1
(7.12)
79.1
(6, 7)
Facility 2
If I I P : < 6 then S = min{EIP l + 9, 12} otherwise no production
2
(8, 8)
(10, 14)
95.45
(8, 9)
(6, %
~7, :0)
90.12
4
{7,11)
(9,11)
111.09
(8, 9)
s = 11 S - - min{EIP~ +6, 12}
18,9)
(10,14)
95.75
{8, I0)
If lIP2 < 6, then s =10 S = mi={EIP 1 +6, t2} otherwise no production
{12,16)
121.33
(10, 12)
tltL 12j
(lO, t4j
s--6 s = 8 production production quantity = 3 quantity = 3
82.20
110.04
95.26
110.89
121.52
140.56
99.25
I00.40
(11, 12} except if EIP2 = 14 then (11, 11)
If liP s < 6, th-n 130.58 s =14 S = min{ElPx * 9 . 14} otherwise no productmn
t38.03
114.80
tlO, 12)
s ~= lO 120.47 production quantity ~ 3
t 27.30
(11,14) 143.12 except if
{1t, 13;
,- = ~3 I53.32 pr~tuction quantit? -- 3
lg0.79
except if
EIP2=10 then (9, 10)
then{10,12) 7
81.65
If IIP2 < 6 then .*=6 S =~ min{EIP~ +% 15} 106.70 otherwise no production
3
00,11) exeeptif
Experiment c (unExp.cost/period constrained policy in a constr, environment) Exp.cost/period
then {I1, 15)
happen if the optimal unconstrained policy was applied in a capacity constrained environment (simulation experiment). For the experiments a and b the Markov Decision Model was used to find the optimal policies. Afterwards, a simulation experiment was run (covering each time 7000 time periods) to identify the amount of safety stock and safety time. The simulation was done for all test problems in the three experimental settings. The results are summarized in Table 9.
B. Analysis of the results The examples introduced above can be split up i n two groups depending on the cost structure. Cost structure 1 (problems 3, 40 7 and 8) results in
collapsed po.,cies. A collapsed policy means that the available units from facility 2, are all pushed to facility I, resulting in equal lot sizes at both fad-
TabLe I0 Average order quantities for problem5 3, 4, 7 and 8 Experiment
Problem
a. No capacity constraint~ b. With ca~cJty constrainta c. Unconstrained policyin a constraine.d environment
Fac, l 2 Fac. t 2 Fac. I 2
3
4
7
4.74 4.74 3 3 2.459 4.73
3.878 3.878 2.86 2.86 2.82t 3.878
3.738 3.738 3 3 2.577 3.738
5.¢~, 5.64 3 3 2.755 5.646
326
M~K Lambrecht et al, / Protective stocks and bottlenecks it. producli~n s rstcm_~
lites. Maxwell and Muckstadt [161 have shown that in this case the lot sizes should be identical for al! items (in a deterministic environment). The simulation experiment shows that this is indeed the case even in a stochastic environment. The avecage order quantities for both facilities ~:e given in Table 10 for the three experiments. It is interesting to observe whzt happens in experiment c. In this experiment we examine ,he behavior (,f an optimal unconstrained rmlicy ~n an environment in which there actually ~s limited capacity. The result is that the collapsed policy can no longer be follo~ed and this explains the important cost differehce of on the average 13.91% between experiment b and c. Cost szructare 2 results in totally different policies (examples 1, 2, 5 and 6). They are (s, S ) policies constrained by the fact that there is never a production order at facility 2 when the installation inventcry position at facility 2 is 6 or more. The logic for this, at first sight, strange relation between tae installation inventory position at 2 and the fact whether or not it might be optimal to produce at fa,:ility 2 can be explained rather easily ~a.~ing an indi:ect demonstration. Assume it would be optimal Io produce X units at facility 2 when the installation inventory position at 2 is 6 (or more). The first facility can never pr(~luce more than three units, (the capacity constraint}, thus the installatJon inventory position at 2 is at least 3 ( = 6 - 3 ) utfits at the end of this period In the beginning of the next period, X units arrive as we decided to produce X units and the lead time at ~acility 2 i., one period. The first facility again cannot produce more than 3 units, so the installation inventory position at 2 is at least (0 + X) units. T;ais implies that the X units are, at least during one iaeriod, not used. Tl-,us, it can never be optimal to produce them now as we have less holding costs when we delay this production one period. More generally, it can never be optimal to produce v, henever the installation inventory position ~t the facility considered is greater than or equal to the lead time at that facility augmented wit? one review period, muhipl;ed by the capacity of the succeeding facility, i.e. in onr example whenever the installation inve~atot3' pcsition at facility 2 is greater than or equal to (I + 1)(3)= 6 unit:,. Th.s concludes our demonstration. The production quantity at all facilities is dominated by the bottlet:eck process. The product,on quantity at
Table I I Average order quantities for problems 1. 2, 5 and 6 Experiment
a b c
Probk-m
Fa¢. 1 2 Fae. 1 2 Fac. 1 2
1
2
5
6
2.624 5.815 2.451 5.816 2.451 5.805
2.977 6.798 2.845 5.761 2.55 6.80
3.016 5.81 3 6 2.5-39 5.816
3.869 6.798 2.611 6.96 2.736 6.890
facility 2 is a multiple of the capacity constraint at facility 1. The resulting average order quantities are summarized in Table 11. From Table 11 it can be seen that the impact of the capacity constraint in experiment c is not so great. The average cost difference between experiments b and c is 2.667~. The cost incurred by not taking into account the capacity constrained is not very high in this ease. Next, we examine the behavior of the safety stock and safety time, We recall that safety stock is defined as the expected amount of on hand inventory at the time an order or lot arrives. Safety time is defined as the expected time between the completion of a lot at the pre~ceding stage and the time at which units from the lot ~re u.~cd in the production of a lot at the ;~ucceeding stage. The result.~ for problems 1, 2. 5 and 6 are given in Table 12. In all four cases the average number of backorders increases in a constrained environment. This explains for a great deal the increased costs per period in the b experiments. The increased backorder position is associated with an increase in safe~y stocks and safety time at one or two facilities (problem 5 is an exception). The service level obtained (measured by the fill rate) in the capacitated case is slightly lower, that means that more safet) stock would be required to obtain the same service level as in the uncapacitated case. In Table 13 the results are given for problems 3, 4, 7 and 8. Remember that problems 3, 4, 7 and 8 are characterized by a collapsed policy. The cost difference between the a and b experiments can no longer be explained by the increased shortage costs as was the case for r.he other problems. The cost difference here is caused by the lower order quantties (increased setup costs). As can be seen from
327
M.P,. Lambrecht et al. / Protecti¢~e st~cks and bo:tlenecks in production st'stems
Table I2 Safety stocks and time for problems l. 2, 5 and 6 Problem
1
Experiment
a
b
a
Safety stock at t Safety stock at 2 Safety time at 2 Average number of backorders Fill rate
1.995 0.1389 0.1355 0.0509 0.974
1.831 0.2951 0.1082 0.0795 0.960
2.621 0.099 0.2409 0.0346 0.983
5 b 2.8274 0.2383 0.3940 0.0741 0.962
6
a 2.097 0 0.196 0.0795 0.960
b i .8615 0 0.1707 0.101 0.949
a 3.056 0 0.239 0.083 0.958
b 3.42 1.066 0.225 0.117 0.942
Table 13 Safety stocks and time for problems 3, 4. 7 and 8 Problem
3
4
7
8
Experiment
a
b
a
b
a
b
a
b
Safety stock at 1 Safety stock at 2 Safety time at 2 Average number of backorders Fill rate
1.23 0 0 0.101 0,949
1.731 0 0.I60 0.075 0.962
1.931 0 0.574 0.0735 0.963
2.25 0.139 0.42 0.0735 0.963
2.216 0 0 0.068 0.956
1.99 0 0 0.129 0.935
2.56 0 0.25 0.154 0.932
3.452 0 0.27 0.119 0.940
Table 14 Average number of backorders for experiment c Problem
1
2
3
4
5
6
7
8
Average number of backorders
0.0812
0.1568
0.18,~1
0.239
0.126
0.246
0.129
f1358
4. SummaD' and conclusion Table t3, the fill rate in a capacitated environment improves in a n u m b e r o f cases (because of more safety stock). It appears as if collapsed policies o f f e r a b e t t e i protection against uncertainty even in the case of capacity constraints. A collapsed policy means that the available units from facility 2 are all pushed to facility 1, consequently the p r o t e c t i o n against uncertainty is realized at ;acility I. In case o f capacity constraints this results in a faster response to changes in demand. F o r the c experiments we can conclude that the cost difference with e.g. e x p e r i m e n t b can be explained by the increased b a c k o r d e r position and c o n s e q u e n t l y the increased shortage costs. T h e average n u m b e r of b a c k o r d e r s (experiment c) is given in T a b l e 14. F o r the problems 3, 4, 7 and 8 we have the additional p r o b l e m that the collapsed policies are no longer followed which causes additional setups at stage t.
This p a p e r studies the allocation of s; fct'/~,tock~ and safety times in multi-echelon prqdaction system. Optimal policies are d e t e r m i n e d by representing the p r o b l e m as a Markovian Decision Process. This allocation process heavily depend,~ on the problem characteristics such as d e m a n d variability, c~st structure and lead time structure. General conclusions such as a safety stock only at the end p r o d u c t level or only at interm~-diate stages are invalid. T h e introduction ol capacity constraints d e m o n s t r a t e s that the bottleneck proce~se~ are dictating the planning. A n o t h e r i m p o r t a n t issue is the cost structure. This dictates whether p r o d u c t i o n poficies t',etween stages are collapsed or not. O u r analy~i~, demonstrates that collapsed policies offer a better protec:tion against uncertainty. F u r t h e r research shou[d be aimed at obtaining easy to c o m p u t e policies b? making use o f the theoretical resutts obtained in this paper.
32g
M,R. Lambrecht et aL / Pr~,tectivextocl~ and b~ttlenex-ks in praductiaw ~3v~tem~
References [1] Arrow. K.J.. Karli~ S. and Scarf. H_ Studies in the Math.marital Theory of Inventory and Production. Stanford Uni~'ersity Pre~s. Stanford, California, 1958. 12] Bancrjee. A.. "Buffer inveratories for material requiremerits planning systems", working paper, University of Delaware. May 1978. [3] Blackburn. J.D.. and Milieu. R.A,. "Improved I~uristlc~ for multistage requirements planning systcm.s". Manage"nent Science 2gO ) f1982) 4,1-56. [4! 8aech.~l. A.. ~An c~,erview of possible procedures for ,tc~:h~,.tic MRP". Engineering Costs and Production Eca"~omiCr~6(1) (t982) 4;3--51. [5] :.~arlson. R.. and Yano. C.A. ""Buffering agai_nst demand Jl~certainty in material requirements planning systerrLS-!;ystems with no emergency ,,.¢tups", Technical Report No. ~¢I-Z Department of Industrial Engineering and Engineering manageraent Stanford University, April 1981. p. 29. [~A Carl~m, R., and Yano. C.A.. "Buffering against demand ,~ncertainty in material requirement~ planning systems-Sygtem.~ ~.i;h emergepcy ~tups tot ~,ome component~°'. Technical geport No. 81-3. Dep~tment of Industrial Engineering znd Engineering Mznagement Stanford University, April 1981, p. 27. [7] Clark. A., and Scarf, H.. "Optimal pclici¢~ for a mvhiechelon i=wentory probiem", Manage~ent Scwnce ~4) t i 9~0) 474.~4c4), iZl (":~rk. A,, and Scarl, I1,, "'Approximate ~:4ut~on~- "~ s:rnffle muhiechelon inventory problem", Chapter 5 in K Ar=cw, S. Karlin and H. ~ a l f (ed.~.i. Studw; tn App,~ed I~rohahtht~, and Management Sctem e, Stanford Univet!,ity Pre.,s, Star, ford, California, 1962. 191 (older. DA.. "Aggregate safety stock level,, ,.rod component part commonality". Manag,,mcm S¢.me 2g¢ll) 19t~2) 1296-1303
[I0] Crowston. W.B., Wagner, M,: aud Wtltlams, J,P., ~Economic lot size Determination in multi-stage asse~biy systems", Management Science 19(.4) 0973) 517-527. [1I] De Bodt. M., Van Wa~senhove, L., and Gddets, L, "Dewand uncertainty", En~neering Co~ls and P r o ~ i o n Econonucs (1982) 67-75. [12] Graves, S.C., "Multistage lot-sizing: An iterative procedure", working paper. M.LT., April 1979. [131 Howard. R_, Dynamic Programming and Marko~ Processes, M.I.T. Press, Cambridge, MA, 1960, [14] Joensson, H.. Lund¢ll, P., and Thorstenson, A.0 "Some a s ~ t s on uncertainty in a multi-level inventory systems", Engineering Costs and Production Economics 6(1) (1982) 14-1-14-6. [15] Lambrecht, M.IL, Mucl~tadL. J., and Luytcn, IL, "Protective s~ocks in multi-stage production systems", International Journal of Production Research 22(6) (1984) 1301-1025. [16] Maxwell, W., and Muckstadt, J., "Establishing reorder intervals in muhi-stage production-distribution systems". Technical Report No. 561, School ot Opef.~tions Research and Industrial Engineering. Cornell University, Ithaca, NY. !982. [17] Meal. H.C. "Safety stocks in MRF systems", Technical Report No. 166. Operations Research Center. Massachusett~ Institute of Technology. Cambridge, MA. [18] Miller, J., "Hedging the master schedule", in: L. Ritzam et al. (¢ds.) Disaggregation Problems in Manufacturing and Serctce Org~Tniz:ttirms, Marlinus Nijhoff, 1979, 237-256. 119] New, C., "Safety ~ttx:ks for requirement,,, planning", Production and lm~ntory Management. 2nd. Qrt. (I 975) 1-18. 1201 Timmer. J., Monhemius. W., and P,ertr;=nd. J. "Production and inventory c~mttol with Ihe ba~;e ~tock ,~ystem", paper pre~ented al the TIMS XXV! Conference, Copenhagen, June 1984 [21] Whybark. ~., ~.qd J. Williamg. "'Material :eqmrement,, plannir, g under uncertainty". Dectswn Sc'tences 7¢4) ¢1976) 595-636.
319
Protective inventories and bottlenecks in production systems M . R . L A M B R E C H T , R. L U Y T E N a n d J. V A N D E R E E C K E N
Operations Management Group, Department of Applied Economics, K.U. L uven, Belgium Abstract: In this paper we study the allocation of safety s t o c ~ and safety t systems. We concentrate on the impact of demand uncertaivty on the presence of capacity constraints. We describe how this problem can be Process. The form of the optimal policies is discussed and illus'~rated witl
rues in multi-echelon production amount of safety stocks in the iewed as a Markovian Decision examples.
Keywords: Inventory, manufacturing industries, Markov processes, stocha tic processes
1. Introduction The purpose of this paper is to study the allocation of safety stocks and safety time in multiechelon production systems with limited capacity. Uncertainty does exist in several forms~variability of demand from the forecast, uncertainty in the supply from stage to stage due to the variability in the yields from each production batch and uncertainty in the lead times for both production and procurement. Protectic.n against the~e disruptions, is required and this results in buffer stocks a n d / o r safety times. Research on protective stocks in multi-stage systems has primarily been experimental, We refer the reader to the work of Banerjee [2], Whybark and Williams [20], Buechel [4], Carlson and Yano [5,6], Collier [91, De Bodt et al. [11], Meal [17], New [191 and Joensson, LundeU and Thorstenson [14]. The authors of this paper have shown iv, a recent paper [15] that the multi-echelon stochastic problem can be viewed as a Markovian Decision Problem. Because of the limited practical value of this approach heuristic solutions methods have been suggested based on the well-known results of Clark and Scarf [7,8]. The authors also have demonstrated that generalized (s, S) policies appear to
ReceivedApril t984; revisedOctober 1984
be optimal. The -esulting (s, S) policies are next simulated in ord:zr to determine the amount of safety ~tock and safety time. The simulation is necessa~ becaus~ it is impossible to determine the exact amount ot safety stock or time by only knowing the valu.~s of the policy variables. Safety stock i~ delermin,:d as the expected arr, ount of on hand inventory at the time an order or a lot arrives Safety tinge is defined as the expected time between the completion of a lot at a preceeding stage until any milts frc,m that lot are :ased in the production of a ot at the succeeding ,,rage. The computar,ional r~sults indicate that both safety time and safety stock can eaist at intermediate stages. The above observations will be illustrated by means of an exaraple in Section 2. The purpose of this paper is to extend the" previous anaIysis [15] for situations where capacity constraints are limiting the production output at certain facilities, In Section 2 the multi-echelon system is ir~troduced together with the underlying assumptions and a description is given of how this problem can be viewed as a Markovian Decision Process. In Section 3 we discuss the form of the optima! operating policies for the case in which there are capacity constraints at the end product level, tn particular we show that generalized (s~ S) polJcie~ appear to be optimal. The discussion is illustrated with examples.
0377-2217/85/$3.30 'co ~985, ElsevierSciencePub}isher~B.V.(North-Holland)
~20
M I ~ i.ambrecht et al. / Protective stocks and bottlenecks in production systems
1. M o d d fommb~tioa
A. Nomenclature In order to facilitate the understanding of the model and its resuits, we recall in this section some generally accepted detrmitions of inventory con,zepts. The problem considered is a single-item multilevel production/inventory system of the serial -ype. Multi-level, or multi-echelon, production/inventory systems have traditionally been subdivided in two types, assembly and arborescence sy~_,ems. An ~sembly system is a system in which each facility, or production stage, requires inputs from a number of immediate predecessor stages while supplying, m turn, at most one immediate successor stage. An arborescence structure, on the other hand, is a system in which each facility supplies a rmmber of immediate successor stages while it gets input from at most one predecessor stage. A serial system is a special c a ~ of both an assembly system and an arbore.w.ence system; every facility has at most one preceeding facility and one succeeding facility. 'Ive installation inventory at facility j, also called the physical inventory, is the number of units pi~ysically on-hand at facility j. The installation invent~,ry position at facility j, also called the economic inventory, is the ;nstallation invento,"y at facility j augmented with5 the quantity on order and decreased with the backorders. The quantity on order reflects replenishment orders which have been relezsed but not yet received whereas backorders reflect orders for the product at facility j which have been received but not yet delivered because of a stockout. The echelon inventory definitions are taken from Tinmmr, Monhemius and Bertrand [20]. They define the echelon inventory at facility j as the number of units in the system which have gone through the operations of facility j but have as yet not been sold. The echelon inventory position at facility j, also called the available echelon inventor3, is defined as the echelon inventory, at facility j plus the quantity on order at facility j minus the backorders for the finished items which contain the item under consideration. Another look at the echelon inventor, position is that it is the installation inventory at facility j plus the installation inventory position at all succeeding facilities.
The symbol liP) will be used to represen: the installation inventory position and the symbol E1P/ will be used to represent the echelon inventory position at facility j.
B. Methodology In this section, we introduce the multi-echelon production/inventory system and its underlying assumptions. We also describe how this system can be modelled as a Markovian Decision Process. Attention is restricted to the situation where the demand process is stationary and the planning horizon is infinite. The product structure we investigate, is depicted in Figure 1. It is a two echelon serial structure, echelon 2 is the component level and echelon 1 is the final assembly level. The model we present is based on the following assumptions: 1. A periodic review policy is followed; 2. The production and procurement lead times, which are measured in periods, are known constants and are independent of the batch size; 3. The demand distributions are known; demand is independent from period to period for each final product; 4. All excess demand is backordered; 5. Prcrduction capacity is limited at the final assembly level and the production of one batch is not allowed to last for more than one period. We also assume that the relevant costs consist of (1) production costs, at each stage, which are composed of a fixed setup cost per production run and a constant cost per unit produced; (2) holding costs which are charged proportional to the installation inventory at the end of the period; (3) linear shortage costs, which are charged proportional to the number of final products backordered at the end of each period.
Eche!on 2
(~) .f
Echelon I
O
.f Final demand Figure 1. A two echelon serial structure
~f.P,. Lambrecht et a L / Protective stocks and bottlenecks ir productto ~ systems
Note that the constant co~t per unit produced can be ignored in our analysis ~.s we assume that costs remain stationary over time and we have an infinite planning horizon. The multi-echelon stochastic production/inventory problem introduced above can be modelled as a Markov Decision Process. It is assumed that the system is observed at time t = 0, 1. . . . . T and classifted into one of a finite number of states i labeled 1, 2 . . . . . N. For every state i, a number of decision alternatives, k, are possible, labeled I, 2 , . . . , K ( i ) . (The number of possible policies might differ from state to state). Whenever the system is in state i and alternative k is chosen the system moves to state j with known ~ransition probability p k/ for all i, j = 1, 2 . . . . . N and k = 1, 2 . . . . . K(i). The cost structure is introdueexi through q~, the expected one-period cost incurred when the system is in state i and decision alternative k is chosen. In terms of the multi-echelon stochastic production planning problem the states i, i = I, 2. . . . . N, of the Markov Decision Process correspond to the echelon inventory position at each facility. Every state thus corresponds to specific inventory positions at each facility; these echelon inventory positions themselves can be positive, negative or zero. The total number of states equals the product of all possible echelon inventory positions at all facilities. The decision alternatives k, k -~ 1. . . . . Kfi), r e p r e ~ n t the production decisions at each facility. In case there are capacity constraints, some of these production decisions are impossible which implies that the number of decision alternatives for state i in the capaeitated case is lower than (or equal to) the number of decision alternatives in the uneapacitated case. The transition probabilities p~, depend on the current state, i, the decision made, k, and the demand distribution, which together with i and k, determines j. The proposed solution mmhod is based on the value iteration or successive approximation method developed by Howard [13]. The recursive relation is given by
I
E(T) = m n.
E j=l
p*,,V/T-1)
}
where V,(0) = 0, Yi. ~ ( T ) stands for the expected total (discounted) cost of a system starting in state i and evolving for T periods when an optimal
321
policy is followed, a denotes the discount factor and q~ and p~) are elements of the cost matrix Q and the transition matrix pk. For large values of T, a stationa:y cost structure and stationary transition probabilities, the system will reach its steady state. This means that the decision to be taken in a certain state i is unaffected by the time period in which the de,'ision has to be taken. After stabilisation, only a limited number of states will be possible; these are the recurrent states. C. A numerical example An example will illustrate the methodology mtroduced in Section 2.B. Consider a two-echelon serial production system, shown in Table 1. Demand is assumed to be independent from period-to-period, all excess demand is backordered. We further assume that the demar:d process is stationary, the costs are time independent and the planning horizon is infinite. In a first step, we restrict the number of states and decision alternatives considered. We therefore determine minimum and maximum levels for the echelon inventory position of each item and a maximum production quantity for each item. These figures are chosen such that no rea:,onabte de:ision ahernatives or states are excluded. For the above mentioned problem the following limits we~c imposed: gable I Two-ezhelon serial production aystem Setupcost
a ~
Echelon holding Lead t;rne coat (perio:l_
Facility 2 30
5
]
Facility 1
9
1
9
t v
Demard Shortage ¢o~t per unit per perie_,d = 150 Demand distribution Demand per period 0 i 2 3 4 MeRR = 2 Variance = 3
Probability 0.4 0 OA rj.2 0.3
322
~tt.R~ ~ t ' ~
el a£ / P r a ~ l i ~ JtockJ' and botdentd~ in production rj~tems
- M a x i m u m production at echelon I, d e n o t e d b y M 1 ,,- 10.
"l'~bl¢ 2 S¢:~p e o ~ at f~,ility I Holding ¢c~t at facility e0A × 3 - 0.1 × 1) 0 4 ) Shortage o ~ t (0.3 × 1) t lSO) Holding c,o~ls at facility 2 ~8-5) (5] Work in process holding cosls (5) (5)
9.00 18.20 45.00 15.00 25~0
To~.al
112.20
- Minimum eche:on inventor3" position at echelon 2=-2.
- Maximum echelon inventory position at echelon 2 = 20.
- Maximum production at 2, denoted b y M 2 = 15. - Minimum echelon inventory position at echelon 1=-6.
- Maximum echelon inventory position at echelon 1 =15,
The decision alternatives, k, are formed by e combination of the production at 1, Z1 {0, 1. . . . . M1}, and the production at 2, Z 2 ~ {0, 1 . . . . . M2}. The cost dements, q~, represent the set-up cost and invt~atory rehted costs. Suppose that the echelon inventory position at facility 2 is 11 units and at facility i we have an echelon inventory position of 3 and that there are no outstanding orders nor backorders (the installation inventory position at facility 2 thus is 8 units). Furthermore, assume that we decide to produce 5 units at facility 1 and nothing at facility 2. We then incur a setup cost at facility 1 and no setup cost at facility 2. We have holding costs at facility 1 whenever demand is less than three and incur shortage costs whenever demand is greater than three m:it_, Each unit at facility 1 leads to an inventory l*olding cost of 14, i.e. the summation of the echelc.w~ holding cost at facility 2 and the
Table 3 The recurrent states Slate t IEIP.~
EIP 1
Steady state
Decisiot k
probabilily
Z2
Z1
E~pected cost~ q~
6 7 7 8 8 8 9 9 10 10 10 10 10 11 11 11 11 12 12 12 12 14 14 14
4 4 5 4 5 6 5 6 2 3 4 6 8 3 4 5 8 4 5 6 8 6 7 8
0.7808 E - 0l 0.4932 E - Ol 0.5205 E--O1 0.2466 E - O I 0.3288 E - O I 0.2603 E - Ol 0.1644 E - O 1 0.1644 E - 0 1 0.2342 E - 02 0.3041E-01 0.9411 E - Ol 0.8219 E - 0 1 0.1041E+00 C.1562 E - 0 1 6.2027 E - 0 1 0.6274 E - O f 0.6575 E - 0 1 0.7808 E - 0 2 ,~.1014 E - 0 1 ¢~.3137 E - 0 1 ('.3288 E - 0 1 C.3123 E - O 1 ~.4055 E - 0 1 01255 E - O I
8 7 7 6 6 6 5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 3 2 4 3 2 3 2 6 5 4 2 0 5 4 3 0 4 3 2 0 2 1 o0
77 82 9t 87 96 105 101 110 180.2 1072 67 85 94 112.2 72 81 99 77 86 95 104 105 I14 I14
(1)
(2)
(3)
~4)
(5)
(6)
Gain = q5.449493, service = 0,974479 fdl rate = 0.983726.
Echelon ~to~k p ~ i t i o n after dexi.~i~m ;z! 2
at 1
14 14 14 14 14 14 14 14 10 10 10 10 10 11 il 11 11 12 12 12 12 I4 14 t4
6 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
(7)
(8)
M.R. Lambrecht et al. / P,otective stocks and bottlenecks in production systems
echelon holding cost at facility I. The installation inventor)' at 2 is 3 units, which is the difference between the beginning installation inventory at facility 2 and the production at facility 1 (there are no open orders at facility 2). The production decision at facility 1 leads to a work in process inventory of 5 units and each unit is charged at five monetary units. Adding all these figures we obtain an expected one period cost of 112.20 as is summarized in Table 2. The solution for the stated problem is given in Table 3. From Table 3 we learn that the production level is zero at the second facility when zhe echelon inventory position at that stage is 10 or more. The production quantity is positive if the echelon inventory position is less than 10. A closer look at the table shows that we always order up to a level of 14. Consequently, at facility 2 we have a pure (s, S) or m i n - m a x policy with s = 10 and S = 14. At stage 1 we also have an (s, S) policy with s = 8 and S = 8. The first three states in Table 3 seem to be in conflict with this policy. This, however, is not true, the (s, S) policy for the first state e.g. suggests we should produce 4 units but we only have 2 units available (the installation inventory at stage 2 is 2 units), so production is 2 units. Sometimes it is impossible to reach th.: order-up-to level because of the unavailability ~,f components at the preceeding facility. The optimal average cost per period is 95.45. The service level defined in term~ of percentage ~Jf periods without stockouts is 97.4~. "Ihe fill rate equals 98.3%. It is interesting to note that all policies are based on the echelon inventory position concept. As was already explained in Section 2.A, the echelon inventory position at facility i can be seen as the installation inventory position at that facility plus the installation inventory position at all succeeding facilities. As a consequence, the echelon holding costs reflect the value added cost s!ructure. The following statistics can be obtained from a simulation experiment:
323
Average order quantity at stage 1 : 2.97, Average order quantity at stage 2: 6.79, Fill rate : 98.3%, Safety stock at stage 1 : 2.62, Safety stock at stage 2 : 0.099, Safety time at stage 2 : 0.24 periods. We can now test, e.g., the impact of the cost structure on the amount of safety stock. In the present example the installation holding cost for facility 1 is 14 and for facility 2, the installation holding cost is 5. The value added ratio is 5/14 × 100 = 35.71%. The optimal policy was computed for tile following value added ratios presented in T~ble 4. The higher the ratio the more expensive the component. The results are given in Figure 2 The impact of the length of the lead time on safety stocks can also be examined. It is a wellknown result that the amount of safety st3ck is much higher when the lead times are longer (see
[15]). We now extend the anaysis to the case in which there is a capacity constraint at the end product level (stage !~. Assume that no more than 3 units can be produced during a specific time pericd and that the production of one batch cannot last for more than one period. The optimal policy is now given by Facilityt Facility 2
s=8, S-9 If the installation inventory position at facility 2 (IIP 2 ) i:~ greater than or equal six units, there is no production. If IIP 2 < 6, we have s=ll, S = m i n { E I P I ~9,15}.
Safety ~teck echelon 2
%
0.3
0.2
i
0." Table 4
F
\
Echelon holding costs Facility 2 Facility 1 Ratio
l 9 t0%
3 9 25%
5 9 35.71%
7 9 43.75%
9 9 50%
L
\
\
t
~0%
25% '>5% 43%
50%
Ratio
Figure 2. Buffer stock as a functicrt of the value added ratio
M. R Lam&tz~ et aL / Pr~rertive uocks and boultnecks in production systtms
324
At first sight, thi~ policy seems difficult t o understand. First of all we repeat that the policy paxameters (s, S) are to be interpreted in 'echelon inventory position' terms. The additional condition for facility 2, namely liP,. < 6, is in that respect not in contradiction with the (s, S) parameters. Simulating the above policy (the simulation experiment always covers 7000 periods) indicates that the average order quantity at facility 1 is 2.85 units and for facility 2 the average order quantity equals 5.'~6. This shows that the production quantity is at-aost 3 for facility 1 and 6 for facility 2. This remit can easily be explained by the fact that all facilities must be coordinated with the bottleneck, which in our case is facility 1. A more detailed analysis in section III will :eveal that the safety stock at echelon 2 is increased. In a multi-echelon production system, the planning depends on the bottleneck processes.
Table 6 Dmnand distributions for the experiment Demand dlqribution A
Deman,t d~tribution B
Demand per period
Probability
Demand per perk~
Probability
0 1 2 3 4 Mean = 2 Variance ~ 1
0 0.4 0.3 0.2 0.1
0 1 2 3 4 Mean -- 2 Variance = 3
0.4 0 0.1 0.2 0.3
Table 7 Lead time combinations used in the experiment Lead time combination l e a d time facility 1 Lead time facility 2 ! 2
t 2
1 1
Table 8 The eight test problems
3. Safety stocks and bottlenecks: Computational results
A. Dewription of the experimant In this section we analyze the results obtained fc)r eight lest problems covering two lead time structures, two demand distributions and two cost combinations. In all cases, a two echelon serial production system was examined. The cost structures we used in the experiment are given in Table 5. In both cases the shortage cost equals 150. Two demand distributions for :he final product are used in the experiment. The expected demand is thz same for both distributions; however, the variance is higher for the second distribution. The specifications are given in Table 6. The lead time combinations are given in Table 4. The resulting eight test problems are summarized in Table 8. For each problem, 3 experiments were con-
Problem number
Cost structure
Demand distribution
Lead time combination
1 2 3 4 5 6 7 g
2 2 1 1 2 2 1 t
A B A B A It A B
1 1 I l 2 2 2 2
ducted: a. The optimal policy is computed in case there are no capacity constraints. b. The optimal operating policies are computed for the same eight examples but with the restriction that production output at facility I (end product level) is limited to 3 units. c. In a third experiment we examine what would
Table Cost data for the experiment Facilit ¢
1 2
Cost structure 1
Cost atructure 2
Setup cost ( K )
Eche'on holding cost ( H )
Setup cost ( K )
Echelon holding cost ( H )
30 9
5 9
9 30
9 5
M.R. Lambrecht ~ af. / Protective stocks and bottlenecks in production system :
325
Table 9 Operating po|icies for each problem Problem Experiment a (no cap. constraints) Facili,t I
1
t6, 7)
Experiment b (cap. constraints)
~'adlity 2 Exp.cost/period Facility 1
(7.12)
79.1
(6, 7)
Facility 2
If I I P : < 6 then S = min{EIP l + 9, 12} otherwise no production
2
(8, 8)
(10, 14)
95.45
(8, 9)
(6, %
~7, :0)
90.12
4
{7,11)
(9,11)
111.09
(8, 9)
s = 11 S - - min{EIP~ +6, 12}
18,9)
(10,14)
95.75
{8, I0)
If lIP2 < 6, then s =10 S = mi={EIP 1 +6, t2} otherwise no production
{12,16)
121.33
(10, 12)
tltL 12j
(lO, t4j
s--6 s = 8 production production quantity = 3 quantity = 3
82.20
110.04
95.26
110.89
121.52
140.56
99.25
I00.40
(11, 12} except if EIP2 = 14 then (11, 11)
If liP s < 6, th-n 130.58 s =14 S = min{ElPx * 9 . 14} otherwise no productmn
t38.03
114.80
tlO, 12)
s ~= lO 120.47 production quantity ~ 3
t 27.30
(11,14) 143.12 except if
{1t, 13;
,- = ~3 I53.32 pr~tuction quantit? -- 3
lg0.79
except if
EIP2=10 then (9, 10)
then{10,12) 7
81.65
If IIP2 < 6 then .*=6 S =~ min{EIP~ +% 15} 106.70 otherwise no production
3
00,11) exeeptif
Experiment c (unExp.cost/period constrained policy in a constr, environment) Exp.cost/period
then {I1, 15)
happen if the optimal unconstrained policy was applied in a capacity constrained environment (simulation experiment). For the experiments a and b the Markov Decision Model was used to find the optimal policies. Afterwards, a simulation experiment was run (covering each time 7000 time periods) to identify the amount of safety stock and safety time. The simulation was done for all test problems in the three experimental settings. The results are summarized in Table 9.
B. Analysis of the results The examples introduced above can be split up i n two groups depending on the cost structure. Cost structure 1 (problems 3, 40 7 and 8) results in
collapsed po.,cies. A collapsed policy means that the available units from facility 2, are all pushed to facility I, resulting in equal lot sizes at both fad-
TabLe I0 Average order quantities for problem5 3, 4, 7 and 8 Experiment
Problem
a. No capacity constraint~ b. With ca~cJty constrainta c. Unconstrained policyin a constraine.d environment
Fac, l 2 Fac. t 2 Fac. I 2
3
4
7
4.74 4.74 3 3 2.459 4.73
3.878 3.878 2.86 2.86 2.82t 3.878
3.738 3.738 3 3 2.577 3.738
5.¢~, 5.64 3 3 2.755 5.646
326
M~K Lambrecht et al, / Protective stocks and bottlenecks it. producli~n s rstcm_~
lites. Maxwell and Muckstadt [161 have shown that in this case the lot sizes should be identical for al! items (in a deterministic environment). The simulation experiment shows that this is indeed the case even in a stochastic environment. The avecage order quantities for both facilities ~:e given in Table 10 for the three experiments. It is interesting to observe whzt happens in experiment c. In this experiment we examine ,he behavior (,f an optimal unconstrained rmlicy ~n an environment in which there actually ~s limited capacity. The result is that the collapsed policy can no longer be follo~ed and this explains the important cost differehce of on the average 13.91% between experiment b and c. Cost szructare 2 results in totally different policies (examples 1, 2, 5 and 6). They are (s, S ) policies constrained by the fact that there is never a production order at facility 2 when the installation inventcry position at facility 2 is 6 or more. The logic for this, at first sight, strange relation between tae installation inventory position at 2 and the fact whether or not it might be optimal to produce at fa,:ility 2 can be explained rather easily ~a.~ing an indi:ect demonstration. Assume it would be optimal Io produce X units at facility 2 when the installation inventory position at 2 is 6 (or more). The first facility can never pr(~luce more than three units, (the capacity constraint}, thus the installatJon inventory position at 2 is at least 3 ( = 6 - 3 ) utfits at the end of this period In the beginning of the next period, X units arrive as we decided to produce X units and the lead time at ~acility 2 i., one period. The first facility again cannot produce more than 3 units, so the installation inventory position at 2 is at least (0 + X) units. T;ais implies that the X units are, at least during one iaeriod, not used. Tl-,us, it can never be optimal to produce them now as we have less holding costs when we delay this production one period. More generally, it can never be optimal to produce v, henever the installation inventory position ~t the facility considered is greater than or equal to the lead time at that facility augmented wit? one review period, muhipl;ed by the capacity of the succeeding facility, i.e. in onr example whenever the installation inve~atot3' pcsition at facility 2 is greater than or equal to (I + 1)(3)= 6 unit:,. Th.s concludes our demonstration. The production quantity at all facilities is dominated by the bottlet:eck process. The product,on quantity at
Table I I Average order quantities for problems 1. 2, 5 and 6 Experiment
a b c
Probk-m
Fa¢. 1 2 Fae. 1 2 Fac. 1 2
1
2
5
6
2.624 5.815 2.451 5.816 2.451 5.805
2.977 6.798 2.845 5.761 2.55 6.80
3.016 5.81 3 6 2.5-39 5.816
3.869 6.798 2.611 6.96 2.736 6.890
facility 2 is a multiple of the capacity constraint at facility 1. The resulting average order quantities are summarized in Table 11. From Table 11 it can be seen that the impact of the capacity constraint in experiment c is not so great. The average cost difference between experiments b and c is 2.667~. The cost incurred by not taking into account the capacity constrained is not very high in this ease. Next, we examine the behavior of the safety stock and safety time, We recall that safety stock is defined as the expected amount of on hand inventory at the time an order or lot arrives. Safety time is defined as the expected time between the completion of a lot at the pre~ceding stage and the time at which units from the lot ~re u.~cd in the production of a lot at the ;~ucceeding stage. The result.~ for problems 1, 2. 5 and 6 are given in Table 12. In all four cases the average number of backorders increases in a constrained environment. This explains for a great deal the increased costs per period in the b experiments. The increased backorder position is associated with an increase in safe~y stocks and safety time at one or two facilities (problem 5 is an exception). The service level obtained (measured by the fill rate) in the capacitated case is slightly lower, that means that more safet) stock would be required to obtain the same service level as in the uncapacitated case. In Table 13 the results are given for problems 3, 4, 7 and 8. Remember that problems 3, 4, 7 and 8 are characterized by a collapsed policy. The cost difference between the a and b experiments can no longer be explained by the increased shortage costs as was the case for r.he other problems. The cost difference here is caused by the lower order quantties (increased setup costs). As can be seen from
327
M.P,. Lambrecht et al. / Protecti¢~e st~cks and bo:tlenecks in production st'stems
Table I2 Safety stocks and time for problems l. 2, 5 and 6 Problem
1
Experiment
a
b
a
Safety stock at t Safety stock at 2 Safety time at 2 Average number of backorders Fill rate
1.995 0.1389 0.1355 0.0509 0.974
1.831 0.2951 0.1082 0.0795 0.960
2.621 0.099 0.2409 0.0346 0.983
5 b 2.8274 0.2383 0.3940 0.0741 0.962
6
a 2.097 0 0.196 0.0795 0.960
b i .8615 0 0.1707 0.101 0.949
a 3.056 0 0.239 0.083 0.958
b 3.42 1.066 0.225 0.117 0.942
Table 13 Safety stocks and time for problems 3, 4. 7 and 8 Problem
3
4
7
8
Experiment
a
b
a
b
a
b
a
b
Safety stock at 1 Safety stock at 2 Safety time at 2 Average number of backorders Fill rate
1.23 0 0 0.101 0,949
1.731 0 0.I60 0.075 0.962
1.931 0 0.574 0.0735 0.963
2.25 0.139 0.42 0.0735 0.963
2.216 0 0 0.068 0.956
1.99 0 0 0.129 0.935
2.56 0 0.25 0.154 0.932
3.452 0 0.27 0.119 0.940
Table 14 Average number of backorders for experiment c Problem
1
2
3
4
5
6
7
8
Average number of backorders
0.0812
0.1568
0.18,~1
0.239
0.126
0.246
0.129
f1358
4. SummaD' and conclusion Table t3, the fill rate in a capacitated environment improves in a n u m b e r o f cases (because of more safety stock). It appears as if collapsed policies o f f e r a b e t t e i protection against uncertainty even in the case of capacity constraints. A collapsed policy means that the available units from facility 2 are all pushed to facility 1, consequently the p r o t e c t i o n against uncertainty is realized at ;acility I. In case o f capacity constraints this results in a faster response to changes in demand. F o r the c experiments we can conclude that the cost difference with e.g. e x p e r i m e n t b can be explained by the increased b a c k o r d e r position and c o n s e q u e n t l y the increased shortage costs. T h e average n u m b e r of b a c k o r d e r s (experiment c) is given in T a b l e 14. F o r the problems 3, 4, 7 and 8 we have the additional p r o b l e m that the collapsed policies are no longer followed which causes additional setups at stage t.
This p a p e r studies the allocation of s; fct'/~,tock~ and safety times in multi-echelon prqdaction system. Optimal policies are d e t e r m i n e d by representing the p r o b l e m as a Markovian Decision Process. This allocation process heavily depend,~ on the problem characteristics such as d e m a n d variability, c~st structure and lead time structure. General conclusions such as a safety stock only at the end p r o d u c t level or only at interm~-diate stages are invalid. T h e introduction ol capacity constraints d e m o n s t r a t e s that the bottleneck proce~se~ are dictating the planning. A n o t h e r i m p o r t a n t issue is the cost structure. This dictates whether p r o d u c t i o n poficies t',etween stages are collapsed or not. O u r analy~i~, demonstrates that collapsed policies offer a better protec:tion against uncertainty. F u r t h e r research shou[d be aimed at obtaining easy to c o m p u t e policies b? making use o f the theoretical resutts obtained in this paper.
32g
M,R. Lambrecht et aL / Pr~,tectivextocl~ and b~ttlenex-ks in praductiaw ~3v~tem~
References [1] Arrow. K.J.. Karli~ S. and Scarf. H_ Studies in the Math.marital Theory of Inventory and Production. Stanford Uni~'ersity Pre~s. Stanford, California, 1958. 12] Bancrjee. A.. "Buffer inveratories for material requiremerits planning systems", working paper, University of Delaware. May 1978. [3] Blackburn. J.D.. and Milieu. R.A,. "Improved I~uristlc~ for multistage requirements planning systcm.s". Manage"nent Science 2gO ) f1982) 4,1-56. [4! 8aech.~l. A.. ~An c~,erview of possible procedures for ,tc~:h~,.tic MRP". Engineering Costs and Production Eca"~omiCr~6(1) (t982) 4;3--51. [5] :.~arlson. R.. and Yano. C.A. ""Buffering agai_nst demand Jl~certainty in material requirements planning systerrLS-!;ystems with no emergency ,,.¢tups", Technical Report No. ~¢I-Z Department of Industrial Engineering and Engineering manageraent Stanford University, April 1981. p. 29. [~A Carl~m, R., and Yano. C.A.. "Buffering against demand ,~ncertainty in material requirement~ planning systems-Sygtem.~ ~.i;h emergepcy ~tups tot ~,ome component~°'. Technical geport No. 81-3. Dep~tment of Industrial Engineering znd Engineering Mznagement Stanford University, April 1981, p. 27. [7] Clark. A., and Scarf, H.. "Optimal pclici¢~ for a mvhiechelon i=wentory probiem", Manage~ent Scwnce ~4) t i 9~0) 474.~4c4), iZl (":~rk. A,, and Scarl, I1,, "'Approximate ~:4ut~on~- "~ s:rnffle muhiechelon inventory problem", Chapter 5 in K Ar=cw, S. Karlin and H. ~ a l f (ed.~.i. Studw; tn App,~ed I~rohahtht~, and Management Sctem e, Stanford Univet!,ity Pre.,s, Star, ford, California, 1962. 191 (older. DA.. "Aggregate safety stock level,, ,.rod component part commonality". Manag,,mcm S¢.me 2g¢ll) 19t~2) 1296-1303
[I0] Crowston. W.B., Wagner, M,: aud Wtltlams, J,P., ~Economic lot size Determination in multi-stage asse~biy systems", Management Science 19(.4) 0973) 517-527. [1I] De Bodt. M., Van Wa~senhove, L., and Gddets, L, "Dewand uncertainty", En~neering Co~ls and P r o ~ i o n Econonucs (1982) 67-75. [12] Graves, S.C., "Multistage lot-sizing: An iterative procedure", working paper. M.LT., April 1979. [131 Howard. R_, Dynamic Programming and Marko~ Processes, M.I.T. Press, Cambridge, MA, 1960, [14] Joensson, H.. Lund¢ll, P., and Thorstenson, A.0 "Some a s ~ t s on uncertainty in a multi-level inventory systems", Engineering Costs and Production Economics 6(1) (1982) 14-1-14-6. [15] Lambrecht, M.IL, Mucl~tadL. J., and Luytcn, IL, "Protective s~ocks in multi-stage production systems", International Journal of Production Research 22(6) (1984) 1301-1025. [16] Maxwell, W., and Muckstadt, J., "Establishing reorder intervals in muhi-stage production-distribution systems". Technical Report No. 561, School ot Opef.~tions Research and Industrial Engineering. Cornell University, Ithaca, NY. !982. [17] Meal. H.C. "Safety stocks in MRF systems", Technical Report No. 166. Operations Research Center. Massachusett~ Institute of Technology. Cambridge, MA. [18] Miller, J., "Hedging the master schedule", in: L. Ritzam et al. (¢ds.) Disaggregation Problems in Manufacturing and Serctce Org~Tniz:ttirms, Marlinus Nijhoff, 1979, 237-256. 119] New, C., "Safety ~ttx:ks for requirement,,, planning", Production and lm~ntory Management. 2nd. Qrt. (I 975) 1-18. 1201 Timmer. J., Monhemius. W., and P,ertr;=nd. J. "Production and inventory c~mttol with Ihe ba~;e ~tock ,~ystem", paper pre~ented al the TIMS XXV! Conference, Copenhagen, June 1984 [21] Whybark. ~., ~.qd J. Williamg. "'Material :eqmrement,, plannir, g under uncertainty". Dectswn Sc'tences 7¢4) ¢1976) 595-636.