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Choosing optimal buffering strategies for dealing with uncertainty in MRP
JOURNAL
OF OPERATIONS
MANAGEMENT-COMRINED
ISSUE
Vol. 7, Nos, I and 2. October 1987
Choosing Optimal Buffering Strategies Dealing with ~n~ertaiuty in MRP EISENHOWER
for
C. ETIENNE*
EXECUTIVE SUMMARY Dealing with uncertainty in the material requirements plan (MRP) is important, because of its importance in meeting the requirements of the master produ~ion schedule (MT’S).in MRP, there is a critical need to make the MPS predictable and consistent. In the early days of MRP, a philosophy of zero safety stock or safety time was rigorously advocated. It was thought to be more advantageous to eliminate the need for buffering by freezing the MPS and managing lead times to make these firm. However, it was soon recognized that it is extremely difficult, in practice, to eliminate all unce~~nty that affects the MPS and its execution. Practitioner know that some level of buffering is necessary (or inevitable) given the vicissitudes of the market place, production system planning and control, and supplier performance. This article accepts these premises and proposes a model and procedures for dealing with the buffering problem and choosing optimal buffering strategies in MRP.
ANATOMY
OF UNCERTAINTY
Strategies for buffering against uncertainty must of necessity consider the nature of the factors that generate uncertainty in a material requirements planning (MRP) system. The framework found in Whybark and Williams’ is one useful way to dissect the character of uncertainty. Figure 1 shows an expansion of the framework. Essentially, the model argues for describing uncertainty along two dimensions: (1) the nature of the uncertainty, i.e., whether it is caused by variations in timing or by variations in quantity, and (2) the source of the uncertainty, i.e., whether it derives from demand or supply phenomena. It is interesting to note that the real distinction between the categories of uncertainty, in terms of their buffering impact, lies in the timing versus quantity classification. Figure 2 shows this for the case of timing for both demand and supply. In the first case, the master production schedule (MPS), and consequently the MRP, moves back two periods, from period six to period four. For safety time to be a valid buffer, the material requirements should be advanced two periods, i.e., rather than plan for the MPS at period six, plan it for period four. On the average, the MPS will hold firm at period six, but it will shift to period four from time to time. If the material is planned for period six, there will be a shortage whenever the MPS shifts * University of Montreal, Montreal, Quebec, Canada. ’ D.C. Whybark and J.G. Williams, “Material Requirements Planning Under Uncertainty,” Decision Sciences, Vol. 7, No. 4, (October 1976).
Journal of Operations
Management
107
FIGURE 1 Anatomy of Uncertainty Sources
Demand Timing
l
SUPPlY
The master production schedule shifts from one period to the next.
Supplier lead times or manufacturing cycle ‘times are longer than planned.
l
Types l
Quantity l
The master production schedule quantity is greater than planned.
The supplier ships less than the quantity ordered.
l
* The inventory records overstate the amount of physical inventory.
The bill of materials quantity, particularly scrap levels, is greater than planned.
* Defect rates than planned.
are
greater
FIGURE 2 Schema of Timing Uncertainty Q4
c
$
I ’\
I\
I
I
; \
I
I ’\
I
i
’
1
2
’I
;
I
\
4
5
MPS shifts to an earlier period
108
\
f \
‘, \
I
0
\
: 6
7
\ \ I
1
\
8
9
10
11
12
13
* Time
The supplier ships at a later period
APICS
to an earlier date-to period four, for example. This situation can be handled by building a two-week safety time in the lead time before time phasing from the MPS due date, The result would be that the material would arrive at period four but would be needed at period six, most of the time. However, in cases when the master schedule did shift to an earlier period, the material would be available to support MPS activity. The same reasoning applies to lead time unce~ainty. Assume that the normal lead time would result in the material being planned for week six, and that it was needed at week six. In the event that there was lead time uncertainty and the material arrived at week eight, a shortage would exist. Buffering the material plan with two weeks safety time would mean that the material would be planned to arrive at week four instead of week six. It would be held idle for two weeks most of the time, but would effectively eliminate out-of-stock situations. The conclusion is that both types of timing uncertainty can be handled using safety time. Timing uncertainty at the level of the MPS can be buffered by planning it earlier than the normal due date. Similarly, supplier delivery unce~ainty can be handled by increasing the lead time used for time phasing and thus plan material deliveries earlier than normally needed. The important thing to note is that the cost structure for buffering against timing uncertainty is the same for both supply and demand. We will now proceed to develop the relevant cost equations for buffering against timing uncertainty. This is a prerequisite to systematic evaluation of buffering options. DEALING
WITH
UNCERTAINTY
There are obviously three general ways to react to uncertainty in material requirements plans. One is to ignore the uncertainty and incur the cost associated with material shortages. Shortage costs can be very high, particularly in a production system that is managed on a time-critical basis as when MRP is used to plan production and materials. Shortages have a pervasive impact on the production system, often resulting in rescheduling or failure to meet delivery promises. Ignoring unce~ainty is very seldom the valid strategy. A second way of dealing with uncertainty is to have firm plans and schedules. In the early days of MRP, this was advocated as the preferred approach. The basic philosophy is that the first reaction to changes in critical events that affect the validity of the MPS should be to replan and reschedule, until one enters a time frame that does not permit replanning. At this point, plans should be firm. One must concede, however, that no matter how much anticipation and replanning is done, there are still events that are unpredictable and more or less uncontrollable. Given the high cost of shortages and the impossibility of eliminating uncertainty, some level of buffering is necessary. Buffering is management’s attempt to incorporate uncertainty in the fo~ulation of plans and schedules. In the case of material requirements plans, two major buffering strategies have received serious attention in the literature-safety time and safety stock. The major problem in buffering a material requirements plan becomes selecting one of these two buffering strategies so as to achieve a target service level, at minimum cost. The search for the optimal strategy is simpli~ed by the fact that there are only two competing options. Whybark and Williams used a simulation procedure to investigate which of the two buffering options was preferable. The best strategy was defined as the one that maximizes the service level for a given investment in inventory. The results of their experiments showed
Journal of Operations
Management
109
that safety time is the preferable strategy for dealing with timing uncertainty while safety stock gave superior results when there was quantity uncertainty. However, some methodological problems remain. First, the choice of buffering strategy was not related to the requirements structure. The term “requirements structure” refers to the relative density (sparsity) of the material requirements plan. Although this issue has not been dealt with in the literature, one can conjecture that the requirements structure may be a significant variable in the choice of buffering strategy. Secondly, the study investigated the choice of buffering mechanism considering either timing or quantity uncertainty. Cases where both types of uncertainty exist simultaneously were not evaluated. And finally, the simulation procedure gives general guidelines for selecting a buffering strategy, but cannot select an optimal strategy for each particular situation. The present study will atttempt to deal with the problems associated with the Whybark and Williams study by doing four things: 1. Modeling the buffering problem schematically and mathematically. 2. Developing decision rules for choosing an optimal buffering strategy. 3. Investigating the impact of the service level and the requirements structure on the choice of optimal buffering strategy. 4. Developing optimal strategies for dealing with simultaneous timing and quantity uncertainty.
BUFFERING The
AGAINST of the
UNCERTAINTY Time Strategy
article investigates option for case of lead is longer planned. According arguments the same demand generated Let us is normal parameters p u. The time is that the service P. Let = holding per unit week (period), Q = size of material to the master The cost incurred using the holding is incurred time the include safety every time is time
where C, = total holding Equation per week.
110
in a
time strategy is ordered and a
uncertainty, the the problem of lead k, is parameter
cycle. be easily The the lead is adjusted order release.
C=hX(k,a)XQ
(1)
hQ(k,d
(2)
cost for a typical scheduling
1 is correct,
generated presented that the to k,a,
cycle.
since the system holds Q units for kc weeks at a cost of h dollars
APICS
The Cost of the Safety Stock Strategy Development of the cost function for the safety stock strategy for dealing with timing uncertainty requires specification of how material requirements are likely to vary. When the parent product is planned in the MPS, the system needs enough material to support normal MPS activity (Q). However, when the parent product is not planned in the MPS, the material requirement is zero, since the material requirement is either Q or 0. The safety stock must be Q if the target service level is 100%. For any service level, P, the required safety stock is given as SS, = PQ (3) where SS, = safety stock for timing uncertainty. According to equation 3, if the material does not arrive on time, the system will need PQ units of safety stock to support normal MPS activity at the target service level, P. With the safety stock option, there is no safety time, which is tantamount to planning for the average lead time. On the basis of the assumption of a normally distributed lead time, 50% of the time the material will arrive on time, and there will be no need for the safety stock. Similarly, the material will not arrive as planned 50% of the time and the system will use some or all of the safety stock. Thus, the expected level of the safety stock, SS,, is given by the following expression. SS,=
SPQ+T) (
=- 3PQ
4 The cost of holding the safety stock during a typical scheduling C
s
=3&h 4 =-
d
Choosing
(utilization)
the Optimal
(6)
3PQ2h (7)
4d and d is the demand
cycle is given by C,, where
per period (week).
Strategy
Equations 2 and 7 form the basis for selecting an optimal buffering strategy for timing uncertainty. To do so, it is sufficient to compare the values of the two cost functions given in equations 2 and 7. The logical place to start is to find the indiffference point between safety time and safety stock. At the point of indifference between the two basic buffering strategies, 3PQ2h hQ(k,d
=
Q=
Journal of Operations
Management
4d
4k,ad 3P
(9)
111
By similar reasoning, we can find when it is best to use safety time because, in order for safety time to be the best buffering strategy, 3PQ2h
hQ(kp4 < 7 Q’
4k,crd 3P
Also, safety stock is the best strategy when 4k,crd Q< 3P
(101 (11)
(12)
The decision rules for choosing between safety time and safety stock as buffering strategies for dealing with timing uncertainty can be summarized as follows: 4k,ud the manager is indifferent between safety time or safety stock as 1. WhenQ=3p, buffering strategy. 4k,od the cost minimizing buffering strategy is safety time. 2. When Q > 3p, 3. WhenQ
4k,cd
safety stock is the optimal buffering strategy.
Some interpretation of the decision rules is in order. Let us look at the case where safety time is the cost minimizing strategy, in which case, 4dk,a (13)
Q’3P
The value of 4d has a decisive impact on the outcome of the decision rule. However, in the case of a material requirements plan based on weekly time buckets, d is equal to the weekly utilization of the material. This means that if Q is greater than some function of 4d, the lot size of the material is greater than four weeks usage, in which case the material is appearing in the schedule less often than every four weeks. Clearly, there will be a lot of holes (empty weeks) in such a schedule, and if the system uses safety stock as a buffering strategy, it will be left idle for long periods (for the length of the typical hole, in fact). To minimize cost, the decision rule will avoid safety stock and use safety time. This holds for any given level of LT. The Service Level Effect The service level has an effect on two of the parameters of the model, k, and P, one in the numerator and the other in the denominator of equation 9. However, when the service level, P, is changed, the value of k, changes much faster. For example, for a 100% service level, the value of P is 1 and the value of k, is 3, approximately. But, if the service level is reduced to 9.5%, the value of P is 0.95, but the value of k, changes to 1.65. Thus, for a 5% change in the value of P, the value of k, goes down by 45%. Hence, when the service level is reduced, the model favors safety time as the cost minimizing strategy. The cost of the safety time strategy is dependent on k,, which changes exponentially with P, but the cost of the safety stock strategy is a linear function of P.
112
APES
TABLE 1 Basic Data for Analyzing Buffering_ Strategies Case A
Observed lead times Order #
I
10.0 9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
2 3 4 5 6 7 8 9 10
I
Standard lot size Average utilization per week
5000 1000
casec
CaseB 10.0 9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
I
2700 1000
10.0
9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
I
1000 1000
A Numerical Example An illustration of the application of the decision rule using a fictitious example is now given. The data in table 1 provide the basis for developing the relevant information. The example is based on variations in lead time as the source of uncertainty, but the same procedure applies to cases where variations in the MPS are the source of uncertainty. The first step is to calculate lead time variation. The standard deviation of the lead times shown in the table is 0.68. The optimal buffering strategy may be selected for the three cases shown in the table. The relevant computations and the optimal strategy for each case are shown in table 2. Table 3 also evaluates the optimal buffering strategies for each of the cases presented in table 1, but on the basis of a 70% service level. The impact of the service level is quite evident. The optimal strategy for case B changed from either safety time or safety stock as calculated in table 2, to safety time only. Schedule Density and Optimal Buffering Strategy The data in table 2 show that the optimal buffering strategy changed from safety time to safety stock from case A to case C. However, the only parameter that changed was the lot TABLE 2 Results of Applying the Decision Rules Data 4k,od 1. Decision parameter 3P 2. Lot size 3. Optimal strategy 4. Safety time 5. Scheduling cycle
Journal of Operations Management
Case A
CaseB
casec
2700
2700
2700
5000 Safety time 2.04 wks 5.0 wks
2700 Indifference 2.04 wks 2.7 wks
1000 Safety stock 2.04 wks 1 wk
113
TABLE 3 Results of Applying the Decision Rules for a 70% Service Level Data I. Decision parameter* 2. Lot size 3. Optimal strategy
*4k,od -= 3P
Case A
Case B
Case C
1300 5000 Safety time
1300 2700 Safety time
1300 1000 Safety stock
4X1x0.68x1000 3 x0.70 = 1300
size, Q, and consequently, the scheduling cycle, Q/d. Q/d is an index of scheduling density, since it is the length of time between two successive launchings of production of the item. When Q/d is small, production of the item appears often in the schedule so that the latter is dense. Moreover, it is clear that d is exogenously determined, so that schedule density is ultimately a function of the lot size. The data in the cases were used to construct the three corresponding schedules. In figure 3, the schedule for case C is indeed more compact than that for cases A and B. In case C, the material is appearing in the schedule continuously, albeit in smaller quantities. The critical thing, however, is that as one changes from a dense to a sparse schedule, the decision rules favor safety time over safety stock. This result is intuitively and logically correct. Safety stock is more or less permanently frozen in the system. In a sparse schedule, as in case A of figure 3, this results in holding the safety stock idle for long periods of time to provide protection for the few weeks it may be needed. Again using case A, out-of-stock conditions can only exist for four of the twenty weeks, since the material is active for only four weeks. The safety stock strategy for the sparse schedule would result in holding the safety stock idle for sixteen weeks, to have protection for four weeks. This is clearly a relatively costly buffering strategy. It is far better to use safety time and thereby minimize the incident of having an idle safety stock. One notes that the more sparse the schedule, the larger the lot size, and consequently, the larger the safety stock. If the target service level is lOO%, the safety stock would be 5000 units in case A, but only 1000 units in case C. The safety stock generates a double benefit when used in a dense schedule situation. The Impact of Firm Lead Times and Schedules Another way to test the theoretical and practical soundness of the model is to apply the decision rules to the case where the requirements plan is based on firm lead times and schedules. A frozen MPS is designed to make requirements plans firm. When lead times and schedules are firm, u = 0. Then, 4k,ad PZZ 0 (14) 3P
Q>O
114
(15)
APICS
FIGURE 3 Schedule Density and Optimal Buffering Strategy
Case A
LLL
0
4 Case
2
4
6
8
10
12
14
16
18
20
B
Q = 2700
and the decision
rule favors safety time. This might appear to be an anomaly after a superficial examination. To demonstrate that there is no anomaly, it is only necessary to compute the amount of safety time. As before, safety time is given by T, where T=3a =0
(16) (17)
which is what we would expect if there are firm lead times or schedules.
Journal of Operations
Management
115
BUFFERING
AGAINST
QUANTITY
UNCERTAINTY
The phenomenon of quantity uncertainty and its impact on the requirements schedule are shown in figure 4. The effect of buffering by safety time is also shown. In the case of’ week 5, the normal amount of materials needed to support MPS activity is Q2 units. However, the MPS level could change and thereby generate a material requirement of Q3 units. Consequently, unless there is buffering, the system will be short Q3 - Q2 units. For purposes of illustration, we assume a safety time of two weeks. If we buffer the materials schedule using safety time, the order would be launched to arrive at week two. Clearly, shifting the normal material requirements to week two to incorporate safety time does not help, because the system would still experience a shortage of (Q3 - Q 2) units when and if the MPS did change and caused a change in requirements. The same reasoning applies to the case where the variation in materials is due to short deliveries by the suppliers. The normal quantity in period nine is Q6 units, and these would have been ordered. However, suppose that the supplier ships only Q, units. If we use safety time and had planned the order for week seven, we would have received Qs = Q7 units from the supplier, and the system would still be short (Q6 - Q7) units. To shed further light on the problem, the author constructed the requirements schedules in table 4. In the first part of the table, there is no variation in requirements so all needs are covered. The second part of the table changes the requirements of week 5 from 250 to 280 and applies a one week safety time. The system will be short thirty units despite the safety time buffer. The lower part of the table holds thirty units of safety stock, which is an effective buffer, despite the absence of safety time.
FIGURE 4 The Impact of Quantity Uncertainty
Q Variation due to MPS
Variation supplier
in delivery
1’\ I’ I A
0
116
I 1
2
’\ 3
4
-5
6
7
8
9
10
11
APICS
TABLE
4
Safety Stock and Safety Time for Quantit: L
2
1 -
Gross requirements Scheduled receipts Planned receipts Available on hand Planned order
Gross requirements Scheduled receipts Planned receipts Available on hand Planned order
-
4
3 200 -
Uncertain
-
200
200 -
250
-
-
200
-
-
220 -
200
-
250
-
Gross requirements
-
-
200
-
Scheduled receipts Planned receipts Available on hand (Safety stock = 30) Planned order
-
-
200 30
-
-
250
250 -
I
6
5
-
220 -
220
220 -
-
220
-
220 -
-
220
250 -
-
220 -
-
220
-
250 -
Lead time = 1 week (280)* 250 250 220
-
Lead time = I week Safety time = 1 week
30 -
30 200
(280)* 250 -
-
Lead time = 1 week * The gross requirement
is 250 but it could be 280.
Evidently, whenever one is dealing with quantity uncertainty, whether the source be variation in MPS quantities or discrepancies in supplier shipments, the safety time strategy does not help much unless there is the unique situation of a very dense schedule, a long lead time, and a safety time that is much greater than the scheduling cycle all occurring simultaneously. In that case, successive lot sizes of the material can be swapped, and material shortages and overages are likely to cancel each other. This situation is likely to be so rare, that very little is lost, from a theoretical and practical standpoint, by eliminating it from consideration. For buffering against quantity uncertainty, one is left with the safety stock strategy, and the important consideration is how to implement it. Let us consider, for the time being, the case where variations in quantity are the only source of uncertainty. The implementation of the safety stock strategy can take one of two forms: carrying the safety stock permanently in the system, or building the safety stock into the order for the material. The latter implementation strategy is evidently superior, particularly when the schedule is sparse, since it eliminates the existence of a safety stock that is idle during periods when the material is inactive.
Journal of Operations
Management
117
Buffering Against Simultaneous
Timing and Quantity Uncertainty
A systematic and logical combination of the optimal buffering strategies for either timing or quantity unce~~nty gives an overall optimal strategy for buffering against the two types of uncertainty simultaneously. The evolution of these optimal strategies is an important benefit of the models proposed here and capitalizes on all the observations developed up to this point. First, the optimal strategy for dealing with timing uncertainty. If, on the basis of the decision rules of equations 9 through 12, the optimal strategy for dealing with timing uncertainty is to use safety time, then the most logical way to deal with both timing and quantity uncertainty is to build the safety stock for quantity uncertainty into the order for the material (i.e., the optimal strategy for dealing with quantity uncertainty) and then apply the safety time to the augmented order. Thus, the augmented order quantity is given by
Q,, where,
Qn=Q+%
(18)
where Q = normal order size for the material, and SS, = safety stock necessary to protect against quantity variation, given the target service level. The effect of this procedure is shown in figure 5. However, if the optimal strategy for dealing with timing is to use safety stock, then we freeze PQ units of safety stock quasi-~~anently in the system, and augment the normal order for the material by the amount of safety stock needed to deal with quantity uncertainty,
FIGURE 5
Safety Time and Safety Stock
4
r
Safety tine
w
w--w
0123456789
118
APICS
given the target service level, and avoid applying safety time. By so doing, the manager would exploit the advantages of the dense schedule, from a buffering viewpoint, and would also minimize the cost of buffering against quantity uncertainty. Figure 6 shows the procedure schematically. What emerges is a strategy that is tantamount to a minimax inventory system. If in any one cycle, the safety stock, SS, , is not used, it will be applied to reduce the augmented order for the next scheduling cycle, i.e., only the normal quantity, Q, will be ordered. Consequently, the maximum inventory on hand is given by the following equation:
Q max=Q+PQ+SSq The order size is always Qmax less the inventory
(19)
on hand.
CONCLUSION This article has presented a systematic way to model and deal in MRP, and minimize the cost of dealing with uncertainty. gained by evaluating and analyzing the model. First, no buffering in all situations. In the case of timing uncertainty, the choice unequivocally dependent on the density of the material schedule,
with the buffering problem Some useful insights were strategy is clearly superior of an optimal strategy is safety time being preferable
FIGURE 6 Parallel Safety Stock Strategies for Buffering Against Uncertainty
5
6
7
8
9
10
SS for timing uncertainty = PQ
Journal of Operations
Management
119
in sparse schedules. This result may prove to be significant to the management of an MRP system. Secondly, the model argues for first setting the target service level and then choosing a buffering strategy that minimizes the cost of achieving that service level. The service level turns out to be a significant factor in the choice of buffering strategy. Low service levels favor safety time, while high service levels favor safety stock. This is contrary to the position taken by Whybark and Williams. Thirdly, the article proposes a systematic way to handle the problem of quantity and simultaneous timing and quantity uncertainty. A major conclusion is that safety time is useless as a buffering strategy for quantity uncertainty. Finally, a procedure was outlined for dealing with both types of uncertainty simultaneously. The critical issue, however, is that a set of simple decision rules have been developed that enables optimization of the buffering decision while capitalizing on the structure of the requirements schedule.
120
APES
OF OPERATIONS
MANAGEMENT-COMRINED
ISSUE
Vol. 7, Nos, I and 2. October 1987
Choosing Optimal Buffering Strategies Dealing with ~n~ertaiuty in MRP EISENHOWER
for
C. ETIENNE*
EXECUTIVE SUMMARY Dealing with uncertainty in the material requirements plan (MRP) is important, because of its importance in meeting the requirements of the master produ~ion schedule (MT’S).in MRP, there is a critical need to make the MPS predictable and consistent. In the early days of MRP, a philosophy of zero safety stock or safety time was rigorously advocated. It was thought to be more advantageous to eliminate the need for buffering by freezing the MPS and managing lead times to make these firm. However, it was soon recognized that it is extremely difficult, in practice, to eliminate all unce~~nty that affects the MPS and its execution. Practitioner know that some level of buffering is necessary (or inevitable) given the vicissitudes of the market place, production system planning and control, and supplier performance. This article accepts these premises and proposes a model and procedures for dealing with the buffering problem and choosing optimal buffering strategies in MRP.
ANATOMY
OF UNCERTAINTY
Strategies for buffering against uncertainty must of necessity consider the nature of the factors that generate uncertainty in a material requirements planning (MRP) system. The framework found in Whybark and Williams’ is one useful way to dissect the character of uncertainty. Figure 1 shows an expansion of the framework. Essentially, the model argues for describing uncertainty along two dimensions: (1) the nature of the uncertainty, i.e., whether it is caused by variations in timing or by variations in quantity, and (2) the source of the uncertainty, i.e., whether it derives from demand or supply phenomena. It is interesting to note that the real distinction between the categories of uncertainty, in terms of their buffering impact, lies in the timing versus quantity classification. Figure 2 shows this for the case of timing for both demand and supply. In the first case, the master production schedule (MPS), and consequently the MRP, moves back two periods, from period six to period four. For safety time to be a valid buffer, the material requirements should be advanced two periods, i.e., rather than plan for the MPS at period six, plan it for period four. On the average, the MPS will hold firm at period six, but it will shift to period four from time to time. If the material is planned for period six, there will be a shortage whenever the MPS shifts * University of Montreal, Montreal, Quebec, Canada. ’ D.C. Whybark and J.G. Williams, “Material Requirements Planning Under Uncertainty,” Decision Sciences, Vol. 7, No. 4, (October 1976).
Journal of Operations
Management
107
FIGURE 1 Anatomy of Uncertainty Sources
Demand Timing
l
SUPPlY
The master production schedule shifts from one period to the next.
Supplier lead times or manufacturing cycle ‘times are longer than planned.
l
Types l
Quantity l
The master production schedule quantity is greater than planned.
The supplier ships less than the quantity ordered.
l
* The inventory records overstate the amount of physical inventory.
The bill of materials quantity, particularly scrap levels, is greater than planned.
* Defect rates than planned.
are
greater
FIGURE 2 Schema of Timing Uncertainty Q4
c
$
I ’\
I\
I
I
; \
I
I ’\
I
i
’
1
2
’I
;
I
\
4
5
MPS shifts to an earlier period
108
\
f \
‘, \
I
0
\
: 6
7
\ \ I
1
\
8
9
10
11
12
13
* Time
The supplier ships at a later period
APICS
to an earlier date-to period four, for example. This situation can be handled by building a two-week safety time in the lead time before time phasing from the MPS due date, The result would be that the material would arrive at period four but would be needed at period six, most of the time. However, in cases when the master schedule did shift to an earlier period, the material would be available to support MPS activity. The same reasoning applies to lead time unce~ainty. Assume that the normal lead time would result in the material being planned for week six, and that it was needed at week six. In the event that there was lead time uncertainty and the material arrived at week eight, a shortage would exist. Buffering the material plan with two weeks safety time would mean that the material would be planned to arrive at week four instead of week six. It would be held idle for two weeks most of the time, but would effectively eliminate out-of-stock situations. The conclusion is that both types of timing uncertainty can be handled using safety time. Timing uncertainty at the level of the MPS can be buffered by planning it earlier than the normal due date. Similarly, supplier delivery unce~ainty can be handled by increasing the lead time used for time phasing and thus plan material deliveries earlier than normally needed. The important thing to note is that the cost structure for buffering against timing uncertainty is the same for both supply and demand. We will now proceed to develop the relevant cost equations for buffering against timing uncertainty. This is a prerequisite to systematic evaluation of buffering options. DEALING
WITH
UNCERTAINTY
There are obviously three general ways to react to uncertainty in material requirements plans. One is to ignore the uncertainty and incur the cost associated with material shortages. Shortage costs can be very high, particularly in a production system that is managed on a time-critical basis as when MRP is used to plan production and materials. Shortages have a pervasive impact on the production system, often resulting in rescheduling or failure to meet delivery promises. Ignoring unce~ainty is very seldom the valid strategy. A second way of dealing with uncertainty is to have firm plans and schedules. In the early days of MRP, this was advocated as the preferred approach. The basic philosophy is that the first reaction to changes in critical events that affect the validity of the MPS should be to replan and reschedule, until one enters a time frame that does not permit replanning. At this point, plans should be firm. One must concede, however, that no matter how much anticipation and replanning is done, there are still events that are unpredictable and more or less uncontrollable. Given the high cost of shortages and the impossibility of eliminating uncertainty, some level of buffering is necessary. Buffering is management’s attempt to incorporate uncertainty in the fo~ulation of plans and schedules. In the case of material requirements plans, two major buffering strategies have received serious attention in the literature-safety time and safety stock. The major problem in buffering a material requirements plan becomes selecting one of these two buffering strategies so as to achieve a target service level, at minimum cost. The search for the optimal strategy is simpli~ed by the fact that there are only two competing options. Whybark and Williams used a simulation procedure to investigate which of the two buffering options was preferable. The best strategy was defined as the one that maximizes the service level for a given investment in inventory. The results of their experiments showed
Journal of Operations
Management
109
that safety time is the preferable strategy for dealing with timing uncertainty while safety stock gave superior results when there was quantity uncertainty. However, some methodological problems remain. First, the choice of buffering strategy was not related to the requirements structure. The term “requirements structure” refers to the relative density (sparsity) of the material requirements plan. Although this issue has not been dealt with in the literature, one can conjecture that the requirements structure may be a significant variable in the choice of buffering strategy. Secondly, the study investigated the choice of buffering mechanism considering either timing or quantity uncertainty. Cases where both types of uncertainty exist simultaneously were not evaluated. And finally, the simulation procedure gives general guidelines for selecting a buffering strategy, but cannot select an optimal strategy for each particular situation. The present study will atttempt to deal with the problems associated with the Whybark and Williams study by doing four things: 1. Modeling the buffering problem schematically and mathematically. 2. Developing decision rules for choosing an optimal buffering strategy. 3. Investigating the impact of the service level and the requirements structure on the choice of optimal buffering strategy. 4. Developing optimal strategies for dealing with simultaneous timing and quantity uncertainty.
BUFFERING The
AGAINST of the
UNCERTAINTY Time Strategy
article investigates option for case of lead is longer planned. According arguments the same demand generated Let us is normal parameters p u. The time is that the service P. Let = holding per unit week (period), Q = size of material to the master The cost incurred using the holding is incurred time the include safety every time is time
where C, = total holding Equation per week.
110
in a
time strategy is ordered and a
uncertainty, the the problem of lead k, is parameter
cycle. be easily The the lead is adjusted order release.
C=hX(k,a)XQ
(1)
hQ(k,d
(2)
cost for a typical scheduling
1 is correct,
generated presented that the to k,a,
cycle.
since the system holds Q units for kc weeks at a cost of h dollars
APICS
The Cost of the Safety Stock Strategy Development of the cost function for the safety stock strategy for dealing with timing uncertainty requires specification of how material requirements are likely to vary. When the parent product is planned in the MPS, the system needs enough material to support normal MPS activity (Q). However, when the parent product is not planned in the MPS, the material requirement is zero, since the material requirement is either Q or 0. The safety stock must be Q if the target service level is 100%. For any service level, P, the required safety stock is given as SS, = PQ (3) where SS, = safety stock for timing uncertainty. According to equation 3, if the material does not arrive on time, the system will need PQ units of safety stock to support normal MPS activity at the target service level, P. With the safety stock option, there is no safety time, which is tantamount to planning for the average lead time. On the basis of the assumption of a normally distributed lead time, 50% of the time the material will arrive on time, and there will be no need for the safety stock. Similarly, the material will not arrive as planned 50% of the time and the system will use some or all of the safety stock. Thus, the expected level of the safety stock, SS,, is given by the following expression. SS,=
SPQ+T) (
=- 3PQ
4 The cost of holding the safety stock during a typical scheduling C
s
=3&h 4 =-
d
Choosing
(utilization)
the Optimal
(6)
3PQ2h (7)
4d and d is the demand
cycle is given by C,, where
per period (week).
Strategy
Equations 2 and 7 form the basis for selecting an optimal buffering strategy for timing uncertainty. To do so, it is sufficient to compare the values of the two cost functions given in equations 2 and 7. The logical place to start is to find the indiffference point between safety time and safety stock. At the point of indifference between the two basic buffering strategies, 3PQ2h hQ(k,d
=
Q=
Journal of Operations
Management
4d
4k,ad 3P
(9)
111
By similar reasoning, we can find when it is best to use safety time because, in order for safety time to be the best buffering strategy, 3PQ2h
hQ(kp4 < 7 Q’
4k,crd 3P
Also, safety stock is the best strategy when 4k,crd Q< 3P
(101 (11)
(12)
The decision rules for choosing between safety time and safety stock as buffering strategies for dealing with timing uncertainty can be summarized as follows: 4k,ud the manager is indifferent between safety time or safety stock as 1. WhenQ=3p, buffering strategy. 4k,od the cost minimizing buffering strategy is safety time. 2. When Q > 3p, 3. WhenQ
4k,cd
safety stock is the optimal buffering strategy.
Some interpretation of the decision rules is in order. Let us look at the case where safety time is the cost minimizing strategy, in which case, 4dk,a (13)
Q’3P
The value of 4d has a decisive impact on the outcome of the decision rule. However, in the case of a material requirements plan based on weekly time buckets, d is equal to the weekly utilization of the material. This means that if Q is greater than some function of 4d, the lot size of the material is greater than four weeks usage, in which case the material is appearing in the schedule less often than every four weeks. Clearly, there will be a lot of holes (empty weeks) in such a schedule, and if the system uses safety stock as a buffering strategy, it will be left idle for long periods (for the length of the typical hole, in fact). To minimize cost, the decision rule will avoid safety stock and use safety time. This holds for any given level of LT. The Service Level Effect The service level has an effect on two of the parameters of the model, k, and P, one in the numerator and the other in the denominator of equation 9. However, when the service level, P, is changed, the value of k, changes much faster. For example, for a 100% service level, the value of P is 1 and the value of k, is 3, approximately. But, if the service level is reduced to 9.5%, the value of P is 0.95, but the value of k, changes to 1.65. Thus, for a 5% change in the value of P, the value of k, goes down by 45%. Hence, when the service level is reduced, the model favors safety time as the cost minimizing strategy. The cost of the safety time strategy is dependent on k,, which changes exponentially with P, but the cost of the safety stock strategy is a linear function of P.
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TABLE 1 Basic Data for Analyzing Buffering_ Strategies Case A
Observed lead times Order #
I
10.0 9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
2 3 4 5 6 7 8 9 10
I
Standard lot size Average utilization per week
5000 1000
casec
CaseB 10.0 9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
I
2700 1000
10.0
9.0 8.5 10.0 9.5 10.0 8.0 9.5 9.0 9.5
I
1000 1000
A Numerical Example An illustration of the application of the decision rule using a fictitious example is now given. The data in table 1 provide the basis for developing the relevant information. The example is based on variations in lead time as the source of uncertainty, but the same procedure applies to cases where variations in the MPS are the source of uncertainty. The first step is to calculate lead time variation. The standard deviation of the lead times shown in the table is 0.68. The optimal buffering strategy may be selected for the three cases shown in the table. The relevant computations and the optimal strategy for each case are shown in table 2. Table 3 also evaluates the optimal buffering strategies for each of the cases presented in table 1, but on the basis of a 70% service level. The impact of the service level is quite evident. The optimal strategy for case B changed from either safety time or safety stock as calculated in table 2, to safety time only. Schedule Density and Optimal Buffering Strategy The data in table 2 show that the optimal buffering strategy changed from safety time to safety stock from case A to case C. However, the only parameter that changed was the lot TABLE 2 Results of Applying the Decision Rules Data 4k,od 1. Decision parameter 3P 2. Lot size 3. Optimal strategy 4. Safety time 5. Scheduling cycle
Journal of Operations Management
Case A
CaseB
casec
2700
2700
2700
5000 Safety time 2.04 wks 5.0 wks
2700 Indifference 2.04 wks 2.7 wks
1000 Safety stock 2.04 wks 1 wk
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TABLE 3 Results of Applying the Decision Rules for a 70% Service Level Data I. Decision parameter* 2. Lot size 3. Optimal strategy
*4k,od -= 3P
Case A
Case B
Case C
1300 5000 Safety time
1300 2700 Safety time
1300 1000 Safety stock
4X1x0.68x1000 3 x0.70 = 1300
size, Q, and consequently, the scheduling cycle, Q/d. Q/d is an index of scheduling density, since it is the length of time between two successive launchings of production of the item. When Q/d is small, production of the item appears often in the schedule so that the latter is dense. Moreover, it is clear that d is exogenously determined, so that schedule density is ultimately a function of the lot size. The data in the cases were used to construct the three corresponding schedules. In figure 3, the schedule for case C is indeed more compact than that for cases A and B. In case C, the material is appearing in the schedule continuously, albeit in smaller quantities. The critical thing, however, is that as one changes from a dense to a sparse schedule, the decision rules favor safety time over safety stock. This result is intuitively and logically correct. Safety stock is more or less permanently frozen in the system. In a sparse schedule, as in case A of figure 3, this results in holding the safety stock idle for long periods of time to provide protection for the few weeks it may be needed. Again using case A, out-of-stock conditions can only exist for four of the twenty weeks, since the material is active for only four weeks. The safety stock strategy for the sparse schedule would result in holding the safety stock idle for sixteen weeks, to have protection for four weeks. This is clearly a relatively costly buffering strategy. It is far better to use safety time and thereby minimize the incident of having an idle safety stock. One notes that the more sparse the schedule, the larger the lot size, and consequently, the larger the safety stock. If the target service level is lOO%, the safety stock would be 5000 units in case A, but only 1000 units in case C. The safety stock generates a double benefit when used in a dense schedule situation. The Impact of Firm Lead Times and Schedules Another way to test the theoretical and practical soundness of the model is to apply the decision rules to the case where the requirements plan is based on firm lead times and schedules. A frozen MPS is designed to make requirements plans firm. When lead times and schedules are firm, u = 0. Then, 4k,ad PZZ 0 (14) 3P
Q>O
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(15)
APICS
FIGURE 3 Schedule Density and Optimal Buffering Strategy
Case A
LLL
0
4 Case
2
4
6
8
10
12
14
16
18
20
B
Q = 2700
and the decision
rule favors safety time. This might appear to be an anomaly after a superficial examination. To demonstrate that there is no anomaly, it is only necessary to compute the amount of safety time. As before, safety time is given by T, where T=3a =0
(16) (17)
which is what we would expect if there are firm lead times or schedules.
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BUFFERING
AGAINST
QUANTITY
UNCERTAINTY
The phenomenon of quantity uncertainty and its impact on the requirements schedule are shown in figure 4. The effect of buffering by safety time is also shown. In the case of’ week 5, the normal amount of materials needed to support MPS activity is Q2 units. However, the MPS level could change and thereby generate a material requirement of Q3 units. Consequently, unless there is buffering, the system will be short Q3 - Q2 units. For purposes of illustration, we assume a safety time of two weeks. If we buffer the materials schedule using safety time, the order would be launched to arrive at week two. Clearly, shifting the normal material requirements to week two to incorporate safety time does not help, because the system would still experience a shortage of (Q3 - Q 2) units when and if the MPS did change and caused a change in requirements. The same reasoning applies to the case where the variation in materials is due to short deliveries by the suppliers. The normal quantity in period nine is Q6 units, and these would have been ordered. However, suppose that the supplier ships only Q, units. If we use safety time and had planned the order for week seven, we would have received Qs = Q7 units from the supplier, and the system would still be short (Q6 - Q7) units. To shed further light on the problem, the author constructed the requirements schedules in table 4. In the first part of the table, there is no variation in requirements so all needs are covered. The second part of the table changes the requirements of week 5 from 250 to 280 and applies a one week safety time. The system will be short thirty units despite the safety time buffer. The lower part of the table holds thirty units of safety stock, which is an effective buffer, despite the absence of safety time.
FIGURE 4 The Impact of Quantity Uncertainty
Q Variation due to MPS
Variation supplier
in delivery
1’\ I’ I A
0
116
I 1
2
’\ 3
4
-5
6
7
8
9
10
11
APICS
TABLE
4
Safety Stock and Safety Time for Quantit: L
2
1 -
Gross requirements Scheduled receipts Planned receipts Available on hand Planned order
Gross requirements Scheduled receipts Planned receipts Available on hand Planned order
-
4
3 200 -
Uncertain
-
200
200 -
250
-
-
200
-
-
220 -
200
-
250
-
Gross requirements
-
-
200
-
Scheduled receipts Planned receipts Available on hand (Safety stock = 30) Planned order
-
-
200 30
-
-
250
250 -
I
6
5
-
220 -
220
220 -
-
220
-
220 -
-
220
250 -
-
220 -
-
220
-
250 -
Lead time = 1 week (280)* 250 250 220
-
Lead time = I week Safety time = 1 week
30 -
30 200
(280)* 250 -
-
Lead time = 1 week * The gross requirement
is 250 but it could be 280.
Evidently, whenever one is dealing with quantity uncertainty, whether the source be variation in MPS quantities or discrepancies in supplier shipments, the safety time strategy does not help much unless there is the unique situation of a very dense schedule, a long lead time, and a safety time that is much greater than the scheduling cycle all occurring simultaneously. In that case, successive lot sizes of the material can be swapped, and material shortages and overages are likely to cancel each other. This situation is likely to be so rare, that very little is lost, from a theoretical and practical standpoint, by eliminating it from consideration. For buffering against quantity uncertainty, one is left with the safety stock strategy, and the important consideration is how to implement it. Let us consider, for the time being, the case where variations in quantity are the only source of uncertainty. The implementation of the safety stock strategy can take one of two forms: carrying the safety stock permanently in the system, or building the safety stock into the order for the material. The latter implementation strategy is evidently superior, particularly when the schedule is sparse, since it eliminates the existence of a safety stock that is idle during periods when the material is inactive.
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117
Buffering Against Simultaneous
Timing and Quantity Uncertainty
A systematic and logical combination of the optimal buffering strategies for either timing or quantity unce~~nty gives an overall optimal strategy for buffering against the two types of uncertainty simultaneously. The evolution of these optimal strategies is an important benefit of the models proposed here and capitalizes on all the observations developed up to this point. First, the optimal strategy for dealing with timing uncertainty. If, on the basis of the decision rules of equations 9 through 12, the optimal strategy for dealing with timing uncertainty is to use safety time, then the most logical way to deal with both timing and quantity uncertainty is to build the safety stock for quantity uncertainty into the order for the material (i.e., the optimal strategy for dealing with quantity uncertainty) and then apply the safety time to the augmented order. Thus, the augmented order quantity is given by
Q,, where,
Qn=Q+%
(18)
where Q = normal order size for the material, and SS, = safety stock necessary to protect against quantity variation, given the target service level. The effect of this procedure is shown in figure 5. However, if the optimal strategy for dealing with timing is to use safety stock, then we freeze PQ units of safety stock quasi-~~anently in the system, and augment the normal order for the material by the amount of safety stock needed to deal with quantity uncertainty,
FIGURE 5
Safety Time and Safety Stock
4
r
Safety tine
w
w--w
0123456789
118
APICS
given the target service level, and avoid applying safety time. By so doing, the manager would exploit the advantages of the dense schedule, from a buffering viewpoint, and would also minimize the cost of buffering against quantity uncertainty. Figure 6 shows the procedure schematically. What emerges is a strategy that is tantamount to a minimax inventory system. If in any one cycle, the safety stock, SS, , is not used, it will be applied to reduce the augmented order for the next scheduling cycle, i.e., only the normal quantity, Q, will be ordered. Consequently, the maximum inventory on hand is given by the following equation:
Q max=Q+PQ+SSq The order size is always Qmax less the inventory
(19)
on hand.
CONCLUSION This article has presented a systematic way to model and deal in MRP, and minimize the cost of dealing with uncertainty. gained by evaluating and analyzing the model. First, no buffering in all situations. In the case of timing uncertainty, the choice unequivocally dependent on the density of the material schedule,
with the buffering problem Some useful insights were strategy is clearly superior of an optimal strategy is safety time being preferable
FIGURE 6 Parallel Safety Stock Strategies for Buffering Against Uncertainty
5
6
7
8
9
10
SS for timing uncertainty = PQ
Journal of Operations
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119
in sparse schedules. This result may prove to be significant to the management of an MRP system. Secondly, the model argues for first setting the target service level and then choosing a buffering strategy that minimizes the cost of achieving that service level. The service level turns out to be a significant factor in the choice of buffering strategy. Low service levels favor safety time, while high service levels favor safety stock. This is contrary to the position taken by Whybark and Williams. Thirdly, the article proposes a systematic way to handle the problem of quantity and simultaneous timing and quantity uncertainty. A major conclusion is that safety time is useless as a buffering strategy for quantity uncertainty. Finally, a procedure was outlined for dealing with both types of uncertainty simultaneously. The critical issue, however, is that a set of simple decision rules have been developed that enables optimization of the buffering decision while capitalizing on the structure of the requirements schedule.
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APES