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MRP with uncertainty: a review and some extensions
International Journal of Production Economics, 25 ( 1991 ) 51-64
51
Elsevier
MRP with uncertainty: a review and some extensions D.N.P. Murthy and L. Ma Department of Mechanical Engineering, The University of Queensland, St. Lucia, Brisbane, Qld. 4072, Austraha (Received June 18, 1991; accepted July 30, 1991 )
Abstract Material Requirement Planning (MRP) is a systems approach used in production processes for planning. Many forms of uncertainty affect the production process and different approaches have been advocated for MRP with uncertainty. In this paper we carry out a review of this literature and discuss the current research of the authors on MRP with uncertainty due to quality variations in the production process.
1. Introduction Material Requirements Planning (MRP) is a systems approach to production planning in complex multistage production systems. The study of MRP has received a lot of attention and the literature on the subject is vast. The early literature dealt with deterministic MRP - that is, MRP in a deterministic framework. In the real world, many forms of uncertainty affect the production process. As such deterministic MRP is inappropriate for most situations, this led to the development of MRP with uncertainty - that is, MRP in a stochastic framework. The literature on MRP with uncertainty is considerable, as different approaches have been advocated to cope with different forms of uncertainty. In this paper, we focus our attention on MRP with uncertainty and carry out a review of the literature and discuss some topics currently being investigated by the authors. The outline of the paper is as follows. We commence with a brief overview of deterministic MRP in Section 2. In Sections 3 and 4, we discuss MRP with uncertainty and the alternate approaches proposed for planning with uncertainty. The review of literature is carried out in Section 5. In Section 6 we discuss the current research being carried out by the authors which deals with MRP with uncertainty due to quality variations in the production
process. This research links the literature on quality control (QC) to MRP. Finally, in the last section we discuss some topics for further research.
2. Material requirements planning The use of MRP is now well established in production control (e.g., Orlicky [ 1 ], Tersine [2] ). In this section, we give a brief description of MRP operation and discuss its role as a management tool.
2.1 Description of MRP operation MRP is designed to provide answers to the following questions - (i) what materials and corn= ponents are needed and (ii) when and how many are needed to meet a specified demand or order. MRP provides a precise scheduling procedure, an efficient material control strategy, and a rescheduling mechanism. Fig. 1 shows the typical MRP system. The inputs to and the output of the system are as follows. Inputs: The three major inputs are as follows: • The master production schedule (MPS). • The product structure records. • The inventory status records. The master production schedule (MPS) is a statement of the orders for end product items. It
0925-5273/90/$03.50 © 1990 Elsevier Science Publishers B.V. All rights reserved.
52 Customer orders
)[Forecasts
Master Production Schedule
( I n d i c a t e s q u a n t i t y and timing of end products)
Product Structure Records (Contains b i l l of material
Inventory Status Records (Contains on-hand balance, open orders, lot sizes, lead times)
and show how product is
produced)
Material Requirements Planning BOM per MPS r e q u i r e m e n t , net out inventory levels, offset lead times and Issue orders)
(Explodes
I
Planned
Order
Releases
I( !
]
I
•
Purchase I order
I
Work orders
Reschedule notices
I Capacity requirements
Fig. 1. TypicalMRP system. indicates the quantity of each end product item that needs to be produced and the time period (or instants) when they are needed. The MPS is developed from end item forecasts and customer orders and is the driving force behind the M R P system. The bill of material (BOM), also known as "product structure", is a list of the different material and component inputs needed to produce a unit of the end product. It contains information on the relationships of parts, components and assemblies and shows the quantity of each and the order (or sequence) in which they are
required to produce a single unit of an end product. An accurate formal bill of material is needed for each different end product. Quantities of items (end product and lower level) in inventory at the start of the planning and available for use are referred to as "on-hand" inventories. The inventory status records contain the status data regarding on-hand inventory and on-order inventory. Output: The output of an M R P system is the planned order release which provides information on the time periods (or instants) and the quantities of input material, parts and compo-
53 nents to purchase and work orders for job shops. Using the output of the M R P system, the capacity requirements of the production system can be determined. The output of the M R P system also allows to compute cost estimates of end products, of work-in-process and inventory costs of raw material, components and subassemblies using appropriate cost figures. This is needed for optimizing the economic performance of the production process. 2.1.1 Operating logic The essential logic of M R P is as follows. Orders for end product items obtained from MPS that is, quantities and due dates for different end products - is exploded into a hierarchy of requirements for subassemblies, components and materials by working through the levels of product structures using the BOM. All the requirements to meet the orders are then aggregated into gross requirements in each time period. Using information about inventory stocks and items in processing, the M R P system computes the net requirements. The requirements are then offset in time to establish the dates by which actions must be initiated so that the components and subassemblies are available at different stages of manufacturing and the orders for end products are met by the times specified by the MPS. 2.2 M R P as a management tool The main function of M R P systems is to trigger purchases and factory orders to meet the ordered production requirements. In this context, it serves a useful role as a management tool (Schnenner [3]) and we discuss this in this subsection. 2.2.1 As a production planning tool As a production planning tool, M R P focuses the management attention on decisions with regards to MPS as it is the driving force for the M R P system. Once this is done, the M R P system converts the MPS into detailed plans which show the timing and quantity requirements for purchased and manufactured components.
2.2.2 As a scheduling device An important feature of M R P is the automating of the decisions involved in the scheduling of the production of component parts and in controlling the inventory levels of these component parts. When some unexpected developments occur - e.g., changes in order and/or in the production process, the M R P system automatically reschedules the production. 2.2.3 As a capacity planning tool The M R P system combines several types of information - (i) the planned production schedule and the materials needed to meet it, (ii) the time requirements for all components, and (iii) the particular work centers in the factory through which those parts and products travel, to predict the "work loads" at different work centers in the factory. Using the "work loads", the M R P system determines the capacity requirements needed. Often, the capacities at different centers are constrained. In this case, feedback of "work loads" from the M R P to the MPS can be used to alter the MPS so as not to violate the capacity constraints. In this "closed-loop" configuration of M R P and MPS, an effective linking of material requirement and production capacity is achieved. 2.2.4 To evaluate system performance The total cost of manufacturing is an indicator of the economic performance of the production process. The total cost is comprised of the following: (i) purchase costs - dependent on the number of times ordered and the quantities ordered, (ii) production costs - dependent on the number of lots and lot sizes, and (iii) inventory holding costs - dependent on the inventory control policy. Using these costs and MRP, one can compute estimates of the total cost. By optimal lot sizing and inventory control one can minimize the total cost and hence improve the economic performance of the process. For a good review of lot siz-
54 ing under dynamic demand conditions, see De Bodt & Van Wassenhove [ 32 ]. 3. MRP with uncertainty The deterministic MRP ignores uncertainty. In real life, production processes are influenced by many forms of uncertainty - e.g., uncertainty in customer orders for end products, uncertain delays due to failure of production process, and uncertainty in quantities delivered by suppliers, to name a few. In this section, we discuss the different types of uncertainty and the effect of uncertainty on the MRP system.
3.1 Types of uncertainty One can categorize the different types of uncertainties that affect production processes into two groups - (i) environmental uncertainty, and (ii) system uncertainty (Ho [5] ), as indicated in Fig. 2. Environmental uncertainty comprises of uncertainties beyond the production process. This includes (i) demand uncertainty due to uncertainty in customer orders and uncertainty in forecasting (also called forecast errors) and, (ii) supply uncertainty due to unreliable vendors. Note that the supply uncertainty can be either in
ENVIRONMENT
the quantities delivered a n d / o r the timing of the delivery. System uncertainty comprises of the uncertainties within the production process. These include operation yield uncertainty, production lead time uncertainty, quality uncertainty, failure of production system, and changes to product structure.
3.2 Impact of uncertainties on MRP system 3.2.1 Impact of environmental uncertainty Demand uncertainty leads to order modifications which in turn impacts on the gross quantities at the end product level. These changes propagate through the different stages of the production process and affect the production in all the remaining periods of the planning horizon. This order instability is called "nervousness" and affects the MRP system operation. Supply uncertainty affects the production of end product items in terms of causing delays a n d / o r the orders not being met in full.
3.2.2 Impact of system uncertainty System uncertainty leads to changes of scheduled receipts of MRP. This can result in the
UNCERTAINTY
•F o r e c a s t i n g e r r o r s •U n c e r t a i n t y in customer orders •U n c e r t a i n t y in v e n d o r supply
MRP SYSTEM •H i g h cost
rescheduling
• Increase in penalty cost •M R P n e r v o u s n e s s
SYSTEM
UNCERTAINTY
•V a r i a t i o n s in p r o d u c t quality •V a r i a t i o n s in p r o d u c t structure •V a r i a t i o n s in production lead time •E q u i p m e n t b r e a k d o w n •D y n a m i c lot s i z i n g
Fig. 2. U n c e r t a i n t y a n d effects on M R P .
55 planned orders not being met in terms of quantity delivered and/or time of delivery.
3.2.3 Cost implications As a consequence of the above-mentioned impacts, the total cost of manufacturing increases due to one or more of the following: (i) more frequent rescheduling, (ii) orders not being serviced effectively, (iii) unplanned setups for unplanned demands, (iv) loss of sales and goodwill, and (v) excess and insufficient inventories.
4. Approaches to dealing with uncertainty in MRP Different approaches to M R P have been advocated for coping with different forms of uncertainty. The four different approaches are as follows: ( 1 ) Safety stocks. (2) Safety lead times. ( 3 ) Hedging and overplanning. (4) Yield factors. We briefly discuss each of these approaches.
4.1 Safety stock In the safety-stock (also called inventorybuffer) approach the inventory levels are increased to provide a buffer against uncertainty. However, in the literature it is still unclear what a safety stock really means in the context of an M R P system. Two questions that need to be resolved are - (i) at what levels (end item level or component level) of the production process should safety stocks be held, and (ii) what should be the size of the safety stock. There is no commonly agreed answer to the first question. Given the levels at which safety stocks are to be held, two methods have been proposed to determine the size of safety stock. These are as follows. Economic approach: The safety stock sizes are selected to minimize the total cost comprising setting up cost, stock holding cost and penalties for possible shortages. Service level approach: Here the management first decides on the level of customer service they wish
to ensure and next the safety stock sizes are determined to ensure the specified service level.
4.2 Safety lead times In the safety lead times approach, order releases are brought forward from that stated in the requirements plan. As a result, the M R P schedules delivery of end products before they are due. Thus, if the process experiences uncertain delay due to any reason, then there is sufficient slack to absorb the delay if it is not too large. The safety lead time approach is mostly used for guarding against supplier uncertainties - that is, uncertainties at the input level where items are purchased from external suppliers who are not very reliable in terms of delivery times. The magnitude of safety time (the amount by which the release date is brought forward) depends on the context and regular adjustment is essential to ensure the desired level of service.
4.3 Hedging and overplanning In the hedging approach, the gross quantities of the MPS are altered. They are no longer simply the values obtained from demand forecasts, but include an extra amount so that the production process can cope with demand uncertainties. The amount of hedging depends on the production horizon and the demand uncertainties. In t h e overplanning approach, one simply makes more items than the orders specified in the MPS, so that the process can meet excess demand should it occur. This approach is similar to "scrap" or "yield loss" allowances at component levels employed in planning for processes where items at component level have to be scrapped when they are defective. Excess overplanning results in meeting the uncertain variations in demand but at the expense of increased inventory holding cost. Too little overplanning reduces the inventory holding costs, but can result in penalty costs due to demand being not met. Hence, the optimal choice of overplanning is critical to the economic performance of the process.
56
4.4 Yield factor The concept of yield factor is used to cope with system uncertainties causing loss of items (at end product and at intermediate levels). For each stage of the operation process, the yield factor is the ratio of the expected number of good items (or nondefective items) at the output to the number of input items. Thus for a single-stage item, the input quantity is determined by dividing the output demand by the yield factor. The yield factor is usually assumed to be constant and obtained from long-run observation of the production process. In a similar manner, a compound yield factor relates the input quantities needed to meet a specified output demand when system uncertainties result in losses occurring at different levels of the production process. The compound yield factor is a function of the yield factors at different stages of the process. 5. Review of the literature on M R P with uncertainty
We classify the literature on MRP with uncertainty into two groups based on: ( 1 ) the source of uncertainty and the method of planning with uncertainty, and (2) the method of analysis used.
5.1 Based on source of uncertainty and the method of planning In Section 3 we discussed the different sources of uncertainty which affect the production process and in Section 4 we discussed alternate approaches to coping with them. As a result, we can classify the papers based on the type of uncertainty and the method of planning as follows: ( 1 ) Environment uncertainty: This comprises of demand uncertainty and supply uncertainty. The approaches advocated for planning with these uncertainties are as follows: (i) Safety stocks: for demand and supply uncertainty in the quantities involved. (ii) Safety lead times: for demand and supply uncertainties in timing. (iii) Hedging: for demand uncertainty in quantities involved.
(iv)
Stochastic usage ratios: for optional parts planning with demand uncertainty (2) System uncertainty: This comprises uncertainties in yield, production quality, and lead time variations in production. Approaches to planning with these types uncertainties are as follows: (i) Safety stocks: for product quantity uncertainty at both end product and different component and subassembly levels. (ii) Safety lead times: for uncertain production lead time. (iii) Yield factor: for uncertain yield. (iv) Overplanning: for uncertainties in product quantity and quality.
5.2 Based on method of analysis Two approaches have been used in the analysis of models of MRP with uncertainty. They are as follows: ( i ) Analytical approach, and (ii) Simulation approach. The analytical approach uses analytical techniques for analysis and various related optimization problems - e.g., use of mathematical programming for optimal scheduling, stochastic optimization methods for optimal inventory control, etc. Most analytical models deal with only one type of uncertainty and simple structure for the production process. For more complex processes with many different end products and more than one type of uncertainty, the analytical approach is intractable. For such situations, simulation approach is the only viable alternative.
5.3 Classification of the literature Table 1 lists for each of the papers on MRP with uncertainty reviewed: (i) the topology of the production process (serial/nonserial; single/ multi-stage), (ii) the type of uncertainty included, (iii) the approach to planning with the uncertainties, (iv) method of analysis, and (v) the scheduling horizon for planning. In the remainder of the section, we review the papers in a chronological order. New [ 6] discusses the advantages and disadvantages of the safety-stock, safety-time and hedging approaches. In the safety stock up-
57 TABLE 1 Classification of literature Author(s)
Year
Topology
Uncertainty
Approach
Method of analysis
Schedule horizon
New
1975
-
E.U. ~, O.Y. 4
Conceptual analysis
finite
Whybark & Williams Melnyk & Piper De Bodt et al.
1976
-
E.U., S.U. 2
Simulation
finite
-
LT.U. 3 E.U.
Simulation Simulation
rolling
Huang et al.
1981 1982 1983a 19838 1982
Safety stock Safety time Hedging Safety stock Safety time Lead time allowance Safety stock
non-serial
LT.U.
finite
Biichel
1983
single
E.U.
Simulation ( Q-GERT ) Statistics
Billington, McClain & Thomas Burstein, Nevison & Carlson
1983
non-serial
E.U., S.U. (timing)
Mixed integer linear program
rolling
1984
serial
E.U. (timing)
Stochastic dynamic program
finite
Hegseth
1984
serial
O.Y.
New & Mapes
1984
serial
O.Y.
Grasso & Taylor III St. John
1984
non-serial
S.U.
1985
non-serial
Wijngaard & Wortmann
1985
single & non-serial
E.U., S.U. (timing) E.U., O.Y.
Wacker Huang, Rees & Taylor Carlson & Yano
1985 1985
non-serial non-serial
E.U., S.U. E.U., LT.U.
1986
non-serial
E.U.
Safety stock
Marlin
1986
non-serial
E.U., O.Y.
Yano
1987
non-serial
LT.U.
Safety stock Safety time Scrap allowance Safety time
Kurtulus & Pentico
1988
non-serial
O.Y.
Anderson & Lagodimos
1989
single
E.U.
E.U. - Environmental uncertainty. 2S.U. - System uncertainty. 3LT.U. - Lead time uncertainty. 40.y. - Operation yield uncertainty.
Stochastic Usage ratio
Compound Yield factor Safety stock Yield factor Safety time Safety stock
finite
finite finite Simulation Simulation
Safety stock Safety time Hedging MPS Safety stock
Yield factor Adjusted yield factor Compound-adjusted factor Safety stock
Statistics
finite
Statistics Simulation
finite finite
Heuristic Simulation Simulation
rolling rolling
nonlinear programming Simulation
Analytical model Simulation
finite
58 proach, the safety stock quantity is pre-set and requirements are triggered whenever forecast stock falls below service level. The safety lead time approach is suggested as an alternative to the safety stock approach and New suggests its use when the final products are produced infrequently. In the hedging approach, the increment to the requirements dependent on the average loss in production. Whybark and Williams [ 7 ] deal with demand, supply and system uncertainties and present a systematic study to provide guidelines for deciding on safety stocks and safety lead times. Using simulation approach, they first show that the quantity uncertainty is best dealt with using safety stock approach at the end product level while the timing uncertainty is best dealt with using safety lead times. They also show that safety stock approach results in increased inventory levels. Melnyk and Piper [ 8 ] deal with lead time uncertainty. They suggest lead time allowance (initially advocated by Whybark and Williams [ 7 ] ) as an effective method. They set planned lead time as the average observed lead time plus a multiple of the standard deviation of the lead time error distributions. Through simulation experiments they show how lead time allowances influence lot-sizing effectiveness and in turn how lot sizing influences the effectiveness of lead time allowances in MRP. De Bodt and Van Wassenhove [4,9,10] study lot sizing and safety stock decisions and the cost increments under demand uncertainty in singlelevel MRP with rolling horizon. They show that forecast errors have a tremendous effect on the cost effectiveness of lot sizing and safety stock decisions. They also report that the differences in the cost estimates are insignificant for different techniques (e.g., Silver-Meal heuristic and The Least-Unit-Cost heuristic) in the presence of forecast errors. For several lot-sizing heuristics, they give estimates of cost increases due to demand uncertainty. Their results show that safetystocks and lot-sizing policies are important to firms using MRP in an uncertain environment. Huang, Clayton and Moore [11] develop a simulation model to incorporate the basic MRP logic into a production process using Q-GERT network modeling and simulation language. Their
objective is to provide information necessary for planning and controlling of material and capacity requirements of a production process with uncertainty. The simulation model yields answers to questions related to production decisions at each station, production capacity and procurement lead times to meet a specified demand for end items. Bfichel [ 12 ] considers planning based on stochastic usage ratios for optional parts when demand for them is stochastic. The usage ratio for a specific component is the ratio of the demand for the component to the total demand for different end product. Small usage ratios (and/or small number of customer orders) results in considerable variations in demand which necessitate either high safety stocks. Biichel first develops a model of usage ratios which reveals the factors influencing the parameters of usage ratio distribution and then demonstrates how stochastic usage ratios may be included in a material requirements planning procedure to reduce demand uncertainty. Billington, McClaim and Thomas [ 13] deal with the interaction of lead times, lot sizing and capacity constraints for a process producing products with complex product structure with demand- and lead-time uncertainty. They propose a mixed integer-linear programming model for coping with capacity constraints and uncertain production lead times. Their model incorporates the following costs - current inventory holding cost, setup and overtime costs, and undertime production costs. The model computes the required production lead times based on the demands on available capacity, thereby reducing in-process inventory. The computational effort to obtain the solution is dependent on the model size and this is reduced by product structure compression. Burstein, Nevison and Carlson [ 14 ] consider a serial multi-stage production process with demand quantities known but with uncertainty in their timing. They propose the dynamic lot sizing approach, with lot sizing at each stage taking into account the subsequent stages of the operation. They use the stochastic dynamic programming technique to obtain the solution. Hegseth [ 15 ] considers a serial production
59 process with yield uncertainty. He uses a deterministic formulation involving different yield factors for different stages of the operation. The BOM is modified using yield factors and the material planning carried out after this modification. Since yield factors are deterministic, the model is deterministic. New and Mapes [ 16 ] also deal with uncertain yield losses. They consider processes with high losses and high variability in the losses - as for example, high-technology precision casting or integrated circuit production. They propose a simple model which relates output and input quantities through a random yield factor. They study safety-stock, safety-time and hedging approaches to coping with such losses. They argue that cost and risk of obsolescence reduce the effectiveness of the use of the safety stocks at the end product level. They suggest that the marketing environment (e.g., make-to-stock or make-to-order) is important for choosing the best approach to handle the variability in the yield rates. The key conclusion of their study is that different approaches are needed for different market conditions. Grasso and Taylor I I I [ 17 ] concentrate on examining the impact of operating policies on the M R P system with uncertainties resulting from lead time errors. Simulation approach is used to assess the impact of four factors - the lead time variability, the safety stock level, the safety lead time and lot-sizing rules, on the total cost. St. John [18] discusses the cost of inflated planned lead time on M R P system performance using a simulation model involving several stochastic variables such as customer demand, processing times (the sum of setup and run times). Their study demonstrates the need for lead time reduction, as long lead times result in high costs. Wijngaard and Wortmann [19 ] study different approaches to stochastic M R P with uncertainties. They introduce three ways to generate interstage slacks: (i) safety stock, (ii) safety time, and (iii) hedging. They consider a multi-stage production process with convergent and divergent nodes. The safety stock is determined using the variance of the forecast error, the yield and the average demand. They compare the relative advantages of safety-stock and safety-time approaches. An advantage of employing safety time
as opposed to safety stock is that it does not generate a safety stock unless there exists a real demand in the near future. A disadvantage of employing safety time is the need to extend the planning horizon at the MPS level. They also discuss the hedging approach to cope with quantity uncertainty in demand and report that it is possible to change hedging without changing the planned order release. This has the advantage of reducing the number of rescheduling messages. Wacker [20] deals with a model which estimates the mean and variances of outputs at the end item and at the component level due to uncertainties. He uses the safety stock approach. In made-to-order production operation, end item safety stock does not alleviate demand uncertainty. The model uses "forecast standard error" for components as an estimate of component safety stock. He comments that M R P system should not involve sophisticated control schemes to monitor environmental and system uncertainties, but should incorporate these variations in the system itself. Huang, Rees and Taylor [21] develop a queueing network simulation model with QGERT, which integrates M R P with control at the shop level in a production process. As such, it adds an extra dimension - control at floor shop level. They conduct several simulation experiments to study the effect of different factors on the total cost. Carlson and Yano [22] investigate an M R P system with rolling-horizon and uncertain demand. They assume that forecasting errors are normally distributed and analyze the need for emergency setups at the component level. They examine a heuristic method and report that there is a significant increase in the cost of lot sizing due to the need for emergency setups and that safety stocks at the component level reduce the need for emergency setups and hence are cost effective. Marlin [23] develops a MRP/job-shop stochastic simulation model coded in SIMSCRIPT II.5. The model closes the loop between order launching via M R P and order execution by an in-house manufacturing facility and by outside vendors. The model simulates two mutually exclusive types of demand uncertainty, viz., quan-
60 tity uncertainty and timing uncertainty. To cope with quantity uncertainty Marlin uses the yield factor approach developed by Hegseth [ 15 ]. Yano [24] deals with the stochastic lead time problem using an analytical approach. He first addresses the problem of determining optimal planned lead times in an assembly type operation with uncertain processing times. This is done using a complex nonlinear programming formulation, with the objective function involving inventory holding costs and tardiness costs. Kurtulus and Pentico [25] deal with yield uncertainty and extend the yield factor approach of Hegseth [ 15 ]. They propose three different versions of the "expected value" rule. In the first version (EXP1) the planned requirements X* (i.e., the number of units started of the item with i stages remaining) is obtained by dividing the net requirements X*_1 of the item by the yield factor Pi. The second version of the rule (EXP2) plans requirement in a similar manner. However, the actual yield factor is continuously monitored during simulation at regular intervals and is updated with new yield factor. The third version (EXP3) uses compound yield factors developed by Hegseth [15 ] with adjustment of the yield factor. Anderson and Lagodimos [26] consider the problem of predicting customer service levels in a single-stage MRP system where demand quantity is uncertain. They derive analytic expressions for service level measures often used in make-to-stock environments and study how service levels are affected by the level of safety stock held. They first develop a dynamic model of the MRP system which takes planned safety stock into account and then derive expressions for different service level measures. The results of the analysis are validated using a simulation model.
2.
3.
4.
5.
6.
7.
Some comments
8. Based on the literature review, the following comments can be made: 1. There are many different types of uncertainties which affect MRP and many different approaches have been advocated for coping with them. There is no paper which deals with MRP
with all different types of uncertainties in an integrated manner. The bulk of the literature deals, at a conceptual level, with specific methods to cope with specific types of uncertainty. There is very little comparative evaluation of the different methods for different types (or forms) of production systems. MRP with demand uncertainty has received more attention than MRP with other types of uncertainty. Most models assume that the demand variations are gaussian in distribution. This is a valid assumption when the quantities involved are large. Most papers dealing with the yield factor approach assume a constant yield factor for each stage of the production process. In real life this is seldom true. The yield factor depends on many factors such as age and condition of machines used in the production, quantities produced, maintenance and quality control actions, etc. In the safety stock approach, there are still many unresolved issues - e.g., the location of safety stocks for a process with fairly general product structure. Optimal decisions (e.g., optimal overplanning in the overplanning approach, optimal location of safety stock) depend critically on the stochastic characterization (i.e., type of distribution, parameters of the distribution, etc. ) of the uncertainty. Misspecification can have a significant impact on the optimal solution. There is no work on the robustness and sensitivity aspects of optimal solutions and this is an open topic for research. Although many papers deal with the use of simulation approach for study of MRP with uncertainty, there is very little work on the comparative evaluation of the advantages and disadvantages of different simulation languages. There are very few papers on real case studies, i.e., papers dealing with the application of different approaches to MRP with uncertainty in the real world. Such studies are important not only to evaluate the effectiveness of the different approaches but also to trigger further theoretical research.
61 9. Regarding optimization problems in the context of M R P with uncertainty, they are, in general, computationally very complex. As such, one needs to use heuristic methods to obtain solutions close to the optimal. Although this topic has received some attention, there is considerable scope and need for further research.
6. MRP with uncertain quality In real life, often a fraction of the end product items are defective. This could be due to defective components a n d / o r quality variations inthe production process. Let D denote the specified gross requirement at the end product level. One can use the classic M R P to compute the quantities of various components to be ordered to meet this requirement. If the input quantities ordered are such that only D end items are produced, then the number of nondefective end product items produced, 37, is a random variable assuming values in [0,D ]. If the defective items are not repairable, or too expensive to repair, they need to be scrapped. As a result, there is a possibility of demand not being met, the shortfall being the number of defective items produced. One way of coping with this form of uncertainty is the yield factor approach. The yield factor accounts for such losses in determining the input quantities to be ordered. As mentioned earlier, the yield factor is determined by long-run observations and roughly equals E [57]/D. Better material planning with uncertainty needs to incorporate quality variations in the process more effectively. In this context, the vast literature on Quality Control dealing with modeling of quality variations is of particular importance. Over the last two years the authors have been studying M R P with uncertain quality in a framework which links M R P with the quality control literature. In this section, we give an overview of this research. We commence with a brief discussion of the modeling of quality variations and then proceed to discussing alternate approaches to M R P with uncertain quality.
6.1 Modeling quality variations In a multi-stage production process, the output quality at each stage of the process can be characterized in terms of (i) individual item being defective or not, or (ii) a batch of items, with quality denoting either the number or the fraction of nondefective items in the batch. Also, a multistage production process can be characterized as a network of one or more of the three block types (called Type-l, Type-2 and Type-3 Blocks) shown schematically in Fig. 3. Type-1 Block has r (r>_ 2) different inputs and a single output and represents an assembly type operation where two or more components are assembled. The output quality of the block is a random variable and is a function of the input quality of the r different inputs and the uncertainty in operations involved. A type-2 Block represents a distribution stage and is characterized by a single input feeding to s ( s > 2 ) different branches on the output side. The output quality o f b r a n c h j ( 1 <_j
6.20verplanning approach In this approach, D, the order quantity for end items, is increased to D( 1 + ~ ) where ~ ( > O) is
62
input,/ --
output 1
input 1
output input j
)
) output j
input r
output s TYPE- 1 BLOCK
TYPE-2 BLOCK
output
input
)
TYPE-3 BLOCK
Fig. 3. Basic block types.
called the overplanning factor and the material planning is carried out using the classic M R P in the usual way. The critical factor is 3. If it is small, then the probability of the specified demand being not met (i.e., the number of nondefective items at the end level < D) is high and can result in heavy penalty cost. On the other hand, if 3 is large, then the probability of the demand being not met is low but this is achieved at the expense of greater manufacturing and/or inventory cost. In Murthy and Ma [27], ~ is selected optimally to achieve a sensible tradeoff between the two conflicting costs mentioned above.
6.30verplanning approach with lot sizing In Murthy and Ma [27 ] the quality variations at each stage are assumed to be independent of the quantities processed. In real life, the process quality at each stage depends on the lot size used. Smaller lot size implies better quality (e.g., see Porteus [28 ]. However, this results in increased costs due to the need for more setups. In Murthy and Ma [29 ], we consider two different models linking quality with lot size and develop an M R P system based on optimal lot sizing and overplanning.
6.40verplanning approach with inspection Often not all defective items need to be scrapped, as some can be rectified through some form of corrective action. This has an impact on the overplanning factor and in turn on the M R P system. In Ma and Murthy [ 30 ], we deal with the case where the items are inspected only at the end product level and defective items can be repaired. Depending on the nature of the defect, the defective item needs to be sent one or more stages back in the production process for rectification and hence the cost of rectification depends on the nature of the defect. As a result, the inspection scheme and the rectification strategy (e.g., decision to repair is not based on a cost limit policy) influence the number of defective items produced. This in turn affects the overplanning factor and the material requirement. In Ma and Murthy [30], we study optimal overplanning with different inspection and repair strategies.
7. Conclusion and topics for further study In this paper we have carried out a review of M R P with uncertainty and outlined the research
63 of the authors in M R P with uncertain quality. There are many topics which need further research. A variety of topics needing further study were identified in Section 5 (see "Some Comments" ). Hence, we shall not repeat them here. As commented earlier, there is no study of M R P incorporating the different types of uncertainties in an integrated manner. An effective M R P system must incorporate all types of uncertainties, the different actions to cope with them, and be automated to indicate best action. There is need for such an automated system (involving concepts from Expert Systems and Artificial Intelligence) and this should be of particular interest to the practitioners. In the remainder of the section, we discuss some extensions to our work reported in Section 6 and further topics for study in M R P with uncertain quality. Our research has so far been confined to production systems with no commonality at the component level. As a result, we could model quality variations for each end product type independent of the other. When there is component commonality, we need to model them together using multivariate formulations. This makes the problem more complex. Lead times can vary significantly when defective items are sent back to earlier stages for rectification. As a result, the lead times are statistically correlated with the product quality. This implies that yield and lead times are correlated. The study of M R P with this correlation is an open topic for research. In Ma and Murthy [30], we deal with inspection at the end item level. One can carry out the inspections at item level to either weed out defective items or repair them. The inspection and rectification scheme has an effect on the final output quality and this in turn affects the overplanning at the end item level. There is a need to study the effect of location of inspection points on the operation of the M R P system. In Murthy and Ma [29 ], we considered M R P with uncertain quality and the quality related to lot sizing. Moreover, maintenance actions affect not only quality but lead times as well. The study of maintenance actions on the performance of
M R P is an interesting topic for future research. As seen from the short list, there are many topics of both theoretical as well as practical interest that need attention in the future. References 1 Orlicky, J., 1975. Material Requirements planning. McGraw-Hill, New York. 2 Tersine, R.J., 1985. Production/Operations Management: Concepts, Structure and Analysis. North-Holland, New York. 3 Schmanner, R.W., 1984. Production/Operations Management: Concepts and Situations, 2nd edition. Science Res. Associates, Chicago. 4 De Bodt, M.A., Van Wassenhove, L.N. and Gelders, L.F,, 1982. Lot sizing and safety stocks decisions in an MRP system with demand undertainty. Eng. Costs Prod. Econ., 6: 67-75. 5 Ho, C., 1989. Evaluating the impact of operating environments on MRP system nervousness. Int. J. Prod. Res., 27: 1115-1135. 6 New, C., 1975. Safety stocks for requirements planning. Prod. Inventory Manage., 16(2): 1-18. 7 Whybark, D.C. and Williams, J.G., 1976. Material requirements planning under uncertainty. Decision Sci.. 7: 595-606. 8 Melnyk, S.A. and Piper, C.J., 1985. Lead time errors in MRP: The lot-sizing effect. Int. J. Prod. Res., 23: 253264. 9 De Bodt, M.A. and Van Wassenhove, L.N., 1983a. Cost increases due to demand uncertainty in MRP lot sizing. Decision Sci., 14: 345-362. 10 De Bodt, M.A. and Van Wassenhove, E.N., 1983b. Lot sizes and safety stocks in MRP: A case study. Prod. Inventory Manage., 24( 1 ): 1-16. 11 Huang, P.Y., Clayton, E.R. and Moore, L.J., 1982. Analysis of material and capacity requirements with Q-GERT. Int. J. Prod. Res., 20: 701-713. 12 Biichel, A., 1983. Stochastic Material Requirements Planning for Optional Parts. Int. J. Prod. Res., 21, 511527. 13 Billington, P.J., McClain, J.O. and Thomas, L.J., 1983. Mathematical programming approaches to capacityconstrained MRP systems: Review, formulation and problem reduction. Manage. Sci., 29:1126-1141. 14 Burstein, M.C., Nevison, C.H. and Carlson, R.C., 1984. Dynamic lot-sizing when demand timing is uncertain. Oper. Res., 32: 362-379. 15 Hegseth, M.A., 1984. The challenge of operation yield. Prod. Inventory Manage., 25 (1): 4-10. 16 New, C. and Mapes, J., 1984. MRP with high uncertain yield losses. J. Oper. Manage., 4:315-330. 17 Grasso, E.T. and Taylor, B.W., 1984. A simulation-based experimental investigation of supply/timing uncertainty in MRP systems. Int. J. Prod. Res., 22: 485-497. 18 St. John, R., 1985. The cost of inflated planned lead times in MRP systems. J. Oper. Manage., 5:119-128.
64 19 20
21
22
Wijngaard, J. and Wortmann, J.C., 1985. MRP and inventories, Eur. J. Oper. Res., 20: 281-293. Wacker, J.G., 1985. A theory of material requirements planning (MRP): An empirical methodology to reduce uncertainty in MRP systems. Int. J. Prod. Res., 23: 807824. Huang, P.Y., Rees, L.P. and Taylor, B.W., 1985. Integrating the MRP-based control level and the multistage shop level of a manufacturing system via network simulation. Int. J. Prod. Res., 23:1217-1231. Carlson, R.C. and Yano, C.A., 1986. Safety stocks in MRP Systems with emergency setups for components. Manage. Sci., 32: 403-412. Marlin, P.G., 1986. An MRP/job shop stochastic simulation model. In: B. Lev (ed.), Production Management: Methods and Studies. North-Holland, Amsterdam, pp. 163-172. Yano, C.A., 1987. Stochastic lead times in two-level assembly systems. IIE Trans., 19:371-377. Kurtulus, I. and Pentico, D.W., 1988. Material Requirement Planning when there is scrap loss. Prod. Inventory Manage., 29(2): 18-21, -
23
24 25
26
Anderson, E.J. and Lagodimos, A.G., 1989. Service levels in single-stage MRP systems with demand uncertainty. Eng. Costs Prod. Econ., VI 7:125-133. 27 Murthy, D.N.P. and Ma, L., 1991a. MRP with uncertain product quality. Report TMC-91/1, Technology Management Centre, The University of Queensland. 28 Porteus, E.L, 1986. Optimal lot sizing, process quality improvement and setup cost reduction. Oper. Res., 34(1): 137-144. 29 Murthy, D.N.P. and Ma, L., 199 lb. MRP and lot sizing with uncertain production quality. Report TMC-91/2, Technology Management Centre, The University of Queensland. 30 Ma, L. and Murthy, D.N.P., 1991. MRP with uncertain product quality and quality control. In preparation. 31 Blackburn, J.D., Kropp, D.H. and Millen, R.A., 1985. MRP system nervousness: Causes and cures. Eng. Costs Prod. Econ., 9: 141-146. 32 De Bodt, M.A., Gelders, LF. and Van Wassenhove, L.N., 1984. Lot-sizing under dynamic demand conditions: A review. Eng. Costs Prod. Econ., 8: 165-187.
51
Elsevier
MRP with uncertainty: a review and some extensions D.N.P. Murthy and L. Ma Department of Mechanical Engineering, The University of Queensland, St. Lucia, Brisbane, Qld. 4072, Austraha (Received June 18, 1991; accepted July 30, 1991 )
Abstract Material Requirement Planning (MRP) is a systems approach used in production processes for planning. Many forms of uncertainty affect the production process and different approaches have been advocated for MRP with uncertainty. In this paper we carry out a review of this literature and discuss the current research of the authors on MRP with uncertainty due to quality variations in the production process.
1. Introduction Material Requirements Planning (MRP) is a systems approach to production planning in complex multistage production systems. The study of MRP has received a lot of attention and the literature on the subject is vast. The early literature dealt with deterministic MRP - that is, MRP in a deterministic framework. In the real world, many forms of uncertainty affect the production process. As such deterministic MRP is inappropriate for most situations, this led to the development of MRP with uncertainty - that is, MRP in a stochastic framework. The literature on MRP with uncertainty is considerable, as different approaches have been advocated to cope with different forms of uncertainty. In this paper, we focus our attention on MRP with uncertainty and carry out a review of the literature and discuss some topics currently being investigated by the authors. The outline of the paper is as follows. We commence with a brief overview of deterministic MRP in Section 2. In Sections 3 and 4, we discuss MRP with uncertainty and the alternate approaches proposed for planning with uncertainty. The review of literature is carried out in Section 5. In Section 6 we discuss the current research being carried out by the authors which deals with MRP with uncertainty due to quality variations in the production
process. This research links the literature on quality control (QC) to MRP. Finally, in the last section we discuss some topics for further research.
2. Material requirements planning The use of MRP is now well established in production control (e.g., Orlicky [ 1 ], Tersine [2] ). In this section, we give a brief description of MRP operation and discuss its role as a management tool.
2.1 Description of MRP operation MRP is designed to provide answers to the following questions - (i) what materials and corn= ponents are needed and (ii) when and how many are needed to meet a specified demand or order. MRP provides a precise scheduling procedure, an efficient material control strategy, and a rescheduling mechanism. Fig. 1 shows the typical MRP system. The inputs to and the output of the system are as follows. Inputs: The three major inputs are as follows: • The master production schedule (MPS). • The product structure records. • The inventory status records. The master production schedule (MPS) is a statement of the orders for end product items. It
0925-5273/90/$03.50 © 1990 Elsevier Science Publishers B.V. All rights reserved.
52 Customer orders
)[Forecasts
Master Production Schedule
( I n d i c a t e s q u a n t i t y and timing of end products)
Product Structure Records (Contains b i l l of material
Inventory Status Records (Contains on-hand balance, open orders, lot sizes, lead times)
and show how product is
produced)
Material Requirements Planning BOM per MPS r e q u i r e m e n t , net out inventory levels, offset lead times and Issue orders)
(Explodes
I
Planned
Order
Releases
I( !
]
I
•
Purchase I order
I
Work orders
Reschedule notices
I Capacity requirements
Fig. 1. TypicalMRP system. indicates the quantity of each end product item that needs to be produced and the time period (or instants) when they are needed. The MPS is developed from end item forecasts and customer orders and is the driving force behind the M R P system. The bill of material (BOM), also known as "product structure", is a list of the different material and component inputs needed to produce a unit of the end product. It contains information on the relationships of parts, components and assemblies and shows the quantity of each and the order (or sequence) in which they are
required to produce a single unit of an end product. An accurate formal bill of material is needed for each different end product. Quantities of items (end product and lower level) in inventory at the start of the planning and available for use are referred to as "on-hand" inventories. The inventory status records contain the status data regarding on-hand inventory and on-order inventory. Output: The output of an M R P system is the planned order release which provides information on the time periods (or instants) and the quantities of input material, parts and compo-
53 nents to purchase and work orders for job shops. Using the output of the M R P system, the capacity requirements of the production system can be determined. The output of the M R P system also allows to compute cost estimates of end products, of work-in-process and inventory costs of raw material, components and subassemblies using appropriate cost figures. This is needed for optimizing the economic performance of the production process. 2.1.1 Operating logic The essential logic of M R P is as follows. Orders for end product items obtained from MPS that is, quantities and due dates for different end products - is exploded into a hierarchy of requirements for subassemblies, components and materials by working through the levels of product structures using the BOM. All the requirements to meet the orders are then aggregated into gross requirements in each time period. Using information about inventory stocks and items in processing, the M R P system computes the net requirements. The requirements are then offset in time to establish the dates by which actions must be initiated so that the components and subassemblies are available at different stages of manufacturing and the orders for end products are met by the times specified by the MPS. 2.2 M R P as a management tool The main function of M R P systems is to trigger purchases and factory orders to meet the ordered production requirements. In this context, it serves a useful role as a management tool (Schnenner [3]) and we discuss this in this subsection. 2.2.1 As a production planning tool As a production planning tool, M R P focuses the management attention on decisions with regards to MPS as it is the driving force for the M R P system. Once this is done, the M R P system converts the MPS into detailed plans which show the timing and quantity requirements for purchased and manufactured components.
2.2.2 As a scheduling device An important feature of M R P is the automating of the decisions involved in the scheduling of the production of component parts and in controlling the inventory levels of these component parts. When some unexpected developments occur - e.g., changes in order and/or in the production process, the M R P system automatically reschedules the production. 2.2.3 As a capacity planning tool The M R P system combines several types of information - (i) the planned production schedule and the materials needed to meet it, (ii) the time requirements for all components, and (iii) the particular work centers in the factory through which those parts and products travel, to predict the "work loads" at different work centers in the factory. Using the "work loads", the M R P system determines the capacity requirements needed. Often, the capacities at different centers are constrained. In this case, feedback of "work loads" from the M R P to the MPS can be used to alter the MPS so as not to violate the capacity constraints. In this "closed-loop" configuration of M R P and MPS, an effective linking of material requirement and production capacity is achieved. 2.2.4 To evaluate system performance The total cost of manufacturing is an indicator of the economic performance of the production process. The total cost is comprised of the following: (i) purchase costs - dependent on the number of times ordered and the quantities ordered, (ii) production costs - dependent on the number of lots and lot sizes, and (iii) inventory holding costs - dependent on the inventory control policy. Using these costs and MRP, one can compute estimates of the total cost. By optimal lot sizing and inventory control one can minimize the total cost and hence improve the economic performance of the process. For a good review of lot siz-
54 ing under dynamic demand conditions, see De Bodt & Van Wassenhove [ 32 ]. 3. MRP with uncertainty The deterministic MRP ignores uncertainty. In real life, production processes are influenced by many forms of uncertainty - e.g., uncertainty in customer orders for end products, uncertain delays due to failure of production process, and uncertainty in quantities delivered by suppliers, to name a few. In this section, we discuss the different types of uncertainty and the effect of uncertainty on the MRP system.
3.1 Types of uncertainty One can categorize the different types of uncertainties that affect production processes into two groups - (i) environmental uncertainty, and (ii) system uncertainty (Ho [5] ), as indicated in Fig. 2. Environmental uncertainty comprises of uncertainties beyond the production process. This includes (i) demand uncertainty due to uncertainty in customer orders and uncertainty in forecasting (also called forecast errors) and, (ii) supply uncertainty due to unreliable vendors. Note that the supply uncertainty can be either in
ENVIRONMENT
the quantities delivered a n d / o r the timing of the delivery. System uncertainty comprises of the uncertainties within the production process. These include operation yield uncertainty, production lead time uncertainty, quality uncertainty, failure of production system, and changes to product structure.
3.2 Impact of uncertainties on MRP system 3.2.1 Impact of environmental uncertainty Demand uncertainty leads to order modifications which in turn impacts on the gross quantities at the end product level. These changes propagate through the different stages of the production process and affect the production in all the remaining periods of the planning horizon. This order instability is called "nervousness" and affects the MRP system operation. Supply uncertainty affects the production of end product items in terms of causing delays a n d / o r the orders not being met in full.
3.2.2 Impact of system uncertainty System uncertainty leads to changes of scheduled receipts of MRP. This can result in the
UNCERTAINTY
•F o r e c a s t i n g e r r o r s •U n c e r t a i n t y in customer orders •U n c e r t a i n t y in v e n d o r supply
MRP SYSTEM •H i g h cost
rescheduling
• Increase in penalty cost •M R P n e r v o u s n e s s
SYSTEM
UNCERTAINTY
•V a r i a t i o n s in p r o d u c t quality •V a r i a t i o n s in p r o d u c t structure •V a r i a t i o n s in production lead time •E q u i p m e n t b r e a k d o w n •D y n a m i c lot s i z i n g
Fig. 2. U n c e r t a i n t y a n d effects on M R P .
55 planned orders not being met in terms of quantity delivered and/or time of delivery.
3.2.3 Cost implications As a consequence of the above-mentioned impacts, the total cost of manufacturing increases due to one or more of the following: (i) more frequent rescheduling, (ii) orders not being serviced effectively, (iii) unplanned setups for unplanned demands, (iv) loss of sales and goodwill, and (v) excess and insufficient inventories.
4. Approaches to dealing with uncertainty in MRP Different approaches to M R P have been advocated for coping with different forms of uncertainty. The four different approaches are as follows: ( 1 ) Safety stocks. (2) Safety lead times. ( 3 ) Hedging and overplanning. (4) Yield factors. We briefly discuss each of these approaches.
4.1 Safety stock In the safety-stock (also called inventorybuffer) approach the inventory levels are increased to provide a buffer against uncertainty. However, in the literature it is still unclear what a safety stock really means in the context of an M R P system. Two questions that need to be resolved are - (i) at what levels (end item level or component level) of the production process should safety stocks be held, and (ii) what should be the size of the safety stock. There is no commonly agreed answer to the first question. Given the levels at which safety stocks are to be held, two methods have been proposed to determine the size of safety stock. These are as follows. Economic approach: The safety stock sizes are selected to minimize the total cost comprising setting up cost, stock holding cost and penalties for possible shortages. Service level approach: Here the management first decides on the level of customer service they wish
to ensure and next the safety stock sizes are determined to ensure the specified service level.
4.2 Safety lead times In the safety lead times approach, order releases are brought forward from that stated in the requirements plan. As a result, the M R P schedules delivery of end products before they are due. Thus, if the process experiences uncertain delay due to any reason, then there is sufficient slack to absorb the delay if it is not too large. The safety lead time approach is mostly used for guarding against supplier uncertainties - that is, uncertainties at the input level where items are purchased from external suppliers who are not very reliable in terms of delivery times. The magnitude of safety time (the amount by which the release date is brought forward) depends on the context and regular adjustment is essential to ensure the desired level of service.
4.3 Hedging and overplanning In the hedging approach, the gross quantities of the MPS are altered. They are no longer simply the values obtained from demand forecasts, but include an extra amount so that the production process can cope with demand uncertainties. The amount of hedging depends on the production horizon and the demand uncertainties. In t h e overplanning approach, one simply makes more items than the orders specified in the MPS, so that the process can meet excess demand should it occur. This approach is similar to "scrap" or "yield loss" allowances at component levels employed in planning for processes where items at component level have to be scrapped when they are defective. Excess overplanning results in meeting the uncertain variations in demand but at the expense of increased inventory holding cost. Too little overplanning reduces the inventory holding costs, but can result in penalty costs due to demand being not met. Hence, the optimal choice of overplanning is critical to the economic performance of the process.
56
4.4 Yield factor The concept of yield factor is used to cope with system uncertainties causing loss of items (at end product and at intermediate levels). For each stage of the operation process, the yield factor is the ratio of the expected number of good items (or nondefective items) at the output to the number of input items. Thus for a single-stage item, the input quantity is determined by dividing the output demand by the yield factor. The yield factor is usually assumed to be constant and obtained from long-run observation of the production process. In a similar manner, a compound yield factor relates the input quantities needed to meet a specified output demand when system uncertainties result in losses occurring at different levels of the production process. The compound yield factor is a function of the yield factors at different stages of the process. 5. Review of the literature on M R P with uncertainty
We classify the literature on MRP with uncertainty into two groups based on: ( 1 ) the source of uncertainty and the method of planning with uncertainty, and (2) the method of analysis used.
5.1 Based on source of uncertainty and the method of planning In Section 3 we discussed the different sources of uncertainty which affect the production process and in Section 4 we discussed alternate approaches to coping with them. As a result, we can classify the papers based on the type of uncertainty and the method of planning as follows: ( 1 ) Environment uncertainty: This comprises of demand uncertainty and supply uncertainty. The approaches advocated for planning with these uncertainties are as follows: (i) Safety stocks: for demand and supply uncertainty in the quantities involved. (ii) Safety lead times: for demand and supply uncertainties in timing. (iii) Hedging: for demand uncertainty in quantities involved.
(iv)
Stochastic usage ratios: for optional parts planning with demand uncertainty (2) System uncertainty: This comprises uncertainties in yield, production quality, and lead time variations in production. Approaches to planning with these types uncertainties are as follows: (i) Safety stocks: for product quantity uncertainty at both end product and different component and subassembly levels. (ii) Safety lead times: for uncertain production lead time. (iii) Yield factor: for uncertain yield. (iv) Overplanning: for uncertainties in product quantity and quality.
5.2 Based on method of analysis Two approaches have been used in the analysis of models of MRP with uncertainty. They are as follows: ( i ) Analytical approach, and (ii) Simulation approach. The analytical approach uses analytical techniques for analysis and various related optimization problems - e.g., use of mathematical programming for optimal scheduling, stochastic optimization methods for optimal inventory control, etc. Most analytical models deal with only one type of uncertainty and simple structure for the production process. For more complex processes with many different end products and more than one type of uncertainty, the analytical approach is intractable. For such situations, simulation approach is the only viable alternative.
5.3 Classification of the literature Table 1 lists for each of the papers on MRP with uncertainty reviewed: (i) the topology of the production process (serial/nonserial; single/ multi-stage), (ii) the type of uncertainty included, (iii) the approach to planning with the uncertainties, (iv) method of analysis, and (v) the scheduling horizon for planning. In the remainder of the section, we review the papers in a chronological order. New [ 6] discusses the advantages and disadvantages of the safety-stock, safety-time and hedging approaches. In the safety stock up-
57 TABLE 1 Classification of literature Author(s)
Year
Topology
Uncertainty
Approach
Method of analysis
Schedule horizon
New
1975
-
E.U. ~, O.Y. 4
Conceptual analysis
finite
Whybark & Williams Melnyk & Piper De Bodt et al.
1976
-
E.U., S.U. 2
Simulation
finite
-
LT.U. 3 E.U.
Simulation Simulation
rolling
Huang et al.
1981 1982 1983a 19838 1982
Safety stock Safety time Hedging Safety stock Safety time Lead time allowance Safety stock
non-serial
LT.U.
finite
Biichel
1983
single
E.U.
Simulation ( Q-GERT ) Statistics
Billington, McClain & Thomas Burstein, Nevison & Carlson
1983
non-serial
E.U., S.U. (timing)
Mixed integer linear program
rolling
1984
serial
E.U. (timing)
Stochastic dynamic program
finite
Hegseth
1984
serial
O.Y.
New & Mapes
1984
serial
O.Y.
Grasso & Taylor III St. John
1984
non-serial
S.U.
1985
non-serial
Wijngaard & Wortmann
1985
single & non-serial
E.U., S.U. (timing) E.U., O.Y.
Wacker Huang, Rees & Taylor Carlson & Yano
1985 1985
non-serial non-serial
E.U., S.U. E.U., LT.U.
1986
non-serial
E.U.
Safety stock
Marlin
1986
non-serial
E.U., O.Y.
Yano
1987
non-serial
LT.U.
Safety stock Safety time Scrap allowance Safety time
Kurtulus & Pentico
1988
non-serial
O.Y.
Anderson & Lagodimos
1989
single
E.U.
E.U. - Environmental uncertainty. 2S.U. - System uncertainty. 3LT.U. - Lead time uncertainty. 40.y. - Operation yield uncertainty.
Stochastic Usage ratio
Compound Yield factor Safety stock Yield factor Safety time Safety stock
finite
finite finite Simulation Simulation
Safety stock Safety time Hedging MPS Safety stock
Yield factor Adjusted yield factor Compound-adjusted factor Safety stock
Statistics
finite
Statistics Simulation
finite finite
Heuristic Simulation Simulation
rolling rolling
nonlinear programming Simulation
Analytical model Simulation
finite
58 proach, the safety stock quantity is pre-set and requirements are triggered whenever forecast stock falls below service level. The safety lead time approach is suggested as an alternative to the safety stock approach and New suggests its use when the final products are produced infrequently. In the hedging approach, the increment to the requirements dependent on the average loss in production. Whybark and Williams [ 7 ] deal with demand, supply and system uncertainties and present a systematic study to provide guidelines for deciding on safety stocks and safety lead times. Using simulation approach, they first show that the quantity uncertainty is best dealt with using safety stock approach at the end product level while the timing uncertainty is best dealt with using safety lead times. They also show that safety stock approach results in increased inventory levels. Melnyk and Piper [ 8 ] deal with lead time uncertainty. They suggest lead time allowance (initially advocated by Whybark and Williams [ 7 ] ) as an effective method. They set planned lead time as the average observed lead time plus a multiple of the standard deviation of the lead time error distributions. Through simulation experiments they show how lead time allowances influence lot-sizing effectiveness and in turn how lot sizing influences the effectiveness of lead time allowances in MRP. De Bodt and Van Wassenhove [4,9,10] study lot sizing and safety stock decisions and the cost increments under demand uncertainty in singlelevel MRP with rolling horizon. They show that forecast errors have a tremendous effect on the cost effectiveness of lot sizing and safety stock decisions. They also report that the differences in the cost estimates are insignificant for different techniques (e.g., Silver-Meal heuristic and The Least-Unit-Cost heuristic) in the presence of forecast errors. For several lot-sizing heuristics, they give estimates of cost increases due to demand uncertainty. Their results show that safetystocks and lot-sizing policies are important to firms using MRP in an uncertain environment. Huang, Clayton and Moore [11] develop a simulation model to incorporate the basic MRP logic into a production process using Q-GERT network modeling and simulation language. Their
objective is to provide information necessary for planning and controlling of material and capacity requirements of a production process with uncertainty. The simulation model yields answers to questions related to production decisions at each station, production capacity and procurement lead times to meet a specified demand for end items. Bfichel [ 12 ] considers planning based on stochastic usage ratios for optional parts when demand for them is stochastic. The usage ratio for a specific component is the ratio of the demand for the component to the total demand for different end product. Small usage ratios (and/or small number of customer orders) results in considerable variations in demand which necessitate either high safety stocks. Biichel first develops a model of usage ratios which reveals the factors influencing the parameters of usage ratio distribution and then demonstrates how stochastic usage ratios may be included in a material requirements planning procedure to reduce demand uncertainty. Billington, McClaim and Thomas [ 13] deal with the interaction of lead times, lot sizing and capacity constraints for a process producing products with complex product structure with demand- and lead-time uncertainty. They propose a mixed integer-linear programming model for coping with capacity constraints and uncertain production lead times. Their model incorporates the following costs - current inventory holding cost, setup and overtime costs, and undertime production costs. The model computes the required production lead times based on the demands on available capacity, thereby reducing in-process inventory. The computational effort to obtain the solution is dependent on the model size and this is reduced by product structure compression. Burstein, Nevison and Carlson [ 14 ] consider a serial multi-stage production process with demand quantities known but with uncertainty in their timing. They propose the dynamic lot sizing approach, with lot sizing at each stage taking into account the subsequent stages of the operation. They use the stochastic dynamic programming technique to obtain the solution. Hegseth [ 15 ] considers a serial production
59 process with yield uncertainty. He uses a deterministic formulation involving different yield factors for different stages of the operation. The BOM is modified using yield factors and the material planning carried out after this modification. Since yield factors are deterministic, the model is deterministic. New and Mapes [ 16 ] also deal with uncertain yield losses. They consider processes with high losses and high variability in the losses - as for example, high-technology precision casting or integrated circuit production. They propose a simple model which relates output and input quantities through a random yield factor. They study safety-stock, safety-time and hedging approaches to coping with such losses. They argue that cost and risk of obsolescence reduce the effectiveness of the use of the safety stocks at the end product level. They suggest that the marketing environment (e.g., make-to-stock or make-to-order) is important for choosing the best approach to handle the variability in the yield rates. The key conclusion of their study is that different approaches are needed for different market conditions. Grasso and Taylor I I I [ 17 ] concentrate on examining the impact of operating policies on the M R P system with uncertainties resulting from lead time errors. Simulation approach is used to assess the impact of four factors - the lead time variability, the safety stock level, the safety lead time and lot-sizing rules, on the total cost. St. John [18] discusses the cost of inflated planned lead time on M R P system performance using a simulation model involving several stochastic variables such as customer demand, processing times (the sum of setup and run times). Their study demonstrates the need for lead time reduction, as long lead times result in high costs. Wijngaard and Wortmann [19 ] study different approaches to stochastic M R P with uncertainties. They introduce three ways to generate interstage slacks: (i) safety stock, (ii) safety time, and (iii) hedging. They consider a multi-stage production process with convergent and divergent nodes. The safety stock is determined using the variance of the forecast error, the yield and the average demand. They compare the relative advantages of safety-stock and safety-time approaches. An advantage of employing safety time
as opposed to safety stock is that it does not generate a safety stock unless there exists a real demand in the near future. A disadvantage of employing safety time is the need to extend the planning horizon at the MPS level. They also discuss the hedging approach to cope with quantity uncertainty in demand and report that it is possible to change hedging without changing the planned order release. This has the advantage of reducing the number of rescheduling messages. Wacker [20] deals with a model which estimates the mean and variances of outputs at the end item and at the component level due to uncertainties. He uses the safety stock approach. In made-to-order production operation, end item safety stock does not alleviate demand uncertainty. The model uses "forecast standard error" for components as an estimate of component safety stock. He comments that M R P system should not involve sophisticated control schemes to monitor environmental and system uncertainties, but should incorporate these variations in the system itself. Huang, Rees and Taylor [21] develop a queueing network simulation model with QGERT, which integrates M R P with control at the shop level in a production process. As such, it adds an extra dimension - control at floor shop level. They conduct several simulation experiments to study the effect of different factors on the total cost. Carlson and Yano [22] investigate an M R P system with rolling-horizon and uncertain demand. They assume that forecasting errors are normally distributed and analyze the need for emergency setups at the component level. They examine a heuristic method and report that there is a significant increase in the cost of lot sizing due to the need for emergency setups and that safety stocks at the component level reduce the need for emergency setups and hence are cost effective. Marlin [23] develops a MRP/job-shop stochastic simulation model coded in SIMSCRIPT II.5. The model closes the loop between order launching via M R P and order execution by an in-house manufacturing facility and by outside vendors. The model simulates two mutually exclusive types of demand uncertainty, viz., quan-
60 tity uncertainty and timing uncertainty. To cope with quantity uncertainty Marlin uses the yield factor approach developed by Hegseth [ 15 ]. Yano [24] deals with the stochastic lead time problem using an analytical approach. He first addresses the problem of determining optimal planned lead times in an assembly type operation with uncertain processing times. This is done using a complex nonlinear programming formulation, with the objective function involving inventory holding costs and tardiness costs. Kurtulus and Pentico [25] deal with yield uncertainty and extend the yield factor approach of Hegseth [ 15 ]. They propose three different versions of the "expected value" rule. In the first version (EXP1) the planned requirements X* (i.e., the number of units started of the item with i stages remaining) is obtained by dividing the net requirements X*_1 of the item by the yield factor Pi. The second version of the rule (EXP2) plans requirement in a similar manner. However, the actual yield factor is continuously monitored during simulation at regular intervals and is updated with new yield factor. The third version (EXP3) uses compound yield factors developed by Hegseth [15 ] with adjustment of the yield factor. Anderson and Lagodimos [26] consider the problem of predicting customer service levels in a single-stage MRP system where demand quantity is uncertain. They derive analytic expressions for service level measures often used in make-to-stock environments and study how service levels are affected by the level of safety stock held. They first develop a dynamic model of the MRP system which takes planned safety stock into account and then derive expressions for different service level measures. The results of the analysis are validated using a simulation model.
2.
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7.
Some comments
8. Based on the literature review, the following comments can be made: 1. There are many different types of uncertainties which affect MRP and many different approaches have been advocated for coping with them. There is no paper which deals with MRP
with all different types of uncertainties in an integrated manner. The bulk of the literature deals, at a conceptual level, with specific methods to cope with specific types of uncertainty. There is very little comparative evaluation of the different methods for different types (or forms) of production systems. MRP with demand uncertainty has received more attention than MRP with other types of uncertainty. Most models assume that the demand variations are gaussian in distribution. This is a valid assumption when the quantities involved are large. Most papers dealing with the yield factor approach assume a constant yield factor for each stage of the production process. In real life this is seldom true. The yield factor depends on many factors such as age and condition of machines used in the production, quantities produced, maintenance and quality control actions, etc. In the safety stock approach, there are still many unresolved issues - e.g., the location of safety stocks for a process with fairly general product structure. Optimal decisions (e.g., optimal overplanning in the overplanning approach, optimal location of safety stock) depend critically on the stochastic characterization (i.e., type of distribution, parameters of the distribution, etc. ) of the uncertainty. Misspecification can have a significant impact on the optimal solution. There is no work on the robustness and sensitivity aspects of optimal solutions and this is an open topic for research. Although many papers deal with the use of simulation approach for study of MRP with uncertainty, there is very little work on the comparative evaluation of the advantages and disadvantages of different simulation languages. There are very few papers on real case studies, i.e., papers dealing with the application of different approaches to MRP with uncertainty in the real world. Such studies are important not only to evaluate the effectiveness of the different approaches but also to trigger further theoretical research.
61 9. Regarding optimization problems in the context of M R P with uncertainty, they are, in general, computationally very complex. As such, one needs to use heuristic methods to obtain solutions close to the optimal. Although this topic has received some attention, there is considerable scope and need for further research.
6. MRP with uncertain quality In real life, often a fraction of the end product items are defective. This could be due to defective components a n d / o r quality variations inthe production process. Let D denote the specified gross requirement at the end product level. One can use the classic M R P to compute the quantities of various components to be ordered to meet this requirement. If the input quantities ordered are such that only D end items are produced, then the number of nondefective end product items produced, 37, is a random variable assuming values in [0,D ]. If the defective items are not repairable, or too expensive to repair, they need to be scrapped. As a result, there is a possibility of demand not being met, the shortfall being the number of defective items produced. One way of coping with this form of uncertainty is the yield factor approach. The yield factor accounts for such losses in determining the input quantities to be ordered. As mentioned earlier, the yield factor is determined by long-run observations and roughly equals E [57]/D. Better material planning with uncertainty needs to incorporate quality variations in the process more effectively. In this context, the vast literature on Quality Control dealing with modeling of quality variations is of particular importance. Over the last two years the authors have been studying M R P with uncertain quality in a framework which links M R P with the quality control literature. In this section, we give an overview of this research. We commence with a brief discussion of the modeling of quality variations and then proceed to discussing alternate approaches to M R P with uncertain quality.
6.1 Modeling quality variations In a multi-stage production process, the output quality at each stage of the process can be characterized in terms of (i) individual item being defective or not, or (ii) a batch of items, with quality denoting either the number or the fraction of nondefective items in the batch. Also, a multistage production process can be characterized as a network of one or more of the three block types (called Type-l, Type-2 and Type-3 Blocks) shown schematically in Fig. 3. Type-1 Block has r (r>_ 2) different inputs and a single output and represents an assembly type operation where two or more components are assembled. The output quality of the block is a random variable and is a function of the input quality of the r different inputs and the uncertainty in operations involved. A type-2 Block represents a distribution stage and is characterized by a single input feeding to s ( s > 2 ) different branches on the output side. The output quality o f b r a n c h j ( 1 <_j
6.20verplanning approach In this approach, D, the order quantity for end items, is increased to D( 1 + ~ ) where ~ ( > O) is
62
input,/ --
output 1
input 1
output input j
)
) output j
input r
output s TYPE- 1 BLOCK
TYPE-2 BLOCK
output
input
)
TYPE-3 BLOCK
Fig. 3. Basic block types.
called the overplanning factor and the material planning is carried out using the classic M R P in the usual way. The critical factor is 3. If it is small, then the probability of the specified demand being not met (i.e., the number of nondefective items at the end level < D) is high and can result in heavy penalty cost. On the other hand, if 3 is large, then the probability of the demand being not met is low but this is achieved at the expense of greater manufacturing and/or inventory cost. In Murthy and Ma [27], ~ is selected optimally to achieve a sensible tradeoff between the two conflicting costs mentioned above.
6.30verplanning approach with lot sizing In Murthy and Ma [27 ] the quality variations at each stage are assumed to be independent of the quantities processed. In real life, the process quality at each stage depends on the lot size used. Smaller lot size implies better quality (e.g., see Porteus [28 ]. However, this results in increased costs due to the need for more setups. In Murthy and Ma [29 ], we consider two different models linking quality with lot size and develop an M R P system based on optimal lot sizing and overplanning.
6.40verplanning approach with inspection Often not all defective items need to be scrapped, as some can be rectified through some form of corrective action. This has an impact on the overplanning factor and in turn on the M R P system. In Ma and Murthy [ 30 ], we deal with the case where the items are inspected only at the end product level and defective items can be repaired. Depending on the nature of the defect, the defective item needs to be sent one or more stages back in the production process for rectification and hence the cost of rectification depends on the nature of the defect. As a result, the inspection scheme and the rectification strategy (e.g., decision to repair is not based on a cost limit policy) influence the number of defective items produced. This in turn affects the overplanning factor and the material requirement. In Ma and Murthy [30], we study optimal overplanning with different inspection and repair strategies.
7. Conclusion and topics for further study In this paper we have carried out a review of M R P with uncertainty and outlined the research
63 of the authors in M R P with uncertain quality. There are many topics which need further research. A variety of topics needing further study were identified in Section 5 (see "Some Comments" ). Hence, we shall not repeat them here. As commented earlier, there is no study of M R P incorporating the different types of uncertainties in an integrated manner. An effective M R P system must incorporate all types of uncertainties, the different actions to cope with them, and be automated to indicate best action. There is need for such an automated system (involving concepts from Expert Systems and Artificial Intelligence) and this should be of particular interest to the practitioners. In the remainder of the section, we discuss some extensions to our work reported in Section 6 and further topics for study in M R P with uncertain quality. Our research has so far been confined to production systems with no commonality at the component level. As a result, we could model quality variations for each end product type independent of the other. When there is component commonality, we need to model them together using multivariate formulations. This makes the problem more complex. Lead times can vary significantly when defective items are sent back to earlier stages for rectification. As a result, the lead times are statistically correlated with the product quality. This implies that yield and lead times are correlated. The study of M R P with this correlation is an open topic for research. In Ma and Murthy [30], we deal with inspection at the end item level. One can carry out the inspections at item level to either weed out defective items or repair them. The inspection and rectification scheme has an effect on the final output quality and this in turn affects the overplanning at the end item level. There is a need to study the effect of location of inspection points on the operation of the M R P system. In Murthy and Ma [29 ], we considered M R P with uncertain quality and the quality related to lot sizing. Moreover, maintenance actions affect not only quality but lead times as well. The study of maintenance actions on the performance of
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