Using Fuzzy Logic to Model MRP Systems under Uncertainty

Transcription

1 Using Fuzzy Logic to Model MRP Systems under Uncertainty G. Daria, V. Cruz Machado Universidade Nova de Lisboa, FCT DEMI, Caparica PORTUGAL Fax: ; Abstract The aim of this paper is to handle uncertainty in a Material Requirements Planning (MRP) model using Fuzzy Logic. This approach can intuitively be explained via the strong relationship between uncertainty and Fuzzy Logic: uncertainty is in the intrinsic meaning of Fuzzy Logic (it is fuzzyness). Particularly, the model is applied to dependent demand in which a MRP method is used. The materials planning model is a combination of basic sources of uncertainty that affects the MRP system, namely, demand and supply uncertainty, and a Fuzzy Logic Inference system. The model is used to control the main parameter to buffer against uncertainties: safety stock. The model performance is tested for historic data and compared with real inventory levels and a MRP reference model. Keywords: Materials Requirements Planning, Fuzzy Logic, Uncertainty. 1. Introduction The issue of uncertainty, associated with material requirements planning systems (MRP), has attracted considerable attention over the last few years. MRP is still regarded as one of the most commonly used production planning and control systems. In this paper, MRP specifications and weaknesses are reviewed; in particular, the MRP systems under uncertainty are investigated. In the presence of uncertainty, MRP systems do not have suitable performance. This weak performance explains the need for extensive research on MRP under uncertainty. The literature review revealed that research on MRP under uncertainty is being carried out [1, 2]; however, the use of a Fuzzy Logic approach is not well documented. It is our thesis that Fuzzy Logic can be used to prevent uncertainty (affecting demand and supply), and it should be more investigated. The Fuzzy Logic approach suggests that there is no need to know the kind of uncertainty and its quantification, to provide an optimized and improved system. Therefore, in this paper a new approach to deal with this problem is presented. The research on fuzzy production planning and inventory control models was started while the fuzzy set theory and its applications were introduced. The works in this area can be divided into two categories, namely the selection of production or process plan, and management of production and inventory plan [3]. The problem of controlling inventory over an infinite planning horizon was investigated and the fuzzy set theory was applied to represent the inventory system, with the fuzzy inventory level as the output and fuzzy replenishment as the input. Demand and constraints on replenishment are also fuzzy. To maximize the membership function for the decision, it is developed an algorithm to find the optimal time-invariant strategy for determining the replenishment to current inventory levels. Fuzzy Logic was also applied to address short-range planning and scheduling problems using a hierarchical structure, which includes three decision levels (higher, middle, lower), each responsible for a different production problem with a different time scale. The higher decision level determines a safety stock level used to compensate for future resource failures. Load rates are computed at the middle level. This is accomplished through a fuzzy controller that tends to minimize the error between the cumulative production and cumulative demand. 2. MRP and Fuzzy Logic Planning and controlling enterprise resources, with a modeling process not delineated and dynamics, with a structural complexity and in reason of the uncertainty variable used in, needs particular tools. The use of conventional tools results inadequate for a complete and accurate description of the model in examination [3]. The system of fuzzy inference allows characterizing the qualitative aspects of the model, through linguistic description of the variable of entry, without having to apply to precise quantitative analysis.

2 Rules are the heart of a Fuzzy Inference System (FIS) and may be provided by experts or extracted from numerical data [4]. In either case, the rules, that we are interested in, can be expressed as a collection of IF-THEN statements, such as If A Then B, when the If part of a rule is its antecedent, and the Then part is its consequent. Interpreting an ifthen rule involves distinct parts: first evaluating the antecedent (which involves fuzzifying the input and applying any necessary fuzzy operators) and second, applying that result to the consequent (known as implication). In the case of two-valued or binary logic, if-then rules do not present much difficulty. If the premise is true, then so is the conclusion. If we relax the restrictions of two-valued logic and let the antecedent be a fuzzy statement, how does this reflect on the conclusion? The answer is a simple one. If the antecedent is true to some degree of membership function, then the consequent is also true to that same extent. For the extremely concise form, the fuzzy rules are employed for simulating the approximate aspects of the human reasoning, that play a fundamental role in the ability to take decisions in an environment very often characterized by inaccuracy and uncertainty [5]. In these terms the Fuzzy Logic is applied, controlling the buffers, calculating the risk and using internal uncertainty that is an intrinsic property, versus uncertainty in MRP. Particularly, Fuzzy Logic handles uncertainties about the meanings of words by using precise membership functions that the user believes to capture the uncertainty of the words. Once membership functions have been chosen, all uncertainty about the words disappears because membership functions are totally precise. On the other hand, Fuzzy Logic handles uncertainties about the meanings of words by modeling the uncertainties. This is accomplished by blurring the boundaries of membership functions into what we call a footprint of uncertainty. Although a membership function will also be totally precise, it includes the footprint of uncertainty that provides new degrees of freedom, to let uncertainties be handled by a Fuzzy Inference System (FIS) in totally new ways. 3. Uncertainty The uncertainty about the meanings of the words that are used in the rules seems to be in accord with fuzziness [6], which results from imprecise boundaries of fuzzy sets. Consequents for rules are obtained from experts, by means of knowledge mining (engineering), or are extracted directly from data [7]. Because experts do not all agree, a survey of experts will usually lead to a histogram of possibilities for the consequent of a rule. Uncertainty about the consequent used in a rule seems to be in accord with strife, which expresses conflicts among the various sets of alternatives. Measurements are usually corrupted by noise. Hence, they are uncertain. All of this is rendered explicit through the frequently made assumption of existing knowledge of a probability model (that is, a probability density function) for either the signal or the noise. Doing this gets around the major shortcoming of a probability-based model. Namely, the assumed probability model, for which results will be good if the data agrees with the model but will not be adequate if it does not. Uncertain measurements can be handled very naturally within the framework of a FIS. Uncertain measurements (such as randomness in the data) can be modeled as fuzzy sets. Hence uncertainty about the measurements that activate the FIS seems to be in accordance with nonspecificity when the latter is associated with information-based imprecision. Finally, a FIS contains many design parameters whose values must be set by the designer before the FIS is operational. There are many ways to do this, and all make use of a set of data, usually called the training set. This set consists of input-output pairs for the FIS, and, if these pairs are measured signals, then they are as uncertain as the measurements that excite the FIS. In this case, one that is quite common in practice but has not received much attention in literature, the FIS must be tuned using unreliable data, which is yet another form of uncertainty that can be handled. The uncertainty about the data that is used to tune the parameters of an FIS also seems to be in accordance with nonspecificity, when it is associated with information-based imprecision. In the production system, uncertainty comes through in many different ways [8]. Recent projects examining the impact of lead-time uncertainty and demand uncertainty have shown that these affect MRP in a number of manners. Two basic sources types of uncertainty affect a MRP system performance directly: demand and supply uncertainty. In either of the two cases, as the demand for the product and its parts is usually assessed through forecasts, uncertainty may arise in quantity and/or in timing between the placement of an order and the replenishment of inventories. This paper deals with the situation in which there are three sources of uncertainty. The first and the second uncertainty depend on demand rate and lead time or both. The third one is the delivery batch quantity. The first uncertainty takes place when there is a variable demand or usage rate and a fixed replenishment lead time. A possible stock out situation can occur during the lead time period, demand may be greater than expected thus stock is reduced to zero before replenishment. On the other hand, a situation in which the demand or the usage rate is fixed but replenishment lead time is variable occurs, when an order is placed at such time that items are received for re-

3 plenishment stock and the constant rate of demand has depleted stocks to zero. This is possible because: lead time is greater than expected; the constant rate of demand has reduced stocks to zero, and; a stock out situation exists before items had obtained the replenish stock. However it is possible that both demand or usage rate and replenishment lead time are variable. It may cause a stock out because of the combination of the two situations described above. Finally the third uncertainty depends on the delivery batch quantity. During a purchasing process it is possible that a different batch quantity may eventually be received into stock even though a particular batch quantity is ordered to replenish stock. In periodic review, in which it is almost universal for MRP systems to be based on, orders are placed to furnish stock to a particular level; consequently the delivery of quantity, different from expectation, can increase the risk of stock out before the next replenishment is received. 4. Model Development The logic of MRP would seem to preclude the use of any buffering mechanism. However, it has also not provided any mechanism to accommodate for uncertainty. Uncertainty is the reason for buffering, and protection is usually in the form of safety stocks or safety lead times. Utilize the first or the second method, or both eventually, depend on many factors. To apply Fuzzy logic to model MRP, the situation suggests the need to design a model of reference to improve confidence in the MRP scheduling plans [9]. The material-planning model, illustrated in Figure 1, proposes an optimal approach to the buffering problem in dealing with uncertainty in MRP systems. The model strategy was based in the idea that the use of safety time to deal with timing uncertainty could be improved by holding safety stock at level one items [10]. Demand uncertainty (σ D, D ) Safety stock (Q 1 ) Other inputs (MPS, CRP, ) Supply uncertainty (σ L, L ) Uncertainty in the MRP scheduling plans Add a fixed extra safety stock (Q 2 ) N Y MRP order plans Does Q 1 protect the MRP scheduling plans against shortages? Figure 1 Model of reference The amount of safety stock should be such that it accounts for demand and lead time characteristics. Hence, safety stock (Q 1 ) would simply be a function of the demand and lead time characteristics and the desired service level. The model used assumes that demand and lead time are independent variables. Q = Zα σ D L + T 1 (1) 2 DL+ T ( L + T ) σ + D σ σ = D L (2) Demand has mean D and standard deviation σ D ; lead time has meant L and standard deviation σ L and T is the order interval. A specific service level of 95% is assumed. Areas under the standardized normal curve are used to obtain the value of Z α, given a desired service level. It should be recognize that the problem of nervousness (i.e., instability) in the MRP plans, responsible for the high percentage of requirements shifted from one period to another within the lead time, increases the instability in production schedules. To compensate the existence of MRP instability, an extra quantity of stock Q 2 added to Q 1 was introduced to deal with demand uncertainty in order to buffer MRP nervousness. This extra quantity of stock, Q 2, is a measure of the forecasting error based on the difference between the actual value (Y t ) and its forecast value (Ŷ t ). The mean square error (MSE) was the technique used to compute the error for each forecast period: MSE = 1 n n t = 1 ( ˆ ) 2 Y t Y t (3)

4 5. Fuzzy Inference System Model The material planning model based on FIS is implemented by the model of reference, where the FIS is place in the decisional phase as in Figure 2. Demand uncertainty (σ, D ) D Safety stock (Q 1 ) Other inputs (MPS, CRP, ) Supply uncertainty (σ L, L ) FIS Fuzzy Inference System Figure 2: MRP FIS Model. In the model of reference to compensate the existence of MRP instability, an extra quantity of stock Q 2 added to Q 1 was introduced to deal with demand uncertainty in order to buffer MRP nervousness. This is unnecessary in the FIS model because it is structured, flexible and adaptive, adjusting itself the discrepancy among the two quantities. In this case, the Safety Stock value is used to determine the average quantity that the system will use in its mem- bership functions. Some experimental tests have underlined that the MRP Model can have substantial differences if the FIS settings change for the Aggregations and Defuzzification phases. Particularly, in these cases we decided to split in two models, called FIS A and FIS B model respectively, that produce a different grade of performances, always effective and stable the first, and more efficient but sometimes unstable, the second. Model A This model applies the sum operator for the Aggregation and the mom (middle of maximum or mean-max membership) operator for the Defuzzification [11]. The mom operator considers the region between maximums, which can be assumed in more points, as the next example: we suppose that the control rules regation produces the followership function µ z, z z z ]. It has two parallel segments to the axis z. We project the segment ing memb ( ) [ 0, q MRP order plans [P 1 P 2 ] (to the maximum height) on the axis z and calculate z* as middle point of the segment projection [p 1 p 2 ] (see Figure 3). z = p 1 + p 2 2 Figure 3: Middle of maximum operator. Model B This model applies the max operator for the Aggregation and the centroid (center of area or gravity) operator for the Defuzzification. Centroid is the me me membership function µ z, z thod most applied: we consider the sa ( ) [ z z ]. We divide the interval [z 0,z q ] in q equal subintervals (or almost equal) through the points z 1, z 2,, z q-1. 0, q The crisp value z* looked for z is the average weighed of the numbers z k (in this case the weight is µ ( z k )): z ( z ) z = k µ µ (4) z k k 6. Simulations and results To test the models, a simulation based on six months historical data from the MRP (used in a refrigerating drinks industry) was carried out. Data is referred to paper package dependent demand; as performance measurement it was considered the average inventory level and the shortage quantity achieved. Obviously, the first objective, much less the conditio sine qua non, was the absence of the stock out situation.

5 In the model of reference, to compensate the existence of MRP instability, an extra quantity of stock Q 2, added to Q 1, was introduced to deal with demand uncertainty in order to buffer MRP nervousness. This is unnecessary in the FIS model, implying that: 1. The model is more efficient starting from a better position. 2. The average quantity, at the end of the simulation, is lesser. Consequently the performance is higher. To confirm, it was selected a material, called M 1, among all those utilized in the simulation. M 1 presents an uncertainty grade of the 64% and is one of the most significant and sold products. Table 1: Material M 1 Real Conventional Model FIS A Model FIS B Model Average Inventory Stock out Yes No No Improvement 42.7% 42.5% 50.8% Q 2 Yes No No Table 1 shows that all models introduced some improvement: The simulation in the Conventional model revealed a stock out situation; even if there was an improvement of 42.7%, which is a good result, th is i s not accepta ble because the main mod el purpose is to prevent a stock out situation. Therefore, it was needed to add an extra stock quantity. Model A achieved 42.5% of improvement, equalizin g the previous model, but without a stock out occurrence. It was a good solution and the stock level was always in a stable situation. A great consumption did not destabilize the system; it replied with a good order to re-establish the equilibrium. Model B achieved 50.8% of improvement. This was the best result; it confirms the right approach in which the model was made. The stock levels took an important decrease, but with an oscillatory course (see Figure 5); the reply to a special consumption should be made only if necessary. It can be dangerous in those cases where the consumption is more irregular and a peak can occur causing a stock out. Generally models based on Fuzzy Logic achieve an improvement on the real and also on the conventional model. Even if Model B produces best performances, it can be a source of stock out situations. In opposition, Model A is more stable, without stock out, and similar performances (see Figure 4). Average Inventory Conventional Real Figure 4: M 1 Model A Simulation Safety Stock Another important aspect to point out in these simulations is the adaptability of the Models based on Fuzzy Logic. The Safety Stock quantity Q 1, calculated by statistical formulas, represent the threshold, under which an order can be analyzed, to avoid the stock out. The model based on Fuzzy Logic can use lower thresholds to achieve optimal results. Figures 4 and 5 show the evolutio n of the safety stock quan tity. In Model A the overall average stock quantity was fairly constant. A stock out occurred in model A at a level which was only half of the level at which occurred the stock out in the conventional model; in the Model B simulation the system had a break only when the safety stock was equal to zero. The gain in this case was until to the 70% behaving a great efficiency for the system and for the enterprise. In this sense the performance could push to a review of the model to make it more stable and effective without compromise the efficiency.

6 Average Inventory Conventional Real Figure 5: M 1 Model B Simulation Safety Stock 7. Conclusions The tool presented in this paper consists of an innovative method to employ Fuzzy Logic as an instrument for the Enterprise Management. I t is b ased on the use o f a knowledg e combination among management, mathematics and co mputer science. As it was demonstrated by the achievement of this work, namely for the company in study, it rarely exists a certainty associated to the supply and demand quantity. This study allowed a better understanding regarding the company s capacity to efficiently adapt itself to market uncertainties; in addition it provides the identification of which goods are more subject to market uncertainty, as well as the identification of uncertainty categories that have an impact on those goods. Therefore the use of Fuzzy Logic increases the performance and prevents the stock out. The results demonstrate that a correspondence between uncertainties can be prevented, without following a particular scheme; rather a specific model can be created on a case by case basis. The correspondence between Fuzzy Logic uncertainty and MRP uncertainty is evident in the results: the FIS Model produced a better result. Besides, when the uncertainty is low, the safety stock quantity can be decrease respect to the statistical calculation because the same FIS is adapted in comparison to the model proposed. References [1] Ghazanfari, M, Murtagh B., Syed, M. and Mathew, P., Safety Stock problem in Modeling Hierarchical Production Plannin under Uncertainty and Soft-Computing, 6 th International Conference on Manufacturing Engineering 1995, Melbourne 29 November 1 December [2] Barroso, A., Machado, H. and Machado, V. C., Production planning uncertainty: characterization and stateof-art, VII Ibero-american Congress of Mechanical Engineering CIBIM7, p. 220 (6p.), México, October [3] Ghazanfari, M., Ahmadvand, A., Using Crisp-Fuzzy Numbers To Model MRP Under Lead Time Uncertainty Conditions Proceedings of the 6 th Annual International Conference on Industrial Engineering Theory, Applications, CA, USA, November 18-20, [4] Mendel, J., Uncertain Rule-Based Fuzzy Logic Systems, Prentice Hall, Dec 22, [5] [6] Berztiss, A., Uncertainty Management, University of Pittsburgh, Department of Computer Science, Pittsburgh PA 1526 USA; SYSLAB, University of Stockholm, Sweden. Internet access: [7] Klir G. J., Wierman M., Uncertainty-based information Physica-Verlag Heidelberg; November 23, [8] Guide, V. and Srivastava, R., A review of techniques for buffering against uncertainty with MRP Systems, Production Planning & Control, vol. 11, no. 3, , [9] Klir G. J., Folger T., Fuzzy sets, uncertainty, and information Prentice Hall, January Buzacott, J. and Shanthikumar, J., Safety Stock versus Safety Time in MRP Controlled Productions Systems, Management Science, 40(12), , [10] Alves, T., Machado, H. and Machado, V. C., Modeling MRP Systems under Uncertainty: Safety stock versus Safety Time, IIE Annual Conference, Houston, USA, p. 111, [11] Lazzerini, B., Introduzione agli insiemi Fuzzy e alla Logica Fuzzy, Università di Pisa, Ing. Informatica, 2004.