An Interval Type-2 Fuzzy multiple echelon supply chain model
Knowledge-Based Systems 23 (2010) 363–368
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
An Interval Type-2 Fuzzy multiple echelon supply chain model Simon Miller *, Robert John Centre for Computational Intelligence, De Montfort University, Leicester, UK
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Article history: Available online 20 November 2009 Keywords: Interval Type-2 Fuzzy Logic Multi-echelon supply chain modelling Genetic algorithms
a b s t r a c t Planning resources for a supply chain is a major factor determining its success or failure. In this paper we build on previous work introducing an Interval Type-2 Fuzzy Logic model of a multiple echelon supply chain. It is believed that the additional degree of uncertainty provided by Interval Type-2 Fuzzy Logic will allow for better representation of the uncertainty and vagueness present in resource planning models. First, the subject of Supply Chain Management is introduced, then some background is given on related work using Type-1 Fuzzy Logic. A description of the Interval Type-2 Fuzzy model is given, and a test scenario detailed. A Genetic Algorithm uses the model to search for a near-optimal plan for the scenario. A discussion of the results follows, along with conclusions and details of intended further work. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction There are a number of definitions of Supply Chain Management (SCM), each having minor variances but describing the same core idea. SCM is the management of material flow in and between facilities including vendors, manufacturing/assembly plants and distribution centres [1]. Planning the allocation of resources within a Supply Chain (SC) has been critical to the success of manufacturers, warehouses and retailers for many years. Mastering the flow of materials from their creation to the point of sale offers considerable advantages to those within a well managed SC. Poorly managed resources result in two main problems: stock outs (a shortage of stock) and surplus stock. The consequence of stock outs is lost sales, and potentially lost customers. Surplus stock causes additional holding cost and the possibility of stock losing value as it becomes obsolete. Holding some surplus stock is advantageous however; safety stock can be used in the event of an unexpected increase in demand or to cover lost productivity. The problem has been addressed numerous times in the literature using different approaches. An overview of traditional and Computational Intelligence approaches to supply-chain resource planning can be found in [2]. In this paper we present a novel approach to modelling the supply chain using Type-2 Fuzzy Logic (T2FL) [3] that is optimised by means of a Genetic Algorithm (GA) [4]. This research is part of a project on demand forecasting and resource planning (Data Storage, Management, Retrieval and Analysis: Improving Customer Demand and Cost Forecasting Methods, * Corresponding author. Tel.: +44 116 2078408. E-mail addresses: [email protected] (S. Miller), [email protected] (R. John). 0950-7051/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2009.11.016
funded by the Technology Strategy Board in the UK). The project aims to: (1) improve the forecasting of demand by using a variety of disparate sources of data and statistical and Machine Learning (ML) methods for analysis; (2) improve the allocation of resources in which the generated forecast is used as an input, the output being is a (long- or short-term) plan of raw materials and resources within the supply chain required in order to meet such demand. The research presented here is intended to address the second aim. Various degrees of uncertainty are present in the different data sources used. This uncertainty is further amplified in the generated forecast by applying methods of analysis with (again) varying degrees of inherent uncertainty. Furthermore, other data that is often used in resource planning such as transportation and other costs, customer satisfaction information, etc. is also uncertain. Therefore, FL and especially T2FL are particularly appropriate for this problem. While Type-1 FL (T1FL) has successfully been used many times for modelling SC operation (see Section 2), T2FL has been shown to offer a better representation of uncertainty on a number of problems (e.g., [5,6]). In [7] the authors applied Interval Type-2 Fuzzy Logic (IT2FL) to a 2 tier distribution resource model and it was shown to work well. This paper presents an IT2FL model of a multiple echelon SC problem which is optimised by a GA to find a near-optimal configuration. The term ‘echelon’ (or ‘tier’) is used to describe a group of nodes that operate at the same stage in a supply chain, e.g., retailers. A multiple echelon supply chain is one that incorporates many stages, e.g., suppliers, manufacturers, warehouses and retailers. The paper is organised as follows: Section 2 discusses Fuzzy Logic and its application to SCM. Section 3 introduces the model and the test case used for evaluation. The results from the experiments on the test case are presented in Section 4. Section 5 concludes the paper with a summary and future research directions.
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2. Fuzzy Logic for supply chain management FL and GAs have been successfully used for supply chain modelling [2] and are particularly appropriate for this problem due to their capacity to tackle the inherent vagueness, uncertainty and incompleteness of the data used. A GA [4] is a heuristic search technique inspired by evolutionary biology. Selection, crossover and mutation are applied to a population of individuals representing solutions in order to find a near-optimal solution. FL is based on fuzzy set theory and provides methods for modelling and reasoning under uncertainty, a characteristic present in many problems, which makes FL a valuable approach. It allows data to be represented in intuitive linguistic categories instead of using precise (crisp) numbers which might not be known, necessary, or may in general be too restrictive. For example, statements such as ‘the cost is about n’, ‘the speed is high’ and ‘the book is very old’ can be described. These categories are represented by means of a membership function which defines the degree to which a crisp number belongs to the category. In this research the aim is to allow linguistic terms to be used by Supply Chain Managers when describing their operation. For example, instead of asking for crisp numbers to describe the current stock level of a product, we may allow them to make statements like ‘Warehouse A has About 500 of product 1’ or even more abstract ‘Warehouse A’s stock level of product 1 is low’. By removing the need for exact information it is possible to produce a system that is much more usable when the information available is vague, uncertain or incomplete. T1FL has been applied to SC modelling numerous times with good results. Some of the research that is considered most relevant for this paper is discussed in the following paragraphs. Petrovic et al. [8] use fuzzy sets to model vagueness and imprecision in customer demand, external supplier reliability and supply within a multiple echelon single product supply chain. The system demonstrates the effect of differing conditions and strategies on fill rate and holding costs of a SC. The results of the evaluation show that there is a slight improvement in the performance of the SC when the inventories are partially co-ordinated. In [9] two-level fuzzy optimisation is employed to find ideal order-up-to quantities in a one warehouse-multiple retailer SC. The problem is decomposed for individual control of the warehouse and retailers; a coordination mechanism provides overall control of the SC. Customer demand, inventory levels, holding cost, and shortage cost are represented by fuzzy sets. A measure of satisfaction is derived from the cost incurred at each element. The solution produced by the system is the best compromise between members, though not necessarily the cheapest. In Aliev et al. [10] a fuzzy system is used with a GA to model a 3 tier SC. The model uses a global policy of management with emphasis on integrating the production and distribution models. The GA searches for a near-optimal configuration; fuzzy sets are used to describe costs, returns, production capacities, storage capacities and forecasts. The proposed fuzzy method, a crisp method and a non-integrated method are compared. The crisp system is unable to produce a feasible configuration if actual demand is lower than the forecast. In contrast, the fuzzy model presented is robust and able to cope with fluctuation in demand and production capacity with little impact on profitability. The non-integrated model performed significantly worse than the fuzzy integrated model. A similar approach is presented by [11] in their multiple echelon SC model. FL is used to represent customer demand, processing time and delivery reliability; a GA finds order-up-to levels. The system attempts to find the configuration that incurs the minimum cost. An optimism–pessimism index is set by the user and passed to the system. When optimistic, the model assumes the best case
scenario for material response time. A pessimistic attitude produces the opposite effect. The results show that more pessimistic strategies increase the fill rate, reducing the sales lost through stock outs, and incur higher inventory cost as more stock is kept. More optimistic strategies result in a drop in fill rate and an increase in sales loss, though inventory cost is also reduced. In these examples we have seen how T1FL has been used to tackle the resource planning problem. However type-1 fuzzy sets represent the fuzziness of the particular problem using a ‘non-fuzzy’ (or crisp) representation – a number in [0, 1]. Dubois and Prade [12] when discussing this issue say: To take into account the imprecision of membership functions, we may think of using type-2 fuzzy sets. . . As [13] point out: . . .it may seem problematical, if not paradoxical, that a representation of fuzziness is made using membership grades that are themselves precise real numbers. This paradox leads us to consider the role of type-2 fuzzy sets as an alternative to the type-1 paradigm. Type-2 fuzzy sets [3] are where the membership grades are not numbers in ½0; 1 but are type-1 fuzzy sets. They can be thought of as ‘fuzzy fuzzy’. Type-2 fuzzy sets have been widely used in a number of applications (see [14,15], for examples) and on a number of problems, T2FL has been shown to outperform T1FL (e.g., [5,6]). In previous work [7] the authors have shown that Interval Type-2 Fuzzy Logic (IT2FL) [16] is an appropriate method of modelling a 2 tier distribution network. IT2FL has been used because it is computationally cheaper than general T2FL as it restricts the additional dimension, referred to as secondary membership function, to only take the values 0 or 1. We believe that the extra degree of freedom will allow a better representation of the uncertain and vague nature of data used in SCM. In this paper we propose an IT2FL model of a multiple echelon SC. Further extensions to the model since previous work published by the authors [7] include a measure of customer service level and maximum capacity that nodes can supply. By adding the notion of customer service level to our model we allow the user to trade-off between customer service level and cost. For example, the user may want to see the effect on the operating cost of a supply chain of altering the acceptable service level from 96% to 95%. It may prove that a small change in service level effects a large change in cost, making it attractive for the supply chain manager. By adding finite capacity to the model we enable users to specify limits on how much stock can pass through nodes. This forces the GA to find solutions that take into account these limits (by spreading supply across nodes or periods) instead of assuming that infinite amounts of stock can be handled. Section 3 describes the created model in detail, as well as the test case used for evaluation. 3. Model The proposed model represents the interaction of nodes within a multi-echelon supply chain. In each echelon there are one or more nodes that supply the subsequent echelon with one or more products, and receive stock from the preceding echelon. The first echelon receives goods from an external supplier which is assumed to have infinite capacity, the final echelon supplies the customer. Below the first echelon, capacity is limited by product. Customer demand is provided by a fuzzy forecast which is given to the model at run-time. This forecast represents the demand placed upon the final echelon in the SC. Echelons above this can see their own demand by looking at the suggested inventory levels at the succeeding echelon, as they will be required to supply these items.
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In order to use the model the following information must be provided:
Number of echelons (not including the end customer). Number of nodes in each echelon. Number of end customers. Number of products. Number of periods. Service level required (as a percentage of orders filled completely). Capacities for each product (amount that can be produced at one node in one period). Distance matrix containing distances between nodes in successive echelons. Forecast of customer demand. Suggested inventory levels. Costs including: – Batch cost. – Production cost. – Transport cost. – Holding cost (as a percentage of purchase price). – Purchase price.
Using this information the model will calculate the cost of the given resource plan. The total cost of a plan is made up of the following: Batch cost. The cost of setting up an order is called the batch cost. This represents the cost of administration, setting up any machines that are required, and picking the items for dispatch. There is a flat fee for each batch which is charged once at each warehouse for the production of a particular item for a particular customer. Production cost. Each product is assigned an individual production cost. The total production cost for each batch is calculated by multiplying the number of items by their production cost. Transport cost. The cost of transporting goods is produced using a matrix of distances between warehouses and customers, and a list of transport costs per km/mile. The product of the relevant cost and mileage gives the overall transport cost for a batch of product. Holding cost. A holding cost is charged if a product is kept at a node for more than one period. The cost is calculated by taking a specified percentage of the purchase price of the goods held, for items carried over from one period to the next. The purpose of the charge is to represent the cost of storing items, the depreciating value of stock and the losses incurred by tying up capital in unsold stock. Stock out cost. Stock out is the shortfall of a product in a particular period. In this model we make the assumption that the end customer is always provided with an item. If it is not in the warehouse, it is purchased at purchase price from a competitor. The stock out cost is the sum of the value of items that had to be purchased in this period. Stock out cost is applied to all but the final echelon that supplies the end customer. In the final echelon service level is used to determine how good a solution is. To apply a stock out penalty as well would be to penalise a solution twice for the same shortfall, leading to the GA being pressured to find solutions that satisfy 100% of customer demand, regardless of the service level required by the user. Echelon cost. The echelon cost is the sum of the batch, production, transport, holding and stock out costs for an echelon. Service penalty. To discourage solutions that do not meet service level requirements, a service penalty is added to the cost of poor solutions in proportion to a solution’s distance from the target service level. Service level is calculated by taking the percentage of customer demand that is satisfied. To measure satisfaction the fuzzy sets for the current stock level and the forecast are compared.
An agreement index is calculated by looking at where the sets intersect, or is set to 1 if the inventory level exceeds the forecast. As stated before, stock out penalties are not applied here. It may appear that a more satisfactory solution would be to simply measure service level throughout the chain and not use stock out penalties. However in practice applying service level throughout the model resulted in the GA finding solutions in which the nodes within the chain placed little or no orders on each other, enabling solutions to achieve a good service level without meeting a significant amount of end customer demand. Stock outs need to be charged within the chain however, else the GA finds solutions in which only the final echelon before the customer supplies any product. 3.1. Interval Type-2 Fuzzy Logic IT2FL [16] has been used to represent some of the values within the model. Other than the authors’ previous work, examples in the literature (as discussed in Section 2) have focused on the use of T1FL; we believe there exists an opportunity to exploit the extra degree of uncertainty provided by IT2FL in a model of this type. As the model operates on IT2 fuzzy numbers, fuzzy arithmetic is used to calculate costs. This involves taking fuzzy sets, discretising them, performing the arithmetic operation, and then reconstructing the fuzzy set. In this model, fuzzy sets are represented using a series of a-cuts. Each set is an array of pairs of intervals. Each pair shows the area of values covered at a particular value of l, the first interval is the left hand side of the set, and the second the right. Storing the sets in this way removes the need to discretise before fuzzy arithmetic is performed, and then reconstruct the result. Operations on the IT2 fuzzy sets are performed at interval level, corresponding intervals (at the same l) are taken from two sets, the operation performed and the result stored in a third fuzzy set. The following values are represented by IT2 fuzzy numbers: forecast demand, inventory level, transportation distances, transportation cost, stock out level, stock out cost, carry over and holding cost. For each of these values we can use the linguistic term ‘about n’, e.g., forecast demand of product 1 for customer 1 in period 1 may be ‘about 200’. Fig. 1 shows how the set ‘about 200’ may look with the a-cut representation used, where x is the scale of values being represented. The set is described using a collection of pairs of intervals. Each pairing represents the left and right hand intervals of the set for a given value of l. As stated before, representing the set in this way considerably simplifies the arithmetic operators that are applied as simple interval arithmetic is used throughout, without discretisation or reconstruction.
1 0.8 0.6 µ
0.4 0.2
198
199
200
x
201
202
203
Fig. 1. Interval representation of IT2 fuzzy set ‘about 200’.
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3.2. Defuzzification We have seen that values in the resource planning model are represented using IT2FL. Within the model this is useful as we can describe the uncertainty present in the supply chain. For the user however it is not explicit enough to state ‘This resource plan will have a total cost of about £1,000,000’ or ‘Warehouse A should stock about 300 of Product 1’. In order to produce an output that can be applied to a real-world supply chain, some of the IT2 fuzzy numbers need to be defuzzified. Defuzzification is the process of taking a fuzzy set (in this instance an IT2 fuzzy set) and deriving a single crisp value from it. To do this, the method proposed by [17] is used. This is a widely used method that finds an interval representing the centroid of a type-2 fuzzy set. The interval can then be used to obtain a crisp number by finding its mean. 3.3. Optimisation The focus of the experiment described here is the validation of the multiple echelon IT2FL model that has been constructed. This is to be achieved by using the model to find a good resource plan for a given forecast, confirming that the model can be used to precisely evaluate potential solutions. A GA has been chosen for this purpose. GAs have been used successfully in previous work (e.g., [18,19]) to find good solutions with T1FL models, and the authors have used them to good effect on an IT2FL model [7]. GAs are useful when a search space is too large to allow evaluation of every solution. In this case the GA is used to search for a resource plan that incurs the minimum cost at a given service level. The setup of the GA has been taken from the authors’ previous related work [7] in which it was shown to work well. The GA has a population of 250 individuals and is executed for 500 generations, in all 125,000 solutions are evaluated out of a total possible search space of 1:1318 10112 solutions. New generations consist of: 1% individuals produced with elitism, 20% copied individuals, 20% individuals created with single point crossover and 59% of individuals created using mutation. A description of the chromosome, operators and processes employed follows. Chromosome. The chromosome used to describe potential solutions is a 5 dimensional matrix of inventory levels. The dimensions are ordered as follows: 1. 2. 3. 4. 5.
Echelon. Period. Source. Destination. Product.
Each element of the matrix contains a value representing the number of items held in an echelon, in a time period, by a source node, for a destination node of a particular product. Initial population. The initial population is randomly generated. Each element of the resource plan matrix can be one of six values between 0 and the capacity limit for a particular product. This has been done to reflect the fact that in industry, products are usually manufactured in round quantities. If the model suggests that a warehouse should make 102 of product 1, this could lead to difficulties and extra expense. Limiting the valid inventory numbers also has the side effect of reducing the search space. In this case six values works well with a capacity of 500 as it can be used to represent the values 0–500 in steps of 100. If this method of dividing the possible values proves to be inappropriate for a given problem, the discretisation can be altered. Fitness evaluation. Fitness is evaluated using the IT2FL model described. Candidate solutions are given to the model which evaluates them, and returns the cost. The cheaper a solution is, the
fitter it is judged to be. In reality, cost may not be the only factor in deciding how much of a product to stock at each warehouse. Other criteria such as customer service level can also be used to prevent the system from choosing solutions that do not meet service requirements. To this end, the service level of each solution is calculated as discussed in Section 3 and a penalty added to solutions with a customer service level that does not meet the specified target. Selection. Selection is performed using a fitness ranking proportionate method similar to roulette wheel selection. First, all solutions in the population are ranked by fitness. They are then given a number of elements of an array in proportion to their fitness ranking. For example, if we have a population of 250 the fittest individual would be allocated 250 elements in the array, the second fittest 249 and so on. An element of the array is then selected at random, and the identification number of the individual it contains is used to retrieve a parent. This tombola style approach ensures that it is possible for any individual to be selected, while weighting in favour of those with greater fitness. Crossover. Crossover is achieved with a single point crossover. Two parents are selected using the method of selection described. Then, a new individual is created with the first half of the first parent and the second half of the second parent. Resource plans have 5 dimensions: echelon, period, source node, destination node and product. For crossover, parents are divided by product. If we have 4 products the first 2 products of the first parent and the final 2 of the second parent will be used to create a new individual. This spans throughout all dimensions, so for each echelon, period, source and destination the first 2 products would be taken from the first parent and the second 2 from the second. Mutation. To create a mutated individual, a parent is selected, then one of the elements of its resource plan is randomly replaced with another valid value to create a new child. 3.4. Test scenario To test the model a scenario has been created. Table 1 shows the configuration of the supply chain. Each customer requires 200 items of each product, and each node can handle a maximum of 500 items in a period. Table 2 gives the operating costs of the supply chain. To calculate transport costs, the model needs to know the distance between nodes. Table 3 shows the distances between nodes in successive echelons. To find a good cost for comparison, a resource plan was created that matched demand in each period, using nodes that are closest
Table 1 Test case supply chain. Echelons
4
Nodes Products Periods Service level required
2 per echelon 2 6 95%
Table 2 Supply chain costs. Product
Batch cost Production cost Transport cost Purchase value Holding cost
1
2
£100 £30 £2 £50 £5
£40 £3 £70 £7
S. Miller, R. John / Knowledge-Based Systems 23 (2010) 363–368 Table 3 Supply chain distances.
1600000 Dest. node 1
2
Echelon 1 Node 1 Node 2
100 200
200 100
Echelon 2 Node 1 Node 2
200 100
100 200
Echelon 3 Node 1 Node 2
100 200
200 100
Cost of best solution found (£s)
Src. node
367
1400000 1200000 1000000 800000 600000 400000 200000
Table 4 Results of first 10 test runs.
0 0
Seed
Cost
Service level (%)
Time (s)
0 1 2 3 4 5 6 7 8 9
£502,496.75 £511,470.63 £511,552.31 £520,084.69 £500,498.69 £510,751.56 £505,909.00 £502,990.06 £505,564.56 £522,328.06
95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83
2000 2080 2018 2049 1996 2061 2021 2001 2017 2052
to one another. When put into the model, a cost of £529,287.31 was produced. It should be noted that this plan will satisfy 100% of customer orders, the test scenario will allow solutions that satisfy 95% of orders and as such may be cheaper. 4. Results To test whether the model could be used to find good resource plans the model was used with the GA described in Section 3.3. The GA was executed 40 times with differing random seeds. Table 4 shows the cost of the best solutions found in each of the first 10 runs along with its service level and time. Table 5 shows the mean cost, time and service level of the best solutions in all 40 tests, and the standard deviation in the costs. The results show that using the model, the GA is able to consistently find solutions that are cheaper than the benchmark solution. All of the solutions found achieve a service level close to the target set, this shows that the penalties imposed by the model encourage discovery of solutions that match the requirements of the user. An extra test was conducted to see how the quality of solutions would be affected by specifying a customer service level of 100%. This would also allow a comparison between the solution found and the ideal solution described earlier. The ideal solution is the perfect solution for a 100% service level costing £529,287.31. In the experiment (with one run only) the GA was able to find a solution that satisfied all end customer demand costing £535,195.75 in 500 generations, just 1.1% more than the ideal solution. The generation limit is applied to constrain the amount time it takes for the GA to run. Previous work by the authors [7] has shown that if the GA is left to run, it will find the optimal solution to a similar problem. Table 5 Overall results of test runs.
50
100 150 200 250 300 350 400 450 500 Number of generations
Fig. 2. Evolution of cost of best solution found.
We can use the results as an indicator of the validity of the multiple echelon model presented. The GA is guided purely by fitness and is blind to the practicalities of the solutions that it finds. Fig. 2 shows the progress of one of the test runs. The fact that it evolved toward cost-effective sensible plans shows that the model is able to differentiate between good and bad solutions to a high level of detail. If this were not the case the GA may find solutions that are cheap but non-sensical, or could result in a situation where the GA is unable to find good resource plans at all. 5. Conclusion In this paper we have presented an IT2FL multiple echelon supply chain model. Using IT2FL allows us to model the fuzziness present in supply chain operation, allowing the user to input data that is uncertain. IT2FL has been chosen as it is believed that the extra level of uncertainty over T1FL offered will benefit a model of this type, while avoiding the computational complexity of a general T2FL model. The model builds on previous work by the authors [7] by representing multiple echelon supply chains, customer service level and finite capacity. Using the model, it was shown that a GA was able to find good multiple echelon resource plans that were both costeffective and sensible. In this case the GA is solely guided by cost and service level, it is essentially ‘blind’ to the practicality of the solutions it finds. That it was able to find good solutions is taken as indication of the model’s validity. Work is ongoing. Future work will include testing the model on more complex problems that have been solved by other methods to allow comparison, and applying the model to real world scenarios. Extensions of the model will include looking at how alternative methods of representing type-2 fuzzy sets (e.g., geometric [20]) could be used to improve the model’s description of the uncertainty in the supply chain. Acknowledgement The research reported here has been funded by the Technology Strategy Board(Grant No. H0254E). References
Mean cost
Std. dev. of cost
Mean service level (%)
Mean time (s)
£508,043.22
6093.19
95.83
2499.68
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Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
An Interval Type-2 Fuzzy multiple echelon supply chain model Simon Miller *, Robert John Centre for Computational Intelligence, De Montfort University, Leicester, UK
a r t i c l e
i n f o
Article history: Available online 20 November 2009 Keywords: Interval Type-2 Fuzzy Logic Multi-echelon supply chain modelling Genetic algorithms
a b s t r a c t Planning resources for a supply chain is a major factor determining its success or failure. In this paper we build on previous work introducing an Interval Type-2 Fuzzy Logic model of a multiple echelon supply chain. It is believed that the additional degree of uncertainty provided by Interval Type-2 Fuzzy Logic will allow for better representation of the uncertainty and vagueness present in resource planning models. First, the subject of Supply Chain Management is introduced, then some background is given on related work using Type-1 Fuzzy Logic. A description of the Interval Type-2 Fuzzy model is given, and a test scenario detailed. A Genetic Algorithm uses the model to search for a near-optimal plan for the scenario. A discussion of the results follows, along with conclusions and details of intended further work. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction There are a number of definitions of Supply Chain Management (SCM), each having minor variances but describing the same core idea. SCM is the management of material flow in and between facilities including vendors, manufacturing/assembly plants and distribution centres [1]. Planning the allocation of resources within a Supply Chain (SC) has been critical to the success of manufacturers, warehouses and retailers for many years. Mastering the flow of materials from their creation to the point of sale offers considerable advantages to those within a well managed SC. Poorly managed resources result in two main problems: stock outs (a shortage of stock) and surplus stock. The consequence of stock outs is lost sales, and potentially lost customers. Surplus stock causes additional holding cost and the possibility of stock losing value as it becomes obsolete. Holding some surplus stock is advantageous however; safety stock can be used in the event of an unexpected increase in demand or to cover lost productivity. The problem has been addressed numerous times in the literature using different approaches. An overview of traditional and Computational Intelligence approaches to supply-chain resource planning can be found in [2]. In this paper we present a novel approach to modelling the supply chain using Type-2 Fuzzy Logic (T2FL) [3] that is optimised by means of a Genetic Algorithm (GA) [4]. This research is part of a project on demand forecasting and resource planning (Data Storage, Management, Retrieval and Analysis: Improving Customer Demand and Cost Forecasting Methods, * Corresponding author. Tel.: +44 116 2078408. E-mail addresses: [email protected] (S. Miller), [email protected] (R. John). 0950-7051/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2009.11.016
funded by the Technology Strategy Board in the UK). The project aims to: (1) improve the forecasting of demand by using a variety of disparate sources of data and statistical and Machine Learning (ML) methods for analysis; (2) improve the allocation of resources in which the generated forecast is used as an input, the output being is a (long- or short-term) plan of raw materials and resources within the supply chain required in order to meet such demand. The research presented here is intended to address the second aim. Various degrees of uncertainty are present in the different data sources used. This uncertainty is further amplified in the generated forecast by applying methods of analysis with (again) varying degrees of inherent uncertainty. Furthermore, other data that is often used in resource planning such as transportation and other costs, customer satisfaction information, etc. is also uncertain. Therefore, FL and especially T2FL are particularly appropriate for this problem. While Type-1 FL (T1FL) has successfully been used many times for modelling SC operation (see Section 2), T2FL has been shown to offer a better representation of uncertainty on a number of problems (e.g., [5,6]). In [7] the authors applied Interval Type-2 Fuzzy Logic (IT2FL) to a 2 tier distribution resource model and it was shown to work well. This paper presents an IT2FL model of a multiple echelon SC problem which is optimised by a GA to find a near-optimal configuration. The term ‘echelon’ (or ‘tier’) is used to describe a group of nodes that operate at the same stage in a supply chain, e.g., retailers. A multiple echelon supply chain is one that incorporates many stages, e.g., suppliers, manufacturers, warehouses and retailers. The paper is organised as follows: Section 2 discusses Fuzzy Logic and its application to SCM. Section 3 introduces the model and the test case used for evaluation. The results from the experiments on the test case are presented in Section 4. Section 5 concludes the paper with a summary and future research directions.
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2. Fuzzy Logic for supply chain management FL and GAs have been successfully used for supply chain modelling [2] and are particularly appropriate for this problem due to their capacity to tackle the inherent vagueness, uncertainty and incompleteness of the data used. A GA [4] is a heuristic search technique inspired by evolutionary biology. Selection, crossover and mutation are applied to a population of individuals representing solutions in order to find a near-optimal solution. FL is based on fuzzy set theory and provides methods for modelling and reasoning under uncertainty, a characteristic present in many problems, which makes FL a valuable approach. It allows data to be represented in intuitive linguistic categories instead of using precise (crisp) numbers which might not be known, necessary, or may in general be too restrictive. For example, statements such as ‘the cost is about n’, ‘the speed is high’ and ‘the book is very old’ can be described. These categories are represented by means of a membership function which defines the degree to which a crisp number belongs to the category. In this research the aim is to allow linguistic terms to be used by Supply Chain Managers when describing their operation. For example, instead of asking for crisp numbers to describe the current stock level of a product, we may allow them to make statements like ‘Warehouse A has About 500 of product 1’ or even more abstract ‘Warehouse A’s stock level of product 1 is low’. By removing the need for exact information it is possible to produce a system that is much more usable when the information available is vague, uncertain or incomplete. T1FL has been applied to SC modelling numerous times with good results. Some of the research that is considered most relevant for this paper is discussed in the following paragraphs. Petrovic et al. [8] use fuzzy sets to model vagueness and imprecision in customer demand, external supplier reliability and supply within a multiple echelon single product supply chain. The system demonstrates the effect of differing conditions and strategies on fill rate and holding costs of a SC. The results of the evaluation show that there is a slight improvement in the performance of the SC when the inventories are partially co-ordinated. In [9] two-level fuzzy optimisation is employed to find ideal order-up-to quantities in a one warehouse-multiple retailer SC. The problem is decomposed for individual control of the warehouse and retailers; a coordination mechanism provides overall control of the SC. Customer demand, inventory levels, holding cost, and shortage cost are represented by fuzzy sets. A measure of satisfaction is derived from the cost incurred at each element. The solution produced by the system is the best compromise between members, though not necessarily the cheapest. In Aliev et al. [10] a fuzzy system is used with a GA to model a 3 tier SC. The model uses a global policy of management with emphasis on integrating the production and distribution models. The GA searches for a near-optimal configuration; fuzzy sets are used to describe costs, returns, production capacities, storage capacities and forecasts. The proposed fuzzy method, a crisp method and a non-integrated method are compared. The crisp system is unable to produce a feasible configuration if actual demand is lower than the forecast. In contrast, the fuzzy model presented is robust and able to cope with fluctuation in demand and production capacity with little impact on profitability. The non-integrated model performed significantly worse than the fuzzy integrated model. A similar approach is presented by [11] in their multiple echelon SC model. FL is used to represent customer demand, processing time and delivery reliability; a GA finds order-up-to levels. The system attempts to find the configuration that incurs the minimum cost. An optimism–pessimism index is set by the user and passed to the system. When optimistic, the model assumes the best case
scenario for material response time. A pessimistic attitude produces the opposite effect. The results show that more pessimistic strategies increase the fill rate, reducing the sales lost through stock outs, and incur higher inventory cost as more stock is kept. More optimistic strategies result in a drop in fill rate and an increase in sales loss, though inventory cost is also reduced. In these examples we have seen how T1FL has been used to tackle the resource planning problem. However type-1 fuzzy sets represent the fuzziness of the particular problem using a ‘non-fuzzy’ (or crisp) representation – a number in [0, 1]. Dubois and Prade [12] when discussing this issue say: To take into account the imprecision of membership functions, we may think of using type-2 fuzzy sets. . . As [13] point out: . . .it may seem problematical, if not paradoxical, that a representation of fuzziness is made using membership grades that are themselves precise real numbers. This paradox leads us to consider the role of type-2 fuzzy sets as an alternative to the type-1 paradigm. Type-2 fuzzy sets [3] are where the membership grades are not numbers in ½0; 1 but are type-1 fuzzy sets. They can be thought of as ‘fuzzy fuzzy’. Type-2 fuzzy sets have been widely used in a number of applications (see [14,15], for examples) and on a number of problems, T2FL has been shown to outperform T1FL (e.g., [5,6]). In previous work [7] the authors have shown that Interval Type-2 Fuzzy Logic (IT2FL) [16] is an appropriate method of modelling a 2 tier distribution network. IT2FL has been used because it is computationally cheaper than general T2FL as it restricts the additional dimension, referred to as secondary membership function, to only take the values 0 or 1. We believe that the extra degree of freedom will allow a better representation of the uncertain and vague nature of data used in SCM. In this paper we propose an IT2FL model of a multiple echelon SC. Further extensions to the model since previous work published by the authors [7] include a measure of customer service level and maximum capacity that nodes can supply. By adding the notion of customer service level to our model we allow the user to trade-off between customer service level and cost. For example, the user may want to see the effect on the operating cost of a supply chain of altering the acceptable service level from 96% to 95%. It may prove that a small change in service level effects a large change in cost, making it attractive for the supply chain manager. By adding finite capacity to the model we enable users to specify limits on how much stock can pass through nodes. This forces the GA to find solutions that take into account these limits (by spreading supply across nodes or periods) instead of assuming that infinite amounts of stock can be handled. Section 3 describes the created model in detail, as well as the test case used for evaluation. 3. Model The proposed model represents the interaction of nodes within a multi-echelon supply chain. In each echelon there are one or more nodes that supply the subsequent echelon with one or more products, and receive stock from the preceding echelon. The first echelon receives goods from an external supplier which is assumed to have infinite capacity, the final echelon supplies the customer. Below the first echelon, capacity is limited by product. Customer demand is provided by a fuzzy forecast which is given to the model at run-time. This forecast represents the demand placed upon the final echelon in the SC. Echelons above this can see their own demand by looking at the suggested inventory levels at the succeeding echelon, as they will be required to supply these items.
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In order to use the model the following information must be provided:
Number of echelons (not including the end customer). Number of nodes in each echelon. Number of end customers. Number of products. Number of periods. Service level required (as a percentage of orders filled completely). Capacities for each product (amount that can be produced at one node in one period). Distance matrix containing distances between nodes in successive echelons. Forecast of customer demand. Suggested inventory levels. Costs including: – Batch cost. – Production cost. – Transport cost. – Holding cost (as a percentage of purchase price). – Purchase price.
Using this information the model will calculate the cost of the given resource plan. The total cost of a plan is made up of the following: Batch cost. The cost of setting up an order is called the batch cost. This represents the cost of administration, setting up any machines that are required, and picking the items for dispatch. There is a flat fee for each batch which is charged once at each warehouse for the production of a particular item for a particular customer. Production cost. Each product is assigned an individual production cost. The total production cost for each batch is calculated by multiplying the number of items by their production cost. Transport cost. The cost of transporting goods is produced using a matrix of distances between warehouses and customers, and a list of transport costs per km/mile. The product of the relevant cost and mileage gives the overall transport cost for a batch of product. Holding cost. A holding cost is charged if a product is kept at a node for more than one period. The cost is calculated by taking a specified percentage of the purchase price of the goods held, for items carried over from one period to the next. The purpose of the charge is to represent the cost of storing items, the depreciating value of stock and the losses incurred by tying up capital in unsold stock. Stock out cost. Stock out is the shortfall of a product in a particular period. In this model we make the assumption that the end customer is always provided with an item. If it is not in the warehouse, it is purchased at purchase price from a competitor. The stock out cost is the sum of the value of items that had to be purchased in this period. Stock out cost is applied to all but the final echelon that supplies the end customer. In the final echelon service level is used to determine how good a solution is. To apply a stock out penalty as well would be to penalise a solution twice for the same shortfall, leading to the GA being pressured to find solutions that satisfy 100% of customer demand, regardless of the service level required by the user. Echelon cost. The echelon cost is the sum of the batch, production, transport, holding and stock out costs for an echelon. Service penalty. To discourage solutions that do not meet service level requirements, a service penalty is added to the cost of poor solutions in proportion to a solution’s distance from the target service level. Service level is calculated by taking the percentage of customer demand that is satisfied. To measure satisfaction the fuzzy sets for the current stock level and the forecast are compared.
An agreement index is calculated by looking at where the sets intersect, or is set to 1 if the inventory level exceeds the forecast. As stated before, stock out penalties are not applied here. It may appear that a more satisfactory solution would be to simply measure service level throughout the chain and not use stock out penalties. However in practice applying service level throughout the model resulted in the GA finding solutions in which the nodes within the chain placed little or no orders on each other, enabling solutions to achieve a good service level without meeting a significant amount of end customer demand. Stock outs need to be charged within the chain however, else the GA finds solutions in which only the final echelon before the customer supplies any product. 3.1. Interval Type-2 Fuzzy Logic IT2FL [16] has been used to represent some of the values within the model. Other than the authors’ previous work, examples in the literature (as discussed in Section 2) have focused on the use of T1FL; we believe there exists an opportunity to exploit the extra degree of uncertainty provided by IT2FL in a model of this type. As the model operates on IT2 fuzzy numbers, fuzzy arithmetic is used to calculate costs. This involves taking fuzzy sets, discretising them, performing the arithmetic operation, and then reconstructing the fuzzy set. In this model, fuzzy sets are represented using a series of a-cuts. Each set is an array of pairs of intervals. Each pair shows the area of values covered at a particular value of l, the first interval is the left hand side of the set, and the second the right. Storing the sets in this way removes the need to discretise before fuzzy arithmetic is performed, and then reconstruct the result. Operations on the IT2 fuzzy sets are performed at interval level, corresponding intervals (at the same l) are taken from two sets, the operation performed and the result stored in a third fuzzy set. The following values are represented by IT2 fuzzy numbers: forecast demand, inventory level, transportation distances, transportation cost, stock out level, stock out cost, carry over and holding cost. For each of these values we can use the linguistic term ‘about n’, e.g., forecast demand of product 1 for customer 1 in period 1 may be ‘about 200’. Fig. 1 shows how the set ‘about 200’ may look with the a-cut representation used, where x is the scale of values being represented. The set is described using a collection of pairs of intervals. Each pairing represents the left and right hand intervals of the set for a given value of l. As stated before, representing the set in this way considerably simplifies the arithmetic operators that are applied as simple interval arithmetic is used throughout, without discretisation or reconstruction.
1 0.8 0.6 µ
0.4 0.2
198
199
200
x
201
202
203
Fig. 1. Interval representation of IT2 fuzzy set ‘about 200’.
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3.2. Defuzzification We have seen that values in the resource planning model are represented using IT2FL. Within the model this is useful as we can describe the uncertainty present in the supply chain. For the user however it is not explicit enough to state ‘This resource plan will have a total cost of about £1,000,000’ or ‘Warehouse A should stock about 300 of Product 1’. In order to produce an output that can be applied to a real-world supply chain, some of the IT2 fuzzy numbers need to be defuzzified. Defuzzification is the process of taking a fuzzy set (in this instance an IT2 fuzzy set) and deriving a single crisp value from it. To do this, the method proposed by [17] is used. This is a widely used method that finds an interval representing the centroid of a type-2 fuzzy set. The interval can then be used to obtain a crisp number by finding its mean. 3.3. Optimisation The focus of the experiment described here is the validation of the multiple echelon IT2FL model that has been constructed. This is to be achieved by using the model to find a good resource plan for a given forecast, confirming that the model can be used to precisely evaluate potential solutions. A GA has been chosen for this purpose. GAs have been used successfully in previous work (e.g., [18,19]) to find good solutions with T1FL models, and the authors have used them to good effect on an IT2FL model [7]. GAs are useful when a search space is too large to allow evaluation of every solution. In this case the GA is used to search for a resource plan that incurs the minimum cost at a given service level. The setup of the GA has been taken from the authors’ previous related work [7] in which it was shown to work well. The GA has a population of 250 individuals and is executed for 500 generations, in all 125,000 solutions are evaluated out of a total possible search space of 1:1318 10112 solutions. New generations consist of: 1% individuals produced with elitism, 20% copied individuals, 20% individuals created with single point crossover and 59% of individuals created using mutation. A description of the chromosome, operators and processes employed follows. Chromosome. The chromosome used to describe potential solutions is a 5 dimensional matrix of inventory levels. The dimensions are ordered as follows: 1. 2. 3. 4. 5.
Echelon. Period. Source. Destination. Product.
Each element of the matrix contains a value representing the number of items held in an echelon, in a time period, by a source node, for a destination node of a particular product. Initial population. The initial population is randomly generated. Each element of the resource plan matrix can be one of six values between 0 and the capacity limit for a particular product. This has been done to reflect the fact that in industry, products are usually manufactured in round quantities. If the model suggests that a warehouse should make 102 of product 1, this could lead to difficulties and extra expense. Limiting the valid inventory numbers also has the side effect of reducing the search space. In this case six values works well with a capacity of 500 as it can be used to represent the values 0–500 in steps of 100. If this method of dividing the possible values proves to be inappropriate for a given problem, the discretisation can be altered. Fitness evaluation. Fitness is evaluated using the IT2FL model described. Candidate solutions are given to the model which evaluates them, and returns the cost. The cheaper a solution is, the
fitter it is judged to be. In reality, cost may not be the only factor in deciding how much of a product to stock at each warehouse. Other criteria such as customer service level can also be used to prevent the system from choosing solutions that do not meet service requirements. To this end, the service level of each solution is calculated as discussed in Section 3 and a penalty added to solutions with a customer service level that does not meet the specified target. Selection. Selection is performed using a fitness ranking proportionate method similar to roulette wheel selection. First, all solutions in the population are ranked by fitness. They are then given a number of elements of an array in proportion to their fitness ranking. For example, if we have a population of 250 the fittest individual would be allocated 250 elements in the array, the second fittest 249 and so on. An element of the array is then selected at random, and the identification number of the individual it contains is used to retrieve a parent. This tombola style approach ensures that it is possible for any individual to be selected, while weighting in favour of those with greater fitness. Crossover. Crossover is achieved with a single point crossover. Two parents are selected using the method of selection described. Then, a new individual is created with the first half of the first parent and the second half of the second parent. Resource plans have 5 dimensions: echelon, period, source node, destination node and product. For crossover, parents are divided by product. If we have 4 products the first 2 products of the first parent and the final 2 of the second parent will be used to create a new individual. This spans throughout all dimensions, so for each echelon, period, source and destination the first 2 products would be taken from the first parent and the second 2 from the second. Mutation. To create a mutated individual, a parent is selected, then one of the elements of its resource plan is randomly replaced with another valid value to create a new child. 3.4. Test scenario To test the model a scenario has been created. Table 1 shows the configuration of the supply chain. Each customer requires 200 items of each product, and each node can handle a maximum of 500 items in a period. Table 2 gives the operating costs of the supply chain. To calculate transport costs, the model needs to know the distance between nodes. Table 3 shows the distances between nodes in successive echelons. To find a good cost for comparison, a resource plan was created that matched demand in each period, using nodes that are closest
Table 1 Test case supply chain. Echelons
4
Nodes Products Periods Service level required
2 per echelon 2 6 95%
Table 2 Supply chain costs. Product
Batch cost Production cost Transport cost Purchase value Holding cost
1
2
£100 £30 £2 £50 £5
£40 £3 £70 £7
S. Miller, R. John / Knowledge-Based Systems 23 (2010) 363–368 Table 3 Supply chain distances.
1600000 Dest. node 1
2
Echelon 1 Node 1 Node 2
100 200
200 100
Echelon 2 Node 1 Node 2
200 100
100 200
Echelon 3 Node 1 Node 2
100 200
200 100
Cost of best solution found (£s)
Src. node
367
1400000 1200000 1000000 800000 600000 400000 200000
Table 4 Results of first 10 test runs.
0 0
Seed
Cost
Service level (%)
Time (s)
0 1 2 3 4 5 6 7 8 9
£502,496.75 £511,470.63 £511,552.31 £520,084.69 £500,498.69 £510,751.56 £505,909.00 £502,990.06 £505,564.56 £522,328.06
95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83
2000 2080 2018 2049 1996 2061 2021 2001 2017 2052
to one another. When put into the model, a cost of £529,287.31 was produced. It should be noted that this plan will satisfy 100% of customer orders, the test scenario will allow solutions that satisfy 95% of orders and as such may be cheaper. 4. Results To test whether the model could be used to find good resource plans the model was used with the GA described in Section 3.3. The GA was executed 40 times with differing random seeds. Table 4 shows the cost of the best solutions found in each of the first 10 runs along with its service level and time. Table 5 shows the mean cost, time and service level of the best solutions in all 40 tests, and the standard deviation in the costs. The results show that using the model, the GA is able to consistently find solutions that are cheaper than the benchmark solution. All of the solutions found achieve a service level close to the target set, this shows that the penalties imposed by the model encourage discovery of solutions that match the requirements of the user. An extra test was conducted to see how the quality of solutions would be affected by specifying a customer service level of 100%. This would also allow a comparison between the solution found and the ideal solution described earlier. The ideal solution is the perfect solution for a 100% service level costing £529,287.31. In the experiment (with one run only) the GA was able to find a solution that satisfied all end customer demand costing £535,195.75 in 500 generations, just 1.1% more than the ideal solution. The generation limit is applied to constrain the amount time it takes for the GA to run. Previous work by the authors [7] has shown that if the GA is left to run, it will find the optimal solution to a similar problem. Table 5 Overall results of test runs.
50
100 150 200 250 300 350 400 450 500 Number of generations
Fig. 2. Evolution of cost of best solution found.
We can use the results as an indicator of the validity of the multiple echelon model presented. The GA is guided purely by fitness and is blind to the practicalities of the solutions that it finds. Fig. 2 shows the progress of one of the test runs. The fact that it evolved toward cost-effective sensible plans shows that the model is able to differentiate between good and bad solutions to a high level of detail. If this were not the case the GA may find solutions that are cheap but non-sensical, or could result in a situation where the GA is unable to find good resource plans at all. 5. Conclusion In this paper we have presented an IT2FL multiple echelon supply chain model. Using IT2FL allows us to model the fuzziness present in supply chain operation, allowing the user to input data that is uncertain. IT2FL has been chosen as it is believed that the extra level of uncertainty over T1FL offered will benefit a model of this type, while avoiding the computational complexity of a general T2FL model. The model builds on previous work by the authors [7] by representing multiple echelon supply chains, customer service level and finite capacity. Using the model, it was shown that a GA was able to find good multiple echelon resource plans that were both costeffective and sensible. In this case the GA is solely guided by cost and service level, it is essentially ‘blind’ to the practicality of the solutions it finds. That it was able to find good solutions is taken as indication of the model’s validity. Work is ongoing. Future work will include testing the model on more complex problems that have been solved by other methods to allow comparison, and applying the model to real world scenarios. Extensions of the model will include looking at how alternative methods of representing type-2 fuzzy sets (e.g., geometric [20]) could be used to improve the model’s description of the uncertainty in the supply chain. Acknowledgement The research reported here has been funded by the Technology Strategy Board(Grant No. H0254E). References
Mean cost
Std. dev. of cost
Mean service level (%)
Mean time (s)
£508,043.22
6093.19
95.83
2499.68
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