Interval Type-2 Fuzzy Modelling and Simulated

Interval Type-2 Fuzzy Modelling and Simulated Annealing for Real-World Inventory Management Simon Miller1 , Mario Gongora2, and Robert John2 1

Horizon Digital Economy Research / Intelligent Modelling and Analysis, University of Nottingham, Nottingham, UK 2 Centre for Computational Intelligence, De Montfort University, Leicester, UK [email protected], {mgongora,rij}@dmu.ac.uk

Abstract. The modelling of real-world complex systems is an area of ongoing interest for the research community. Real-world systems present a variety of challenges not least of which is the problem of uncertainty inherent in their operation. In this research the problem of inventory management was chosen. The goal was to discover a suitable configuration for a Simulated Annealing search with a fuzzy inventory management problem. A hybrid of a series of Simulated Annealing configurations and an Interval Type-2 Fuzzy Logic model were used to identify suitable inventory plans for a large-scale real-world problem supplied by collaborators on a Technology Strategy Board research project (ref: H0254E). Keywords: Interval Type-2 Fuzzy Logic, Simulated Annealing, Realworld problems, Inventory Management.

1

Introduction

Optimising inventory levels within a supply chain is an area of ongoing interest for supply chain managers. Planning the allocation of resources within a supply chain has been critical to the success of manufacturers, warehouses and retailers for many years. Poorly managed resources result in two main problems: stock outs and surplus stock. The consequence of stock outs is lost sales, and potentially lost customers. Surplus stock results in added 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. Various degrees of uncertainty are present in the different data sources used in supply chain management (SCM). This uncertainty is further amplified in demand forecasts by applying methods of analysis which have varying degrees of inherent uncertainty within themselves. Furthermore, other data that is often used in resource planning such as transportation and other costs, customer satisfaction information, etc. is also uncertain. Therefore, Fuzzy Logic (FL) and E. Corchado, M. Kurzy´ nski, M. Wo´ zniak (Eds.): HAIS 2011, Part I, LNAI 6678, pp. 231–238, 2011. c Springer-Verlag Berlin Heidelberg 2011

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especially Type-2 Fuzzy Logic (T2FL) are particularly appropriate for this problem. While traditional (or Type-1) FL (T1FL) has successfully been used many times for modelling supply chain operation (e.g., [16] and [1]), T2FL has been shown to offer a better representation of uncertainty on a number of problems (e.g., [4] and [8]). In this research an Interval Type-2 Fuzzy Logic (IT2FL) model is used, as it benefits from some of the advantages of T2FL, while incurring considerably less computation. Section 2 provides details of the model. The search spaces involved in inventory management are often very large even for a relatively simple problem. As such, it is not possible to find a resource plan using an exhaustive search, a more efficient method needs to be selected. In this research a hybrid of a series of Simulated Annealing (SA) configurations and the IT2FL model mentioned previously were evaluated for this purpose. Section 3 describes SA in more detail, sections 4 and 5 give the test scenario used and the results respectively. Section 6 considers the conclusions that can be drawn from the work, and what form future work might take.

2

Model

The proposed model represents the interaction of nodes within a multi-tier supply chain. Figure 1 provides an example of a typical supply chain. In each tier there are one or more nodes that supply the subsequent tier with one or more products, and receive stock from the preceding tier. The first tier receives goods from an external supplier which is assumed to have infinite capacity, the final tier supplies the customer. Below the first tier, capacity is limited by node and product. Raw Materials

A

B

C

C1

Fig. 1. A typical supply chain

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 tier in the SC. Tiers above this can see their own demand by looking at the suggested inventory levels at the succeeding tier, as they will be required to supply these items. In order to use the model the following information must be provided: the number of tiers not including the end customer, number of nodes in each tier, number of end customers, number of products, number of periods, service level required as a percentage of orders filled completely, capacities for each product at each node (amount that can be produced in one period), lead time in periods for production/supply of each product at each node, minimum order and unit of order quantities for each product at each node, initial stock levels for each node, distance between nodes in successive echelons, forecast of customer demand, suggested inventory levels and costs including: batch cost by node, production

IT2 Fuzzy Modelling and SA Search for Real-World Inventory Management

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cost by product, transport cost, holding cost as a percentage of purchase price and purchase price by product. 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 represents the cost of administration, setting up any machines that are required, and picking the items for dispatch. Production cost is the cost of producing an item of a particular product. Transport cost is a fixed value representative of the cost per mile of transporting a batch of goods from a node to another node in the succeeding echelon. Holding cost is the cost of storing items. Stock out cost is charged for the shortfall of a product in a particular period. Service penalties are added to the cost of sub-optimal solutions that do not meet service level requirements. 2.1

Interval Type-2 Fuzzy Logic

As stated previously T1FL has been used to tackle the resource planning problem. However Type-1 (T1) fuzzy sets represent the fuzziness of the particular problem using a ‘non-fuzzy’ (or crisp) representation - a number in [0, 1]. As Klir and Folger [10] 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 (T2) fuzzy sets as an alternative to the T1 paradigm. T2 fuzzy sets [13] represent membership grades not as numbers in [0, 1], but as T1 fuzzy sets. T2 fuzzy sets have been widely used in a number of applications (see [6] and [11] for examples), and on a number of problems T2FL has been shown to outperform T1FL (e.g., [4] and [8]). Some work has been done regarding the use of optimisation methods to design T2 Fuzzy sets (e.g., [18]) however, in this work we do not optimise the sets; we use an Interval Type-2 (IT2) fuzzy model as the means to evaluate resource plans, as this work focuses on optimising the latter. Resource plans naturally take the format of a matrix of values detailing inventory by time period, node and product. In previous work the authors have shown that Interval Type-2 Fuzzy Logic (IT2FL) [12] is an appropriate method of modelling a multi-echelon supply chain [14]. IT2FL has been used because it is computationally cheaper than general T2FL as it restricts the additional dimension, referred to as the secondary membership function, to only take the values 0 or 1. We believe that the extra degree of freedom offered over a T1FL model will allow a better representation of the uncertain and vague nature of data used in SCM. Fuzzy arithmetic is used to calculate costs. In this model, fuzzy sets are represented using a series of α-cuts, each set is an array of pairs of intervals. Each pair shows the area of values in x covered at a particular value of μ, the first interval is the left hand side of the set, and the second the right. Operations on

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S. Miller, M. Gongora, and R. John 1 0.8 0.6 µ 0.4 0.2

198

199

200 x

201

202

203

Fig. 2. Interval representation of IT2 fuzzy set ‘about 200’

the IT2 fuzzy sets are performed at the interval level, corresponding intervals (at the same μ) are taken from two sets, the operation performed and the result stored in a third fuzzy set. Forecast demand, inventory level, transportation distance, transportation cost, stock out level, stock out cost, carry over and holding cost are represented with IT2 fuzzy numbers. For each of these values we can use the linguistic term ‘about n’, e.g., forecast demand of product a for customer b in period c may be ‘about 200’. Figure 2 shows how the set ‘about 200’ may look with the α-cut representation used, where x is the scale of values being represented. 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 and deriving a crisp value from it. To do this, the Karnik-Mendel method proposed in [7] is used. This is a widely used method that finds an interval representing the centroid of an IT2 fuzzy set. The interval can then be used to obtain a crisp number by finding its centre.

3

Optimisation

The purpose of these experiments is to evaluate the performance of a set of optimisation configurations. For the purposes of these experiments, Simulated Annealing has been chosen to conduct the search as it has been shown to work well in previous experiments (see [15]), and does not require the maintenance of a large population as is the case with some other methods (e.g., a Genetic Algorithm); for this large-scale problem, this is critical. 3.1

Simulated Annealing

Simulated Annealing (SA) [9] is inspired by a real-world phenomenon, in this case the process of heating and cooling (annealing) of metals to reduce defects. An initial solution is created, then a neighbouring solution is selected and compared


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with it. The probability of the algorithm accepting the neighbour as the current solution is based upon a temperature value and the difference in quality between the two solutions. The higher the temperature value, the more likely it is that the algorithm will accept an inferior solution. The process is then repeated using the selected solution as a starting point. Over the course of a run the temperature is gradually decreased, making inferior choices less likely. SA has been successfully applied to a number of optimisation problems. For example, in [17] a series of experiments are described that show SA is an appropriate choice for optimising a production/inventory system. In [2] SA is used to solve production scheduling problems. By using SA and IT2FL together the advantages that both methods offer can be exploited. In [5] a number of real-world problems closely related to inventory management are described including: process planning, assembly line management and dynamic scheduling that have been tackled using hybrid artificial intelligence approaches. T1FL and SA have previously been used for operations management in [3] where SA is used with a fuzzy job shop scheduling problem to find suitable schedules.

4

Test Scenario

The data set is part of a large-scale real-world scenario provided by a UK consultancy that design and manage supply chain operations for other organisations. An overview of the test scenario used can be seen in Table 1. Table 1. Real-world scenario test supply chain setup Tiers 3 Nodes 1,2,100 Products 100 Periods 13 months Batch Cost 100 Distance 100 Stock out Multiplier 25 Holding Cost 10% of purch. price Production Cost 1.2 Purchase Cost 1.2 Transport Cost 0.1 Transport Distance 100 miles Service Level 100%

In the first tier is a manufacturing warehouse based in Newquay, the second tier contains warehouses in Stockport England, Dublin Ireland and Orléans France. The final tier contains customers spread over the UK, Ireland, France and Spain. Forecast demand, capacities, leadtimes, minimum order quantities, unit of order quantities and initial stock levels were produced using information supplied by the UK consultancy.

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5


Results

With the test scenario described, a number of tests were performed. The configurations can be seen in Table 2 along with the results. For these exploratory tests, time was a factor; only a single test was executed for each configuration, and fewer evaluations are performed than in previous experiments (e.g., [15]). Configurations were chosen to result in a comparable number of evaluations for each test. The intention was to use the results of this study to conduct further research with longer experiments, taking advantage of multiple runs. Even with these restrictions, a typical test took approximately 4 hours and 20 minutes. Table 2. Results of real-world data SA tests Stage No. Test No. Temp. Dec. 1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

100,000 50,000 25,000 12,500 125,000 150,000 10,000 15,000 20,000 30,000 40,000 60,000 70,000 80,000 90,000 21,000 22,000 23,000 24,000 26,000 27,000 28,000 29,000

1,000 500 250 125 1250 1500 100 150 200 300 400 700 700 800 900 210 220 230 240 260 270 280 290

Stalls Iterations w/out Improvement w/ Improvement 50 100 50 100 50 100 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50

Cost £608,914,048 £601,184,960 £598,187,840 £602,834,688 £606,111,104 £610,011,648 £619027008 £599,733,632 £612,790,464 £606,846,080 £604,292,352 £604,731,008 £600,298,880 £608,938,368 £619,253,760 £607,766,656 £606,997,952 £600,488,576 £601,860,608 £600,713,984 £606,076,096 £618,071,104 £609,204,032

In stage 1 the intent was to identify an area of the search space to focus on in further tests, however the results were inconclusive. The best result came in test 3 when a starting temperature of 25,000 was used, Table 3 shows the attributes of the plan found. Less than 1% of the total cost was incurred through stockouts, as almost all (99.9%) customer demand was satisfied. Holding cost, however, was a concern, in this plan as 29.4% of the total cost came from holding stock. This suggested that too much stock was being allocated. In subsequent tests (stages 2 – 4), a wider area of the confugration space was explored, before returning the


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Table 3. Real-world scenario - Test 3 - Resource plan attributes Total Cost Service Level Batch Cost Production Cost £598,187,840 99.9% £9,067,901 (1.5%) £407,605,024 (68.1%) Transport Cost Stockout Cost Holding Cost £899,522.19 (0.2%) £4,886,310.50 (0.8%) £175,729,056 (29.4%)

local area (stage 5) around the temperature of 25,000. In all tests, none bettered the result found in the test 3.

6

Conclusion

A large-scale real-world case study was collected from a UK supply chain design and management consultancy, and a series of experiments was conducted to discover a suitable configuration of SA in a hybrid with an IT2FL supply chain model to determine near-optimal solutions. The tests showed that using the model, SA was able to find realistic solutions to the real-world scenario, satisfying 99.9% of customer demand. As SA was guided purely by the model, this suggests that the IT2FL model is a valid representation of the problem. However, all of the tests gave results with high holding costs, showing that there are some limitations that could be considered in additional research. Other future work that could be undertaken includes addressing the scalability of the model. Lack of scalability limited the extent to which SA was able to operate. In these experiments solutions were represented by a 3 dimensional array representing time periods, destinations and products respectively. This caused the computational effort required to increase exponentially as scenarios became more complex, limiting the amount of solutions that could be evaluated in a reasonable time frame. Future work could focus on re-designing the solution representation so that it does not experience exponential increase in size as complexity increases; perhaps by encoding solutions or rules. Another area of interest is a comparison of the IT2FL model with an equivalent T1 model. Using the best settings found here more extensive tests are to be conducted with each type of model, and the results analysed to determine whether one can be said to be ‘better’ than the other. Acknowledgments. The research reported here has been funded by the Technology Strategy Board (Grant No. H0254E).

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