Fuzzy Logic and Optimisation: A Case study in Supply Chain Management

Transcription

1 Fuzzy Logic and Optimisation: A Case study in Supply Chain Management Based on work by Seda Türk, Simon Miller, Ender Özcan, Bob John ASAP Research Group School of Computer Science University of Nottingham, UK syt,sym,rij,exo@cs.nott.ac.uk April / 44

2 Overview 1 A Bit of Background 2 Supply Chain Management (SCM) Supplier Selection Inventory Management 3 Fuzzy Logic and Fuzzy Systems 4 Type-1 Fuzzy Logic Type-1 Fuzzy Sets 5 Type-2 Fuzzy Logic Motivation Type-2 Fuzzy Sets 6 Multi-objective Optimisation 7 Experiments Stage-I: Ranking of Suppliers Stage-II: Inventory Planning with Consideration of Supplier Risk 8 Conclusions 2 / 44

3 A Bit of Background Why this problem Managing the supply chain is (surprisingly still) difficult: There are large amounts of uncertainties There are conflicting objectives Different stake holders have different needs The chains can be huge 3 / 44

4 A Bit of Background A bit of history Bob John Professor of Operational Research and Computer Science ASAP Research Group Been at Nottingham for 3 years Led a Computational Intelligence group at De Monfort University for 13 years Main research interests: Fuzzy logic, particularly type-2 - theory and applications Uncertainty modelling generally - Grey Systems, Rough Sets, Intuitionistic Fuzzy Sets More recently Optimisation, heuristics and fuzzy logic Had a TSB grant to work on demand forecasting in the supply chain 4 / 44

5 A Bit of Background A bit of history Simon Miller My PhD student at De Montfort University working on fuzzy logic and the supply chain Worked at Nottingham for a number of years Now working in industry in data science Seda Türk - Current ASAP student being supervised by myself and Ender Özcan. This is mostly her work being reported here Ender Özcan - an Assistant Professor in ASAP. Expert in Optimisation, particularly hyper-heuristics 5 / 44

6 Supply Chain Management (SCM) Supply Chain Management (SCM) SCM is the management of material flows from the procurement of basic raw materials to final product delivery considering; information flows among whole processes of supply chains, material flows, long-term relations between customers and suppliers. A typical supply chain consists of five main components; Customers, Retailers, Wholesalers/Distributors, Manufacturers, Component/Raw material suppliers. 6 / 44

7 Supply Chain Management (SCM) Supplier Selection Supplier Selection Selection of an appropriate supplier is a crucial and challenging task in the effective management of a supply chain. Importance of supplier selection; Suppliers have a large and direct impact on the cost, quality,technology, and time-to-market of new products. Organization s ability to produce a quality product at a reasonable cost and in a timely manner is heavily influenced by its suppliers capabilities. Supplier selection process is a multi-criteria problem, which includes both qualitative and quantitative factors. 7 / 44

8 Supply Chain Management (SCM) Supplier Selection Supplier Selection - Previous work Ordoobadi described an approach: To build a fuzzy score for each supplier considering various selection criteria The evaluation of the potential suppliers with respect to determined criteria in type-1 fuzzy sets. Fuzzy scores converted to crisp values through the Center of Area (COA) defuzzification approach in order to make the ranking of the suppliers. The aim of the work was to select the supplier with the highest ranking and provide more information about suppliers to the decision maker. 8 / 44

9 Supply Chain Management (SCM) Inventory Management Inventory Management Inventory management is an integrated approach to plan and control inventory while considering the whole network from suppliers to end users. It is essential for good inventory management: to avoid stock outs, to manage surplus stock. Purpose of inventory management is to find out: 1) How many units to order? 2) When to order? 9 / 44

10 Supply Chain Management (SCM) The Purpose of the Work Inventory Management 10 / 44

11 Fuzzy Logic and Fuzzy Systems Philosophical On traditional logic All traditional logic assumes that precise symbols are being employed. It is therefore not applicable to this terrestial life, but only to an imaginary celestial existence. Everything is vague to a degree you do not realise until you have tried to make it precise Bertrand Russell The Australasian Journal of Psychology and Philosophy, 1 (June 1923): / 44

12 Fuzzy Logic and Fuzzy Systems Philosophical On Measurement The indeterminacy which is characteristic in vagueness is present also in all scientific measurement. Vagueness is a feature of scientific as other discourse. Black, Max (1937) Vagueness: An exercise in logical analysis. Philosophy of Science 4: / 44

13 Fuzzy Logic and Fuzzy Systems Practical If only computers were like human beings...the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Familiar examples of such tasks are parking a car; driving in heavy traffic; playing golf; understanding speech, and summarizing a story. Underlying this remarkable ability is the brain s crucial ability to manipulate perceptions - perceptions of size, distance, weight, speed, time, direction, smell, color, shape, force, likelihood, truth and intent, amongst others. Zadeh (1996) Fuzzy Logic=Computing with Words IEEE Transactions on Fuzzy Systems, 4(2), / 44

14 Fuzzy Logic and Fuzzy Systems Motivation - Summary The world is imprecise and vague. Mathematical and Statistical techniques often unsatisfactory. Experts make decisions with imprecise data in an uncertain world. Working with knowledge that is rarely defined mathematically or algorithmically but uses vague terminology with words. 14 / 44

15 Fuzzy Logic and Fuzzy Systems What it s good for.. is particularly good at handling uncertainty, vagueness and imprecision. especially useful where a problem can be described linguistically (using words) where there is data and you are looking for relationships or patterns within that data. uses imprecision to provide robust, tractable solutions to problems. 15 / 44

16 Fuzzy Systems Fuzzy Logic and Fuzzy Systems Fuzzy logic allows for the development of computer systems that employ fuzzy rules They use fuzzy sets, rules and inferencing / 44

17 (Type-1) Fuzzy Sets Type-1 Fuzzy Logic Type-1 Fuzzy Sets Crisp Tall 1 5* 11in Height 17 / 44

18 Type-1 Fuzzy Logic Type-1 Fuzzy Sets 1 Fuzzy Tall defined by a membership func=on x Height 18 / 44

19 Type-1 Fuzzy Logic Type-1 Fuzzy Sets You can be tall and short at the same time: 1 Short Tall x Height 19 / 44

20 Type-1 Fuzzy Logic A Mathematical Definition Original (Zadeh 1965) Type-1 Fuzzy Sets For any fuzzy set, A, the function µ A represents the membership function for which µ A (x) indicates the degree of membership that x, of the universal set X, belongs to set A and is, usually, expressed as a number between 0 and 1: µ A (x) : X [0, 1] 20 / 44

21 Type-1 Fuzzy Logic Type-1 Fuzzy Sets Discrete sets are written as: A = µ 1 /x 1 + µ 2 /x µ n /x n or A = µ i /x i i=1,n where x 1, x 2...x n are members of the set A and µ 1, µ 2,..., µ n are their degrees of membership. A continuous fuzzy set A is written as: A = µ(x)/x. X 21 / 44

22 Type-1 Fuzzy Logic A Mathematical Definition Modern Type-1 Fuzzy Sets A type-1 fuzzy set, A, over X is a defined by the following function A : X [0, 1] Let, F (X ), be the set of all type-1 fuzzy sets on X. 22 / 44

23 Type-1 Fuzzy Logic Type-1 Fuzzy Sets The concept of fuzzy age 1 Young Middle Aged Old Height 23 / 44

24 Type-1 Fuzzy Logic Type-1 Fuzzy Sets Fuzzy logic has a well-established theoretical base. practical implementations require a relatively small number of operations Two particularly important operations are intersection and union which correspond to AND and OR respectively. the building blocks for us to be able to compute with fuzzy if-then rules. 24 / 44

25 Type-1 Fuzzy Logic Type-1 Fuzzy Sets Fuzzy Set Theory In many problems, knowledge can comprise objective knowledge and subjective knowledge. Zadeh (1965) introduced the fuzzy set theory which allows to consider both form of knowledge [Zadeh, 1996]. Crisp Sets; A µ A (x) = { 1 if x A 0 if x A Fuzzy Sets; F : X [0, 1] F = {(x, µ F (x)) x X } 25 / 44

26 Type-1 Fuzzy Logic Type-1 Fuzzy Sets Mmmm... But... Where do the membership functions come from? Where do the rules come from? What operators should we use? The operations are crisp 26 / 44

27 Type-2 Fuzzy Sets That man Zadeh again! Type-2 Fuzzy Logic Type-2 Fuzzy Sets A fuzzy set is of type n, n = 2, 3,... if its membership function ranges over fuzzy sets of type n 1. The membership function of a fuzzy set of type-1 ranges over the interval [0,1]. Zadeh, L.A., The Concept of a Linguistic Variable and its Application to Approximate Reasoning - I, Information Sciences, 8, , / 44

28 Type-2 Fuzzy Sets Traditional Notation Type-2 Fuzzy Logic Type-2 Fuzzy Sets A type-2 fuzzy set, Ã, is characterised by a type-2 membership function µã(x, u), where x X and u J x [0, 1] à = {((x, u), µã(x, u)) x X, u J x [0, 1]} Mendel, J.M. and John, R.I., Type-2 Fuzzy Sets Made Simple, IEEE Transactions on Fuzzy Systems,10(2), / 44

29 Type-2 Fuzzy Logic Type-2 Fuzzy Sets Type-2 Fuzzy Sets Modern Notation A type-2 fuzzy set, Ã, over X is a defined by the following function à :X F([0,1]) : x Ã(x) A x (1) where A x : [0, 1] [0, 1] : u x A x (u x ) Let, F (X), be the set of all T2FSs on X. 29 / 44

30 Type-2 Fuzzy Logic Drawing them is difficult Type-2 Fuzzy Sets x 0.0 x 30 / 44

31 Fuzzy Logic Systems Type-2 Fuzzy Logic Type-2 Fuzzy Sets Type-1 Fuzzy Logic Systems; Example of Fuzzy Rules; IF the quality of product is high AND the service is low THEN the supplier is good [Mendel et al., 2006]. 31 / 44

32 Fuzzy Logic Systems Type-2 Fuzzy Logic Type-2 Fuzzy Sets Type-2 Fuzzy Logic Systems; Example of Fuzzy Rules; IF the quality of product is high AND the service is low THEN the supplier is good [Mendel et al., 2006]. 32 / 44

33 Type-2 Fuzzy Logic Interval Type-2 Fuzzy Systems Type-2 Fuzzy Sets Type-2 Fuzzy Sets Type-1 Fuzzy Sets Interval Type-2 Fuzzy Sets 33 / 44

34 Supply Chain Modelling Supply Chain Modelling Miller, John & Gongora Aims Improve resource planning in a Supply Chain using Computational Intelligence techniques. The work is part of a larger Technology Strategy Board funded project concerned with using Computational Intelligence techniques to improve demand forecasting in a supply chain. 34 / 44

35 Supply Chain Modelling Problem Resource planning Resource planning involves making decisions about how much stock to hold at each node of a supply chain. This is difficult due to the uncertain nature of supply chain operation. Poor allocation of stock can lead to stock outs and holding surplus stock. 35 / 44

36 Methods Supply Chain Modelling In this research we use the following methods: Genetic Algorithm - Search the large solution space presented by a typical SC. Interval Type-2 Fuzzy Logic - Modelling uncertainty inherent in SCs, such as: Customer demand Inventory levels Costs 36 / 44

37 Model Supply Chain Modelling Using a forecast, costs, transport distances and a required service level the model calculates the total cost and service level of a given solution. 37 / 44

38 Supply Chain Modelling Experiments Model has been tested in a variety of scenarios: Multiple Tiers, Improved IT2FS, Improved GA GA configurations, Multiple Tiers, IT2FS Optimisation methods, Multiple Tiers, IT2FS 2 Tiers, IT2FS Initial tests, Crisp model 38 / 44

39 Supply Chain Modelling Results Through experimentation, discoveries include: Users can represent the uncertainty present in their SC. System will discover good/sensible resource plans that meet a specified service level. Randomness in mutation and crossover improves search results. Combining the GA with Simulated Annealing can improve the speed of search. 39 / 44

40 Multi-objective Optimisation Multi-objective Optimisation Nature of the most optimization problems consists of several objectives to be handled, but to simplify their solution, they consider only one objective (the remaining objectives are normally handled as constraints). maximize f (x) = (f 1 (x), f 2 (x),..., f m (x)) subject to: e(x) = (e 1 (x), e 2 (x),..., e j (x)) 0 where x = (x 1, x 2,...x n ) X, y = (y 1, y 2,..., y m ) Y (2) Scalarisation methods; 1 Weighted Sum Method, 2 Tchebycheff Method. 40 / 44

41 Multi-objective Optimisation Weighted Sum Method This method is the simplest approach and probably the most widely used classical method. This method aggregate the set of objectives into a single objective by multiplying each objective with a user supplied weight. m minimise f (x) = w i f i (x) i=1 where x X (3) while w i are non-zero fractional numbers called as weights and w i = 1. minimise TF = w 1 TR + w 2 TC (4) where TR is the total risk associated with a supply chain, TC is the total cost, their respective weights are w 1 and w 2 and w 1 + w 2 = / 44

42 Multi-objective Optimisation Tchebycheff Method The aim of this method is to minimize a function value assuming that the global ideal objective vector named as utopian vector is known. minimise: max[w i (f i (x) z i )] for i = 1,..., k subject to: x X (5) where w W = {w R k 0 < w i < 1, k i=1 w i = 1} and z is the utopian objective vector. minimise TF = max{w 1 (TR µ TR )/(max TR µ TR ), w 2 (TC µ TC )/(max TC µ TC )} (6) where µ TR is the utopian total risk associated with a supply chain, µ TC is the ideal total cost, max T R, max T C are the maximum values in 100 experiments for TR and TC, their respective weights are w 1 and w 2 and 42 / 44

43 Multi-objective Optimisation Simulated Annealing(SA) Simulated Annealing is a well-known iterative meta-heuristic, inspired from the annealing process in the metallurgical industry, for solving computationally difficult optimisation problems. To accept or not to accept? If new objective value is better than current one, If it is not, evaluate with a probability based on Metropolis criteria. 43 / 44

44 Experiments Stage-I: Ranking of Suppliers Stage-I: Ranking of Suppliers 1 The decision makers selected the criteria relevant to the circumstance at hand from a list of criteria. 2 They were evaluated in a linguistic way such as low, moderate, high, very high to generate trapezoidal fuzzy sets for the importance of each criterion based on thoughts of the decision maker. 3 The linguistic terms were converted into fuzzy weights using fuzzy membership functions. 44 / 44

45 Experiments Stage-I: Ranking of Suppliers Stage-I: Ranking of Suppliers (cont.) 4 Performances were determined in the same manner of the criteria using linguistic terms such as excellent, very good, good, poor. 5 The linguistic terms were converted into fuzzy weights using fuzzy membership functions. 6 The aggregate fuzzy score for each supplier was calculated by aggregating fuzzy score all the pertinent criteria. 45 / 44

46 Experiments Stage-I: Ranking of Suppliers Stage-I: Ranking of Suppliers (cont.) 7 Each of them was converted into crisp scores using centroid type-reduction and defuzzification methods. 8 The suppliers were ranked according to their crisp scores. 9 The risk values are calculated based on their scores. Suppliers Crisp Scores Rank Risk Supplier A Supplier B Table : Crisp scores and supplier risks 46 / 44

47 Experiments Stage-II: Inventory Planning with Consideration of Supplier Risk Stage-II: Inventory Planning with Consideration of Supplier Risk 1 Several assumptions are made to determine the form of problem. 2 The problem is formulated defining objectives, constraints and decision variables. 3 Objectives are scaled into one objective using two scalarisation method; weighted sum and tchebycheff approaches. 4 6 different scenario are generated using different weight settings. 47 / 44

48 Experiments Stage-II: Inventory Planning with Consideration of Supplier Risk Stage-II: Inventory Planning with Consideration of Supplier Risk (cont.) 5 The design of several specific components in SA such as solution representation, move operator, algorithmic settings are decided. 6 The experiment is performed to observe trade-off solutions for the given instances. 48 / 44

49 Conclusions Conclusions This work shows; The results indicate that there is a certainly trade-off between risk and cost. This work provides flexibility for the decision makers enabling them to try out different scenarios. Based on the hypervolume values of each method, weighted sum approach produces a slightly better pareto front. For future research; In order to tackle this problem, the performance of other multi-objective optimisation approaches will be investigated. 49 / 44

50 Conclusions Q & A Seda TURK, References Mendel, Jerry M. and John, Robert and Liu, Feilong (2006) Interval Type-2 Fuzzy Logic Systems Made Simple IEEE Fuzzy Systems 14(6), Sharon M. Ordoobadi (2009) Development of a supplier selection model using fuzzy logic International Journal of Supplier Chain Management 14(4), Zadeh, L. A. (1996) Fuzzy Logic = Computing with Words Trans. Fuz Sys. 4(2), Nicholas Metropolis and Arianna W. Rosenbluth and Marshall N. Rosenbluth and Augusta H. Teller and Edward Teller (1953) Equation of State Calculations by Fast Computing Machines Journal of Chemical Physics 21, S. Kirkpatrick and C. D. Gelatt and M. P. Vecchi (1983) Optimization by simulated annealing SCIENCE 220(4598), / 44