Beer game order policy optimization under changing customer demand
Introduction
The Beer Distribution Game (“Beer Game”), developed at the Sloan School of Management in early 1960s [7], is a classic supply chain problem widely used in graduate business programs to teach the concepts of supply chain management [9].
The goal of the participants in the game is to minimize the costs of maintaining sufficient inventories of beer while at the same time avoiding out-of-stock condition that could lead to loss of customers. The Beer Game is notable for its ability to confuse, human players [10], [11], typically giving rise to instabilities in the supply chain and distortion of the expected demand [1]. The game is also used to illustrate how different decision policies influence the dynamics of a distribution system.
Sterman [11] formulated a four parameter discrete model for the order policy, based on the theory of bounded rationality [2], [3] and showed that this model captures the main aspects of the decisions taken in the game by real players. Mosekilde and Larsen [8] were first to show that a time-continuous version of the Beer Game model could produce deterministic chaos and other forms of complex behaviour [15].
In this work we are going to analyse the optimal order policies, i.e. the optimal parameters of the Sterman model, when step-changes of different magnitudes occur in the customer demand. The optimal policy considered is the policy that gives the minimum cost, accounting both for the costs to maintain a stock of goods and for the costs of loosing business when it is not possible to satisfy the demand.
Two scenarios have been analysed: (i) all sectors apply the same order policies, or (ii) different policies are applied from sector to sector.
The search of the optimal solution has been performed using Genetic Algorithms [4] due to the complexity of the objective cost function, which has many local minima and, in the case of different policies, many parameters. The application of GAs optimization techniques represents a new approach in the case of the Beer Game model but Genetic Algorithms have been applied in several managerial problems such as portfolio optimization and job scheduling [14].
A description of the Beer Game and the order policy models is presented in Section 2, and the implementation and application of the Genetic Algorithms to optimise the costs are discussed in Section 3. The results, Section 4, show that it is possible to reduce the score both for the chain and for the individual participants by allowing different order policies, even when the customer demand is changing. Moreover, the advantage of different order policies increases as a function of the step-change in the customer demand. This advantage is obtained when the Factory and the Distributor pay more attention to the stock than to the supply chain, whereas the Wholesaler and the Retailer focus on the supply chain.
Section snippets
The beer game
The Beer Game is a representation of a production–distribution system on four levels: Factory, Distributor, Wholesaler and Retailer, see Fig. 1. The orders starting from the customer go to the Retailer, then to the Wholesaler, the Distributor, and finally reach the Factory. In the meantime, deliveries are shipped from the Factory down through the supply chain until they reach the customer. The Beer Game is widely used in management schools as a means of conveying to the students the causal
Genetic algorithms
Genetic Algorithms (GAs) are optimisation algorithms inspired by the natural selection rules of Darwin's theory of evolution, i.e. the survival of the fittest. Building on this idea, Holland [4] developed the first GAs where an optimisation problem is turned into an evolutionary process, in which a group of individuals evolve to adapt better to a fitness function generation after generation.
Since then, GAs have been applied on many fields, the technique showing itself to be a robust and
Results and discussion
For all the cases analysed, optimisation using GAs was carried out with a population size of 30 possible behaviours, a crossover rate of 90% and a mutation rate of 1% per generation. The maximum number of iterations was 500. Once an order policy was established and a step in the demand was fixed, the GAs was applied ten times in order to explore the search space. The optimal solution, from the point of view of the fitness function, was considered.
In Fig. 4 the optimum policies of the sectors
Conclusions
In the present study, the optimal parameters for the order policy of the Beer Game for a period of 60 weeks were analysed for the case where customer demand changes. Two different situations were considered: (i) all sectors having the same ordering policy or (ii) different policies were allowed for the different sectors.
The optimisation was performed using Genetic Algorithms. The technique of GAs is especially suited for this problem because of the complexity of the objective function with high
Acknowledgments
Insights and criticisms from anonymous referees significantly improved the presentation of the present work. The English revision by Dr. C. N. Murray is greatly appreciated.
Dr. Fernanda Strozzi is a Contract Professor in the Engineering Department at Carlo Cattaneo University (Castellana, Italy). She graduated in Applied Mathematics from Pavia University and holds a PhD in Mathematics and Chemical Engineering from Twente University (Enschede, The Netherlands) on the application of chaos theory to chemical reactors. She was a Marie Curie fellowship from 1993 to 1995. Her current research interests include the application on complex systems theory to logistic chains
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Dr. Fernanda Strozzi is a Contract Professor in the Engineering Department at Carlo Cattaneo University (Castellana, Italy). She graduated in Applied Mathematics from Pavia University and holds a PhD in Mathematics and Chemical Engineering from Twente University (Enschede, The Netherlands) on the application of chaos theory to chemical reactors. She was a Marie Curie fellowship from 1993 to 1995. Her current research interests include the application on complex systems theory to logistic chains and on the analysis of financial time series.
Mr. Jordi Bosch Pagans is a PhD student at Manchester University. He graduated in Chemical Engineering at IQS (Barcelona, Spain) in June 2004. His PhD is focused on the development of an early warning detection system for the initiation of runaway reaction in chemical reactors using techniques from nonlinear time series analysis. He has collaborated with Castellanza University, the Joint Research Center of the European Commission (Ispra, Italy) and the Health and Safety Laboratory (Buxton, UK) in the frame of an EU funded project (AWARD, G1RD-CT-2002-00499).
Dr. José-Manuel Zaldívar Comenges is a Senior scientist at the Institute for Environment and Sustainability of the Joint Research Centre of the European Commission where he initially joined in 1987. He obtained a MS in organic chemistry at IQS (Barcelona, Spain) and a PhD in Chemical Engineering from Twente University (Enschede, The Netherlands). He has published numerous papers in safety of chemical reactors, mathematical modelling and the application of nonlinear dynamical systems theory to several fields. His current research interests include ecological modelling and the interactions between ecosystem and contaminants.