Quel est le nombre d'échantillons à prélever pour analyse bactériologique dans un réseau de distribution d'eau potable afin réaliser un autocontrôle optimal du point de vue économique (coûts analytiques et coût des actions curatives), tout en limitant les risques de dégradation de la qualité ? Pour répondre à cette question, nous proposons un modèle probabiliste qui simule le choix de la décision curative lorsque les analyses indiquent des résultats insatisfaisants ainsi que l'effet de cette décision sur la qualité de l'eau du réseau. Les différentes actions curatives et leur efficacité ont été déterminées empiriquement à partir de l'expertise du gestionnaire du réseau de la Banlieue de Paris et des données collectées de 1992 à 1996. Le modèle s'appuie sur un schéma Markovien d'évolution du couple (Qualité de l'eau, Action curative). Par programmation dynamique, on calcule le coût moyen de la politique décisionnelle de la Banlieue de Paris et le risque généré par cette politique en terme de qualité de l'eau (fréquence des états dégradés), pour différents niveaux d'autocontrôle (nombre d'analyses d'autocontrôle). Le risque d'avoir un état dégradé diminue avec le nombre d'analyses jusqu'au seuil de 140 analyses (autocontrôle et contrôle réglementaire) puis reste quasiment constant, tandis que les coûts continuent d'augmenter.
Drinking water quality is monitored regularly by state officers (DDASS), and also by the water distributor at a level of his own choice. A model has been constructed to simulate decision making after observations of one or more bacteriological-positive samples from the drinking water distribution system of suburban Paris (four million inhabitants in 144 boroughs). In cases of non-conformity, a curative action is taken (rinsing, chlorinating...) that tends to increase the level of water quality for the ensuing weeks. The model compares the trade-offs between the global cost of the policy and the risk of quality failure, based on various sampling plans of different intensity which the quality manager may design to get information from the distribution system. The more weekly analyses he makes, the more money he spends in control, but at the same time, the more valuable is the information that he receives with which to assess the appropriateness of curative actions to increase quality within the system.The state of quality in the distribution system is supposed to be homogeneous, with each sampling station representative of the overall water quality. Three discrete classes of quality (acceptable, poor, unsatisfactory) have been defined, corresponding respectively to an average frequency of 5%, 10% and 15% of coliform-positive samples from the control design. The set of alternatives is composed of eight curative actions presently in use in the distribution system when a defect sample is registered: (1) complementary checking of measurements of the quality parameters, such as chlorine and temperature; (2) additional analyses of bacteriological counts; (3) rinses (water is released during a few hours from certain pipes directly into the sewage system, to allow its replacement by fresh water supposedly of better quality); (4) purges (same as rinses, but for a longer period in larger zones); (5) disinfection (the defective zone is isolated and a specialized truck introduces a large amount of chlorine into the distribution pipes); (6) deep cleaning (a yearly cleaning of 3% of the distribution system); (7) chlorinating (the level of free chlorine injection is increased in the water treatment plant); (8) a change in the treatment plant mode of operation (the complete process is checked to prevent the possible transfer of bacteria from the river to the distribution system). It also includes the standard decision of "doing nothing," i.e., let the system evolve on its own. The cost of each decision has been evaluated according to the economic data available from the distribution company, taking mainly into account controllers' work hours and travel expenses. Water quality dynamics in the distribution system are modeled as a Markov chain controlled by the possible decisions at each stage. For each curative action a (3*3) transition matrix is empirically elicited, using both available data and expertise from the team of quality managers. The present control strategy of the distribution company is embedded in the model, by respecting the observed constraints in the sequence of decisions: for instance, if a previous rinse has not been followed by a decrease in the number of coliform-positive samples in the following week, a stronger action such as a purge or disinfection is enforced, rather than repeating the rinse. The strategy also mimics the empirical rules of the quality manager's behavior in facing bacteriological incidents: for example, a single occurrence of coliforms with no specific curative action taken in a previous week will generally dictate a rinse (84% of time), sometimes demand a purge (12%), and occasionally require a disinfection (1%). The Markov model is run in a simulation mode for the spring-summer period: for a given value of the sample size, the average cost of the quality monitoring policy and a failure index (average frequency in the two lowest quality states) can be evaluated by backward induction. Although the assessment of the parameters has been made empirically, the model exhibits realistic performances with regard to side criteria used as discrepancy measures for model rejection checking: the relative use of each curative action and the average time necessary to escape from a non-acceptable state (resiliency) are of the same order of magnitude as the corresponding real indices. A sensitivity analysis reveals that the results are fairly robust to small changes in the probabilities of transition, but do depend on the way the range of water qualities is divided into discrete classes. With the data chosen, the model showed a satisfactory cost/risk balance at 110-140 analyses per week, for the homogeneous subsystem under study. In the case of more data availability, this model could become a valuable decision tool. Provided that a criterion of joint global utility between risk and cost can be defined, it could be used to design a control policy with a weekly varying sample.
