The economic value of weather forecasts for decision-making problems in the profit/loss situation

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1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 4: (27) Published online in Wiley InterScience ( DOI:.2/met.44 The economic value of weather forecasts for decision-making problems in the profit/loss situation Ki-Kwang Lee a and Joong-Woo Lee b * a Institute of Industrial Management Research, School of Management, Inje University, 67 Obang-dong, Gimhae, Gyungnam , South Korea b Atmospheric Environment and Information Research Center, School of Management, Inje University, 67 Obang-dong, Gimhae, Gyungnam , South Korea ABSTRACT: This article presents a method to estimate the economic value of forecasts for profit-oriented enterprise decision-making problems related to the levels of preparations for goods or services. The sales of goods or services in the study are supposed to be influenced and predicted by meteorological variables. Value is calculated in terms of monetary profit (or benefit) returned from the user s decision under the specific payoff structure, which is represented by a profit/loss ratio model. The decision is determined as a function of the user s subjective reliability of forecasts and forecast probability. The resulting value score (VS) curve shows the scaled economic values relative to the value of a perfect forecast, specified by a function of the profit/loss ratios for different decision makers. The proposed evaluation method, based on the profit/loss ratio model and the VS, is illustrated using hypothetical sets of forecasts, and later verified by applying site-specific probability and deterministic forecasts, each of which is generated from the Korea and China Meteorological Administrations (KMA and CMA). The application results show that decision makers with high subjective reliability of forecasts can receive great benefits from a given forecast, and there are ranges of profit/loss ratios in which each of the forecast sources used for the evaluation is preferred. Copyright 27 Royal Meteorological Society KEY WORDS cost/loss model; profit/loss model; 2 2 contingency table; decision function; value score Received 7 April 27; Revised August 27; Accepted 4 October 27. Introduction A weather forecast has value only if it can be used to make decisions that yield attractive benefits to the user (Katz and Murphy, 997; Mylne, 22). The weather forecast is usually provided to the user with a best estimate of whether a defined climatic event of concern will occur in a deterministic or probabilistic manner. A decision maker tries to follow an optimal action based on the given weather information. Many studies have been conducted evaluating economic values of the applications of weather information based on a user s prototypical decision-making models, most of which assume basic cost/loss ratio situations (Thompson, 952; Thompson and Brier, 955; Murphy, 976, 977; Katz and Murphy, 997; Stewart et al., 24). The simple cost/loss ratio model supposes a decisionmaking situation where a protective action is taken depending on whether or not a particular adverse weather event is forecast. The protective action can save losing expenses with a smaller cost for protection; however, an even larger loss is incurred if the protective action is not taken and adverse weather occurs. Since the cost/loss * Correspondence to: Joong-Woo Lee, Institute of Industrial Management Research, School of Management, Inje University, 67 Obangdong, Gimhae, Gyungnam , South Korea. busiljw@inje.ac.kr ratio model has an advantage in describing realistic situations faced by many forecast-sensitive decision makers with a considerably simple normative structure (Murphy, 976), it can provide a useful framework to investigate the use and the value of weather forecasts. Several studies have been conducted to extend the basic cost/loss ratio situation for the application of various real-world situations. The generalized model with N-actions and N-events was proposed by Murphy (985) to augment a basic situation with two actions and two events. The dynamic or sequential models for repetitive decision making were presented in the context of the basic situation (Murphy et al., 985; Epstein and Murphy, 988; Katz, 993). Moreover, considerable approaches were discussed to investigate the relationship between forecast quality and forecast value for the situation of both static and dynamic versions (Murphy et al., 985; Katz and Murphy, 99; Thornes, 2; Wilks, 2; Mylne, 22; Zhu et al., 22). In addition, Murphy and Ye (99) suggested a time-dependent version of the cost/loss ratio model to describe a situation in which the decision maker may delay the decision until forecasts of greater accuracy become available despite the increased cost of protection caused by the delayed action. This article proposes a modified version of the cost/loss ratio model for estimating the economic value of weather Copyright 27 Royal Meteorological Society

2 456 K.-K. LEE AND J.-W. LEE forecasts of dichotomous events considering the continuous degree of actions. The level of action is determined by the combination of the decision maker s subjective reliability of forecast and the value of the profit/loss ratio reflecting benefit (i.e. profit) instead of expense (i.e. cost or loss). That is, the degree of action is affected by the decision maker s own subjective forecast accuracy from cumulative experience, not the objective accuracy officially presented by a forecast provider. A decision function is developed in this study to define the continuous level of action as a function of the user s subjective reliability of forecasts and the forecast information provided. The decision maker also considers profits returned from the action in addition to cost savings as benefits. In particular, the main purpose of most businesses, in general, industries such as manufacturing, distribution, construction, and electricity generation, is to maximize profit, not to minimize expense. This article presents a profit/loss ratio model modified from the conventional cost/loss ratio model to fit the realistic situations of enterprise decision makers whose concerns are maximizing profit. This approach is directly relevant to enterprise users, and will also be informative to forecast providers who are interested in the value of their forecasts, which may be different for the various types of enterprises or industries. Section 2 explains a profit/loss ratio model combined with a decision function to emphasize profit maximization rather than cost or loss minimization. In Section 3, the economic values of various types of hypothetical weather forecasts are analysed in terms of the value score (VS) discussed by Wilks (2) and Mylne (22). Section 4 describes a practical application of the proposed evaluation method for real forecast data of a probability and its deterministic format. Section 5 gives the conclusion. 2. The profit/loss model and the decision function From the perspective of the decision makers in weathersensitive businesses, forecasts are valuable only if an action is taken as a result of the forecast information, and if the action proves to be profitable. Accordingly, the evaluation of forecasts proposed in this study is based on a simple model of optimal decision making, called the profit/loss model, which is augmented from the basic cost/loss model first introduced by Thompson (952). The profit/loss model is extended to consider the decision function thoroughly, which continuously defines the degree of the action. The proposed approach considers the economic value for all possible profit/loss ratio situations, providing a more universal analysis needed for forecast providers and users. 2.. The 2 2 profit/loss model The profit/loss model assumes that a decision maker of an enterprise is faced with the uncertain prospect of some kind of weather that may affect the principal decision factors. A business in a traditional industry can make a profit on preparing goods or services and selling them with the related costs to customers. In general, every business tries to maximize profit through enlarging the sales volume while minimizing cost. Preparation of excess products past demand results in an inventory or disposal cost, whereas insufficiency of goods in stock incurs an opportunity cost. Accordingly, an exact demand forecast is especially required for industries where demands commonly fluctuate due to environmental factors. Necessarily, the weather forecasts can be valuable to weather-sensitive industries such as the manufacturing of seasonal goods, leisure, distribution, and electricity generation. Assuming the demand in an industry depends on adverse weather as shown in Figure, the basic profit/loss function is specified by the 2 2 contingency table shown in Figure. The decision maker is able to makeabaseprofitb by preparing the minimum level of goods in stock whether an adverse weather event occurs or not. Occurrence of adverse weather, when the size of production or order from the maximum demand is not reduced, results in an inventory loss L, whereasif adverse weather does not occur and enough preparation of stocked goods for the maximum level of demand is taken, an additional profit P is made. Preparation Min Max Demand Max Min Adverse weather Event No Event Base profit (B) Base profit (B) Base profit -Inventory loss (B-L) Base profit +Additional profit (B+P) No Event Event Adverse weather Figure. Demand function for adverse weather event, and contingency table of basic user profits and losses. Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

3 THE ECONOMIC VALUE OF WEATHER FORECASTS 457 The optimal decision for the size of preparation will be the one resulting in the most expected (i.e. probabilityweighted average) profit. The information available to the decision maker is that adverse weather will occur with probability p. If a passive preparation strategy for the minimum demand is chosen, the base profit B will be obtained with probability. With an aggressive preparation strategy designed by expecting maximum demand, the profit will be decreased to B L with probability p, or will be increased to B + P with probability p. Therefore, the passive preparation strategy against adverse weather is optimal whenever or B>p(B L) + ( p)(b + P) P L < p p The passive preparation strategy is optimal when the profit/loss ratio is less than the ratio of probability of adverse weather to that of favourable weather. Note that this result is independent of the base profit and only dependent on the profit/loss ratio and the probability of adverse weather The generalized profit/loss model with a decision function The simple profit/loss model with two extreme action strategies for a binary weather event can be generalized by introducing a decision function which determines the continuous level of preparation strategies between the extremes of strategies. The decision maker decides the level of strategy based on how reliable the forecast provider is to the recipient, and the user s profits and losses associated with various possible forecast outcomes. Consequently, the decision pattern depends on the application domain as well as the subjective skill of the forecast. Assuming that the decision maker considers probability forecasts, an example of the decision pattern is shown in Figure 2, which represents the tendency for the passive strategy given forecast probabilities f for the adverse weather event. The value of the decision function in Figure 2, which is between and, corresponds to the level of cutoff for the decision maker to reduce the stock preparation from the maximum demand as shown in Figure 2. A large decision function value means large reduction of goods in stock. The decision function for the given forecast probability f is defined by a sigmoidal function which is passive dec(f ) Preparation Min+(Max-Min)[-dec(f )] Max Min dec(f ) -dec(f ) aggressive f Forecast probability of adverse weather f Forecast probability of adverse weather Figure 2. Relation between decision function and preparation pattern for forecast probability of adverse weather. dec(f;a,b) y 2 =dec(f;,.5) dec(f;a,b) y 3 =dec(f;,.3) y =dec(f;5,.5) y 4 =dec(f;,.6) f f Figure 3. Decision functions specified by sigmoidal functions for y = dec(f ; 5,.5), y 2 = dec(f ; 2,.5) and y 3 = dec(f ; 5,.3), y 4 = dec(f ; 5,.6). Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

4 458 K.-K. LEE AND J.-W. LEE expressed as: dec(f ; a,b) = () + exp[ a(f b)] where a and b are positive values, and a controls the slope at the crossover point f = b. Figure 3 shows two decision functions, y = dec(f ; 5,.5) and y 2 = dec(f ;,.5). Depending on the value of the parameter a, the sigmoidal decision function has a different slope, and thus, is appropriate for representing concepts of subjective accuracy or reliability of forecast information. Figure 3 more definitely indicates that a decision is either passive or aggressive as the forecast is more subjectively reliable (larger value of a). The decision pattern is also affected by the profit/loss ratio, which is specific to applications, as shown in Figure 3 which represents two decision functions, y 3 = dec(f ;,.3) and y 4 = dec(f ;,.6). Figure 3 describes that the decision related with a large profit/loss ratio, specified by the parameter b, is rather aggressive, even with the same subjective accuracy of the forecast. The amount of reducing products in applications with a large profit/loss ratio is less than that with a small value of parameter b. This means, the larger the profit/loss ratio of the decision domain, the higher the level of goods in preparation is set, even if the same probability of adverse weather is provided. Although the values of the parameters a and b mathematically determine a specific decision function, it is difficult to estimate the exact values of a and b caused by the subjective characteristics of the decision function itself. Accordingly, this study tries to analyse the economic values of weather forecasts according to the relative values of parameter a, supposing the value of parameter b can be substituted by the value of profit/loss ratio. 3. Calculation of forecast value The economic value of forecasts is defined by the additional profits a decision maker can make by using a forecast compared to not using one, so it is assumed that the decision criterion is to maximize the expected profit. The decision maker is postulated to choose the level of preparation for which the expected profit is maximized, where the expected profit of a preparation strategy is the probability-weighted average of the profits, including inventory losses and additional profits associated with the determined preparation strategy. 3.. Climatological reference profit If decisions are based solely on information of the climatological probability of adverse weather, π, then the decision maker s best choice is either the minimum preparation strategy for products if P/L < π/( π) or the maximum preparation strategy otherwise. Assuming the decision maker chooses the best strategy, the maximum expected climatological profit in the simple profit/loss situation shown in Figure is given by E cl = max{b,π(b L) + ( π)(b + P)} (2) Therefore, the expected profit when the climatological information is given can be defined as follows. { B if P/L < π E cl = π (3) B πl + ( π)p otherwise 3.2. Expected profit with perfect forecasts Suppose that the decision maker has access to perfect forecasts. The decision to minimize preparation would be taken only when forecasts informing adverse weather would occur with the frequency of π, making the baseline profit B. Otherwise, the maximum preparation strategy, which could lead to the maximum profit B + P, would be taken with the frequency of π. Consequently, the expected profit following perfect forecasts would be E p = πb + ( π)(b + P) = B + ( π)p (4) 3.3. Expected profit with imperfect forecasts The expected profit when decisions are based on imperfect probability forecasts for adverse weather depends on both the decision patterns specified by the subjective reliabilities of forecasts, specific profit/loss ratio, and performance characteristics of those forecasts. In effect, the decision maker determines the level of preparation based on the decision function in Equation () in relation to the magnitude of the forecast probability f. As a result, it is expected that over a number of forecast occasions, the decision maker will experience a mean profit, which depends on the joint distribution of probability forecasts and the corresponding observations p(f i,o j ) (Murphy and Winkler, 987; Wilks, 2). Here f i,i =,...,I, are the allowable forecasts and o j,j =,, are the possible observations for binary outcomes. This study supposes I =, with f =., f 2 =.,...,andf =., assuming forecast probabilities are rounded to the tenths. Although the number of allowable forecasts may be more or fewer than I = depending on the forecast format, Wilks (2) has shown that the simplified model with I = is sufficient for calculating the economic value of probability forecasts. Thus, the expected profit obtained can be calculated by considering the decision pattern specified by Equation () and the demand pattern according to the dichotomous weather events shown in Figure, which are weighted by the joint probabilities p(f i,o j ), i =,...,,j =,. Therefore, the expected profit from following specific decision patterns based on imperfect forecasts would be E f = B + I p(f i,o ){ dec(f i ; a,b)}p i= I p(f i,o ){ dec(f i ; a,b)}l (5) i= Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

5 THE ECONOMIC VALUE OF WEATHER FORECASTS 459 Under-confident Calibration, p(o f ) Under-forecast Over-confident Probability Density Over-forecast σ=.3 Forecast probability, f..5.9 Forecast probability, f Figure 4. Calibration functions and refinement distributions of beta distribution with µ =.3 and σ =.3 for hypothetical forecasts used to calculate the VS in Figure 5 (Wilks, 2). This expected profit depends on the particular decision maker s decision pattern determined by the subjective reliability of forecasts and the profit/loss ratio, and on the performance characteristics of the forecasts used by the decision maker, as specified by the joint distribution of forecasts and observations p(f i,o j ) The economic value of forecasts in monetary terms is considered as additional profit the user can expect to make from following the forecast, i.e. the difference between expected profits from the baseline climatological information and the forecasts under consideration, E f E cl. For the general utilization of assessments of forecast values, some studies (Wilks, 2; Mylne, 22) introduced the concept of a skill score, or VS, transforming the difference form to a normalized scale representing the relative increase in expected return to that of a perfect forecast. The relative economic value of a forecast, called avs, is defined by VS = E f E cl E p E cl (6) which can be transformed into a form that depends only on the profit/loss ratio by using Equations (3), (4), (5), and (6) as follows. Equation (7) in practice shows that the VS is likely to have a negative value. A negative value indicates that the forecast system has insufficient skill to provide valuable information to the decision maker, while the best strategy is to follow the climatological information as Equation (3) (Mylne, 22) Hypothetical example It is useful to analyse economic value according to the various profit/loss ratios since different decision makers with their particular environment will obtain different levels of monetary values by optimally using the same forecasts. Therefore, this article describes the VS graphically as a function of P/L as it does in Equation (7). The joint distribution of forecasts and observations, p(f i,o j ), in Equation (7), is factored into conditional and marginal distributions, i.e. p(f i,o j ) = p(o j f i )p(f i ) (Murphy and Winkler, 987). The calibration refinement factorization process is conducted by adopting the calibration functions and the refinement distributions in a manner similar to the concept of Wilks (2). Figure 4 shows the four calibration functions p(o f i ): under-confident, over-confident, under-forecast and over-forecast. Each is assigned an equation and a climatological probability, which is summarized in Table I. The under-confident forecasts constructed by using p(o f i ) = 2f i π have conditional biases: for VS = I { dec(f i ; a,b)}{p(f i,o ) P L p(f i,o )} i= I ( π) P L if P L < π π (7) i= { dec(f i ; a,b)}{p(f i,o ) P L p(f i,o )}+π ( π) P L π otherwise From the theoretical definition of Equation (6), the VS can have values ranging from zero for a climatology forecast to (or %) for a perfect forecast. However, larger probabilities, the event occurs more frequently than predicted, and for smaller probabilities, the event occurs less frequently than predicted. The over-confident Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

6 46 K.-K. LEE AND J.-W. LEE forecasts with p(o f i ) = (f i + π)/2 exhibit the opposite conditional biases: the higher forecast probabilities are associated with over-forecast and the lower forecast probabilities are associated with under-forecast. The remaining two calibration functions, expressed by p(o f i ) = f i ±., show unconditional biases, or consistent under-forecasts and over-forecasts. The refinement distribution p(f i ) is defined by using beta distributions with a mean of.3 and a standard deviation of.3 assuming short-range weather forecasts (Murphy and Wilks, 998), which is shown in Figure 4. A more detailed explanation is provided by Wilks (2). Figure 5 shows VS curves for the user s subjective reliabilities of forecasts, which is specified by the parameter a in Equation (). Each panel of Figure 5 represents the under-confident, the over-confident, (c) the under-forecast, and (d) the over-forecast calibration functions in Figure 4 and the refinement distribution in Figure 4. The under-confident forecasts in Figure 5 Table I. Summary of the characteristics of the calibration functions. Type of calibration function Equation Climatological probability, π Under-confident p(o f i ) = 2f i π.3 Over-confident p(o f i ) = (f i + π)/2.3 Under-forecast p(o f i ) = f i + o.4 Over-forecast p(o f i ) = f i o.2 provide the user with the most valuable information, whereas the over-confident, or no skill forecasts, can impart the least economic value. In particular, the overconfident and under-forecast offer negative values to the decision makers with small profit/loss ratios, since users whose losses for the remaining inventories are comparatively larger than profit for sales are unlikely to reduce Under-confident Over-confident (c) Under-forecast (d) Over-forecast Figure 5. VS curves for the under-confident, over-confident, (c) under-forecast, and (d) over-forecast calibration functions. The values of parameter a (the subjective reliability) are 5 (dotted line), (dashed), and 2 (solid). Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

7 THE ECONOMIC VALUE OF WEATHER FORECASTS 46 their stock levels when the probability of adverse weather p(o f i ) is actually larger than the forecast probability f i, which is smaller than the climatological probability π =.3. Moreover, the assumption of a short-range forecast, where small f i happens more frequently as shown in Figure 4, results in a rapid slope for the small profit/loss ratio as shown in Figure 5 and (c). In the case of a negative forecast value, the user s best strategy is to prepare either minimally or maximally, whichever is better, rather than to follow the forecasts (Mylne, 22). Each of the VS curves has the maximum value for P/L whose value is near π/( π). Note that all the four types of forecasts have larger economic values as the value of parameter a in a decision function of Equation () increases, implying that the user s subjective reliability of the provided forecast as well as the objective accuracy of the forecast itself determines the economic value of the weather forecast. Consequently, an effective method to increase the economic values of forecasts other than by improving the skill of the forecast system is by keeping the user s subjective reliability high, since in most cases subjective reliability is smaller than the actual objective accuracy of the forecast caused by the negative evidence effects, which was psychologically verified by Wright (974). 4. Applications to real datasets The proposed skill score model was applied to real forecast datasets to illustrate two simple verification examples with probability and deterministic forecasts, and was used to compare the verification results. The application was based on probability forecasts of precipitation at Seoul, South Korea, made by the Korea Meteorological Administration (KMA), and deterministic or non-probabilistic forecasts of precipitation at Shanghai, China, made by the China Meteorological Administration (CMA) for the period 23 through 25. In both cases, the event is defined as an occurrence if at least.254 mm of precipitation is recorded over a forecast lead time of 2 h. Table II shows the joint distribution of forecasts and observations, p(f i,o ), for the probability forecasts of,.,.2,...,.9, and provided at Seoul, and for the deterministic forecasts of (no precipitation forecast) and (precipitation forecast) provided at Shanghai. The summation of the joint probabilities p(f i,o ) for all i =, 2,...,I, is considered as the climatological probability π. The economic values of forecasts as a function of the profit/loss ratio, P/L, for the two types of forecasts at Seoul and Shanghai are depicted in Figure 6 for selected values of subjective reliability a in the decision function dec(f;a,b). It can be seen from Figure 6 that there are ranges of profit/loss ratios where each of the two forecast sources is more valuable. The probability forecasts in Seoul are more valuable for decision makers whose profit/loss ratios are small, while those with large profit/loss ratios can benefit more from the deterministic forecasts in Shanghai. However, the value of Table II. Joint distribution of forecasts and observations for Seoul and Shanghai. Seoul Shanghai f i p(f i,o ) p(f i,o ) p(f i,o ) p(f i,o ) Sum Seoul, a= Seoul, a=2 Seoul, a=5 Shanghai, a= Shanghai, a=2 Shanghai, a= Figure 6. VS curves for precipitation forecasts in a probabilistic (at Seoul, South Korea) and non-probabilistic (at Shanghai, China) manner selected subjective reliability a = 2 and a = during the period a forecast is strongly dependent on the profit/loss ratio, regardless of the forecast sources being probabilistic or non-probabilistic. It is interesting to note in Figure 6 that the VS curves of deterministic forecasts are less contingent on the parameter a of the decision functions rather than those of probability forecasts, since the decision makers with non-probabilistic forecasts are presumed to ignore the decision levels for the probability of forecasts between. and.9, where the decision levels are sharply Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

8 462 K.-K. LEE AND J.-W. LEE different according to the value of the parameter a. However, this does not imply that the VS of the deterministic forecasts is independent of the user s subjective reliability of forecasts; rather, the subjective reliability may have more influence over the VS of the probability forecasts than the non-probabilistic ones due to the characteristics of the subjective reliability, which is closely related to the probability forecasts. 5. Conclusion This article proposes a method of estimating forecast value by introducing the modified profit/loss ratio situation combined with a decision function. The profit/loss ratio model was discussed to generalize the 2 2cost/ loss ratio contingency table and to fit the evaluation model to a real enterprise decision-making situation. The standard cost/loss ratio situation with two actions and two events was generalized to a profit/loss ratio with the continuous degree of actions for two events of adverse weather, considering profit and loss incurred by the weather and the strategy for its preparatory action. The profit/loss ratio situation assumes that the decision maker will try to maximize expected monetary returns obtained by optimally balancing the additional profits and the incurred costs, which is very similar to the assumption of general business decision-making problems. The action for maximizing benefits is determined based on the market demand size affected by the weather events. Thus, this article suggests the decision function, which specifies the relationship between the determined levels of action and probability forecasts of weather events for the selected user s subjective reliabilities on forecasts and profit/loss ratios. The decision function was devised to demonstrate that a particular decision is made based on the forecast information, the subjective reliability of the forecast information, and the specific profit loss structure. In the proposed profit/loss ratio situation, the concept of the VS was used to represent the variations of scaled economic values as a function of the profit/loss ratio. The VS curve was derived for the four types of hypothetical forecast systems with a short lead time, i.e. under-confident, over-confident, under-forecast, and over-forecast. The result indicates that decision makers with different profit/loss ratios will receive different benefits from a given set of forecasts. The under-confident forecast system can impart the most valuable information among the four types of forecasts, while decision makers with P/L = π/( π) or high reliabilities of the forecasts will obtain the relatively great benefits regardless of the type of forecast system used. The proposed economic valuation method with the VS curve in the profit/loss situation was applied to the real-field forecast data for precipitation at Seoul and Shanghai with probability and deterministic formats. The verification result revealed that there are ranges of P/L where each of the forecast sources is more valuable. The result of the analysis shows that the VS evaluation as a function of P/L is a useful way to analyse the value of forecast for various users. In conclusion, the practical application to maximize returns by using forecast information will require some prerequisites to be met by forecast users and providers. The users must define their subjective reliability of forecast information, and estimate the payoff structure (i.e. profits and losses) expected in each of the contingencies. The forecast providers are required not only to improve the forecast system s skill but also to strengthen publicity activities to increase the user s subjective reliability of forecasts. This study can also help decision makers of various enterprises extract the greatest benefits from forecasts through possible extensions of the profit/loss model. For example, we can elaborate the payoff relationship structure between the forecast-based decisions and events by subdividing the two-event situation supposed in this article into N-events or continuous degrees of events. Another possible extension would be needed to derive not only a theoretical but also a practical parameter a in the decision function, reflecting an actual user s subjective reliability. Acknowledgements We would like to express our appreciation to Mr K.-T. Park (President & CEO, CJ Group China Headquarters) and Mr M. H. Jung (President, Shanghai E-Mart Supercenter Co., LTD.) for providing us with invaluable background information on the business decisionmaking problem with meteorological forecast. This work was funded by the Korea Meteorological Administration Research and Development Program under Grant CATER References Epstein ES, Murphy AH Use and value of multiple-period forecasts in a dynamic model of the cost-loss ratio situation. Monthly Weather Review 6: Katz RW Dynamic cost-loss ratio decision-making model with an autocorrelated climate variable. Journal of Climate 6: 5 6. Katz RW, Murphy AH. 99. Quality/value relationship for imperfect weather forecasts in a prototype multistage decision-making model. Weather and Forecasting 8: Katz RW, Murphy AH Economic Value of Weather and Climate Forecasts. Cambridge University Press: Cambridge, New York. Murphy AH Decision-making models in the cost-loss ratio situation and measures of the value of probability forecasts. Monthly Weather Review 4: Murphy AH The value of climatological, categorical and probabilistic forecasts in the cost-loss situation. Monthly Weather Review 5: Murphy AH Decision making and the value of forecasts in a generalized model of the cost-loss ratio situation. Monthly Weather Review 3: Murphy AH, Winkler RL A general framework for forecast verification. Monthly Weather Review 5: Murphy AH, Ye Q. 99. Optimal decision making and the value of information in a time-dependent version of the cost-loss ratio situation. Monthly Weather Review 8: Murphy AH, Wilks DS A case study of the use of statistical models in forecast verification: precipitation probability forecasts. Weather and Forecasting 3: Murphy AH, Katz RW, Winkler RL, Hsu W-R Repetitive decision making and the value of forecasts in the cost-loss ratio situation: a dynamic model. Monthly Weather Review 3: Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met

9 THE ECONOMIC VALUE OF WEATHER FORECASTS 463 Mylne KR. 22. Decision-making from probability forecasts based on forecast value. Meteorological Applications 9: Stewart TR, Pielke R, Nath R. 24. Understanding user decision making and the value of improved precipitation forecasts-lessons from a case study. Bulletin of the American Meteorological Society 85: Thompson JC On the operational deficiencies in categorical weather forecasts. Bulletin of the American Meteorological Society 33: Thompson JC, Brier GW The economic utility of weather forecasts. Monthly Weather Review 83: Thornes JE. 2. How to judge the quality and value of weather forecast products. Meteorological Applications 8: Wilks DS. 2. A skill score based on economic value for probability forecasts. Meteorological Applications 8: Wright P The harassed decision maker: Time pressures, distractions, and the use of evidence. Journal of Applied Psychology 59: Zhu Y, Toth Z, Wobus R, Richardson D, Mylne K. 22. The economic value of ensemble-based weather forecasts. Bulletin of the American Meteorological Society 83: Copyright 27 Royal Meteorological Society Meteorol. Appl. 4: (27) DOI:.2/met