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i i “M19N43” — 2008/11/5 — 17:14 — page 589 — #1 i International Journal of Information and Management Sciences Volume 19, Number 4, pp. 589-600, 2008 Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context: A Scheduling Example R. Bryan Kethley Middle Tennessee State University U.S.A. Abstract Many multiple criteria scheduling problems reach a complexity level that can be diﬃcult, if not impossible, to capture in a mathematical model. Most of these problems are classiﬁed as NP-hard. Woolsey [21] points out that “scheduling is almost never an activity in which there is just one goal”. For this reason scheduling was chosen for the example but in reality Taguchi Loss Functions could be used to develop a single objective function for almost any function in a Multi-Criteria Context. For example, if one heuristic measure, such as minimizing tardiness, is used to determine a heuristic’s utility, a scheduling policy may be implemented that results in a signiﬁcant number of early jobs. This policy may not be appropriate if the organization is also interested in limiting the amount of completed inventory on hand. In this paper we suggest that one possible solution to the multiple criteria scheduling problem is using Taguchi loss functions as an objective function for the scheduling algorithm or heuristic. Keywords: Scheduling, Heuristics, Algorithms, Taguchi Loss Functions. 1. Introduction Many times in “real world” applications the decision-maker is concerned with gener- ating a production schedule that must address multiple criteria and the utility of compet- ing schedules are often judged using several measures. Little research has been concen- trated on the single machine multiple criteria problem. One literature review [11] listed some of the research concerning the single machine, multiple criteria problem. Previous research has followed two general scenarios. In the case of bicriterion problems, both of the measures of interest are either programmed as part of a dual objective function or with one measure as the objective function and the second measure as a constraint Received October 2006; Revised July 2007; Accepted October 2007. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 590 — #2 i International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 [16, 19]. Other multiple criterion decision making problems have been structured as a marginal objective function with weights assigned for each criteria [19]. The lack of re- search is not surprising because, except for the simplest problems, the majority of these problems are classiﬁed as NP-hard [4]. In order to gain a solution, that may not be opti- mal, many large-scale problems are being “solved” using heuristics or algorithms. In this paper Taguchi loss functions (TLFs) are oﬀered as a means to combine several diﬀerent criteria into a single, simple objective function that can be used as part of virtually any search algorithm. 2. Taguchi Loss Functions Traditionally, characteristics of products are evaluated using a step function ap- proach [19]. A design target value is developed and speciﬁcation limits are set to indicate the maximum deviation allowed from the target value. If the characteristic measurement falls within this speciﬁcation range, the product is deemed acceptable. If the measure- ment of the characteristic falls outside this range, the product is rejected. Figure (I) illustrates this relationship. Taguchi indicated that any deviation from a characteristic’s target value results in a loss and a higher quality characteristic measurement is one that will result in minimal variation from the target value. Speciﬁcally, if a characteristic measurement is the same as the target value the loss is zero, otherwise the loss can be measured using a quadratic function. Primarily Taguchi loss functions have been used to measure physical charac- Figure I. Traditional speciﬁcation function. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 591 — #3 i Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context 591 teristics of a manufactured product. Caporaletti et al. [2] provide a good example of an application of Taguchi methods, that incorporates design of experiments and Taguchi loss functions, in a manufacturing process environment. Chan et al. [3] and Heredia et al. [7] also using TLfs in a manufacturing environment but in a multiple decision making context. Other non-manufacturing processes have beneﬁted from the application of the Taguchi loss function as well. Taguchi loss functions have been used to rank employee performance in a management by objectives (MBO) appraisal system [15] and to evaluate product quality as an aid in the selection of suppliers [18]. Kethley et al. [8] and Fester- vand et al. [6] used TLFs as a ranking methodology to reduce a larger set of available properties to a more manageable subset of properties available to the potential buyer. Snow [18] lists four types of loss functions that may be used to determine a metric’s utility. In the case of the two sided equal speciﬁcation function and the two sided with speciﬁcation preference function, variation is allowed in both directions from the target value. For example, if a shaft has a diameter target value of .010”, a two sided equal speciﬁcation function might set the lower speciﬁcation at .008” and the upper speciﬁcation limit could be set at .012”. These settings would allow equal deviation from the target value in both directions. The two sided with speciﬁcation preference function is appropriate when deviation is allowed in both directions from the target value but less variation is allowed in one direction. Using the shaft diameter target value of .010” again as an example, we could set the upper speciﬁcation limit at .014” and the lower speciﬁcation at .008”. These settings would allow more deviation from the target value in the upper speciﬁcation limit direction. In each of these ﬁgures the target indicates the nominal value. USL and LSL indicates the upper speciﬁcation limit and the lower speciﬁcation limit respectively With the one sided minimum speciﬁcation function and the one sided maximum speciﬁcation function, variation is allowed in one direction only from the target value. If a shaft has a minimum diameter target value of .010”, the upper speciﬁcation limit could be set at .012”. If the shaft has a maximum diameter target value of .010” then the lower speciﬁcation limit could be set at .008”. In all the scenarios previously identiﬁed, zero loss will occur at the target value and any deviation from the target value will generate a loss that will follow a quadratic function up to a 100% loss at the speciﬁcation limits. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 592 — #4 i 592 International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 Figure II. Two sided equal speciﬁcations Taguchi loss function. Figure III. Two sided with speciﬁcation preference Taguchi loss function. Parameters are determined by the decision maker. The ideal value is set as the target, the limits are set as the upper and lower limits. After these values are known, using the Taguchi formulas, a constant “k” is calculated that sets 100% loss at the limits. In manufacturing the target value could be represented by the optimum diameter of a bolt and the upper and lower limit would be the tolerances assigned to the measurement. An non manufacturing example could be Kethley et al. [8] that allows the decision maker to determine target and speciﬁcation limits such as house price and square footage. The decision maker determines the price of the home that is optimal (target value) and the i i i i i “M19N43” — 2008/11/5 — 17:14 — page 593 — #5 i Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context 593 Figure IV. One-sided minimum speciﬁcation Taguchi loss function. Figure V. One-sided maximum speciﬁcation Taguchi loss function. maximum the decision maker is willing to spend (upper speciﬁcation). 3. Taguchi Loss Functions as a Search Algorithm Objective Function In this context, search algorithms are deﬁned as heuristics or algorithms that, through an iterative process, search for an improved solution. Some examples of search algorithms are simulated annealing [10], and the genetic algorithm [1]. In each of these algorithms, sequences are evaluated against a single objective function, and if a sequence results in a better solution then the sequence is retained. A weighted Taguchi loss (WTL) function encompassing several algorithm performance measures can be substituted for the single objective function in these algorithms. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 594 — #6 i 594 International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 Problem Deﬁnition and Performance Measurements For this evaluation there are n jobs available for processing on a single machine. Each job is assigned a processing time pi and a due date di . Each job is available for processing at time zero. The processing times and due dates for ten jobs are as follows. jobi 1 2 3 4 5 6 7 8 9 10 Pi 20 3 8 18 14 20 7 7 9 14 di 51 10 88 52 27 26 9 76 31 92 For example purposes three diﬀerent measurements are used to develop the WTL func- tion that will be used to evaluate the utility of possible schedule sequences. The ﬁrst measure is a common penalty function in scheduling known as the Total Tardiness penalty (Equation 1). n min Ti i=1 where: Ti = max(Ci − di , 0). (1) We can deﬁne Ci as the completion time for job i. If a job is completed after its due date then a penalty is accrued that is equal to the diﬀerence between the completion date and the due date. If the diﬀerence is a negative number the job is early and no penalty is assigned. The second measure used is the number of tardy jobs generated by the schedule (Equation 2). In this function any job is considered to be tardy if it is completed after the assigned due date. n min Ui i=1 1 if Ci > di ; else where: Ui = . (2) 0 The last measure is the number of early jobs generated by the schedule (Equation 3). In the previous two measurements early jobs are desirable because they result in no penalty. In practice, especially in a Just-in-Time environment, many organizations are interested in limiting the amount of ﬁnished inventory. This could result in conﬂicting scheduling objectives. n min Ei i=1 1 if Ci < di ; else where: Ei = . (3) 0 i i i i i “M19N43” — 2008/11/5 — 17:14 — page 595 — #7 i Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context 595 In this function any job is considered to be early if it is completed before the assigned due date. Weighted Taguchi Loss Example To use the Taguchi loss functions two values are needed. A target or desired value must be identiﬁed and a speciﬁcation limit must be set. The constant “K” is developed such that, when the calculated value of “K” is entered into the loss function equation, the loss will be zero at the target value and 100% at the speciﬁcation limit. Two ways that weights may be assigned are using the decision maker’s judgment or the analytic hierarchy process (AHP). The AHP [13] uses a pairwise comparison to determine a weight. A recent trend of published journals is to use the decision makers input when generating the model weights [6], [8], [14], [18]. Stewart [19] indicates that as the process progresses it may become an iterative process. As the decision maker’s gains additional information the weights may be adjusted. Each of the measurements previously identiﬁed will use the one-sided maximum speciﬁcation Taguchi loss function (see Figure V). In the case of tardy jobs and early jobs, the best performance possible by the heuristic is a schedule that results in no tardy or early jobs. To reﬂect this, the speciﬁcation limit is set at 10, indicating that at worst case, the rule results in tardy or early jobs 100% of the time and will result in the maximum loss of 100%. The target value is set at 0, indicating the best performance that can be expected is no tardy or early jobs, which results in a loss of zero. Actual values reﬂecting the number of early or tardy jobs will be inserted into equation IV and the loss calculated will fall between 0% and 100% loss for that characteristic. Calculating the Total Tardiness penalty loss is not as straightforward. Many times the problems under evaluation are NP-hard and the optimal solution is rarely known. If the heuristic resulted in a perfect schedule then the penalty assigned to the schedule would be zero so we can set our target value at zero. To set the speciﬁcation limit we can generate an initial sequence by randomly selecting a job sequence or by using a sequencing rule such as the shortest processing time [17]. The penalty generated by the initial sequence can be used as a surrogate speciﬁcation limit. “K” is calculated to reﬂect 100% loss at the maximum penalty and 0% loss will be assigned at the target value of zero penalty. Again, actual loss will be calculated using equation 4 and will fall between 0% and 100% loss for that characteristic. L = Kx2 (4) i i i i i “M19N43” — 2008/11/5 — 17:14 — page 596 — #8 i 596 International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 K = 100%/(U SL)2 where: L = Loss generated by the process for the characteristic measured. x = Characteristic measurement. U SL = Upper Speciﬁcation Limit. K = A constant calculated to return a 100% loss at the speciﬁcation limit. Using 10 as the upper speciﬁcation limit for tardy jobs and early jobs results in a “K” value of 1. For the total tardiness problem an initial penalty of 250 was generated. Setting 250 as the upper speciﬁcation limit for the total tardiness penalty results in a “K” value of .0016. Two sequences, S1 and S2 , are given below. As previously stated, the sequences can be generated using any search algorithm. The WTL function simply gives the practitioner a method to include many diﬀerent objectives in a single objective function. S1 = 2 7 8 3 9 5 10 4 6 1 S2 = 7 2 6 5 9 1 4 8 3 10 The ﬁrst step is to determine the number of early jobs, the number of tardy jobs and the tardiness penalty for S1 and S2 . The performance measurements for sequences S1 and S2 are as follows. S1 S2 Early Jobs = 4 Early Jobs = 1∗ Tardy Jobs = 6 Tardy Jobs = 8∗ Tardiness Penalty = 196 Tardiness Penalty = 172 * One job is completed on time The next step is to transform the raw performance measurements into Taguchi loss functions. This transformation illustrates two valuable features of Taguchi loss functions. First, all of the raw measurements are transformed into the common Taguchi unit of measure, the percentage of loss for that characteristic. Any measurement, regardless of unit of measure or magnitude of scale, can be compared using TLFs. Secondly, because the loss function is not linear the loss becomes increasingly larger as the value moves away from the target value. This places a higher premium on those measurements exhibiting lower variation from the target value. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 597 — #9 i Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context 597 Using equation IV and the constant “K” calculated for the number of early jobs, the number of tardy jobs and the tardiness penalty, losses are determined using Taguchi loss functions. The Taguchi loss function output for sequences S1 and S2 are as follows. TaguchiLoss S1 S2 Early Jobs = 16% Early Jobs = 1% Tardy Jobs = 36% Tardy Jobs = 64% Tardiness Penalty = 61% Tardiness Penalty = 47% At this point the TLFs result in three separate measurements. Losses generated by TLFs can be weighted to represent the relative value of each measurement (Equation V). The weighted TLF is then used as the single objective function for the search algorithm. n min Wi λi (5) i=1 where: Wi = Weight assigned to characteristic i λi = Taguchi loss of characteristic i To illustrate how setting the characteristic weights can impact the decision regarding a particular schedule’s utility consider the following scenarios. In the ﬁrst scenario each characteristic being measured is of equal importance to the decision-maker. WeightedTaguchiLoss(equalweights) S1 S2 Early Jobs = .33(16) Early Jobs = .33(1) Tardy Jobs = .33(36) Tardy Jobs = .33(64) Tardiness Penalty = .33(61) Tardiness Penalty = .33(47) 37% 37% Equal weights are assigned to each characteristic to reﬂect the relative importance. Even though the individual losses generated vary between the two sequences, the overall weighted Taguchi loss indicates that the two sequences generate approximately the same penalty. In the second scenario the organization is more interested in limiting the amount of completed inventory on hand. To represent this preference the early job penalty is weighted more heavily than the number of tardy jobs and the tardiness penalty. Using i i i i i “M19N43” — 2008/11/5 — 17:14 — page 598 — #10 i 598 International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 the same characteristic Taguchi loss as before, but with diﬀerent weights, results in sequence 2 being preferred WeightedTaguchiLoss(limitingearlyjobs) S1 S2 Early Jobs = .60(16) Early Jobs = .60(1) Tardy Jobs = .20(36) Tardy Jobs = .20(64) Tardiness Penalty = .20(61) Tardiness Penalty = .20(47) 29% 23% 4. Summary and Discussion In this paper we suggest using the Taguchi loss functions as a simple method to incorporate multiple objectives into a single objective function for search algorithms. A weighted Taguchi loss function can be easily incorporated into any search algorithm that uses a single objective function and oﬀers several beneﬁts. TLFs place a higher premium on those measurements that result in less variation from the target value and can transform characteristics having diﬀerent units of measure and varying magnitude of scale into a common measurement. Higher weights may be assigned to those char- acteristics deemed more important and lower weights to those characteristics of lower importance. Multiple criteria can be incorporated into weighted Taguchi loss function that can be easily utilized as a single objective function within a search algorithm. Some multi-criteria manufacturing applications include [3], [7]. Some non-manufacturing ap- plications that use Taguchi Loss Functions in a multiple criteria decision making context include Supplier evaluation and selection [12], [13], [18]. Others include employee perfor- mance appraisal [15], and evaluation of domestic air travel industry [9] There are literally over a hundred applications of Taguchi Loss Functions in ABM/Inform but there is no documented use of TLFs as proposed in this manuscript. Given these factors the application of Taguchi loss functions can be an excellent tool when faced with determining the utility of competing scheduling policies or practices. References [1] Bean, J. C., Genetic Algorithms and Random Keys for Sequencing and Optimization, ORSA Journal on Computing, Vol. 6, No. 2, pp.154-160, 1994. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 599 — #11 i Using Taguchi Loss Functions to Develop a Single Objective Function in a Multi-Criteria Context 599 [2] Caporaletti, L., Gillenwater, E. and Jaggers, J., The Application of Taguchi Methods to a Coil Spring Manufacturing Process, Production and Inventory Management Journal, Vol. 34, No. 4, pp.22-27, 1993. [3] Chan, W. M. and Ibrahim, R. N., Evaluating the Quality Level of a Product with Multiple Quality Characteristics, Vol. 24, pp.738-742, 2004. [4] Chen, C. and Bulﬁn, R. L., Complexity of Single Machine, Multi-Criteria Scheduling Problems, European Journal of Operational Research, Vol. 70, pp.115-125, 1993. [5] Dileepan, P., Bicriterion Static Scheduling Research for a Single Machine, OMEGA, Vol. 16, No. 1, pp.53-59, 1988. [6] Festervand, T. A, Kethley, R. B. and Waller, B. D., The Marketing of Industrial Real Estate: Appli- cation of Taguchi Loss Functions, Journal of Multi-Criteria Decision Analysis, Vol. 10., pp.219-228, 2001. [7] Herdia, J. A. and Romero, F., Using Economic Criteriato Establish Manufacturing Tolerances, In- ternational Journal of Advanced Manufacturing Technology, Vol. 26., pp. 78-85, 1981. [8] Kethley, R. B., Waller, B. D. and Festervand T. A., Improving Customer Service in the Real Estate Industry: A Property Selection Model Using Taguchi Loss Functions, Total Quality Management, Vol. 13, No. 6, pp.739-748, 2002. [9] Li, C. and Chen A., Quality Evaluation of Domestic Airline Industry using Modiﬁed Taguchi Loss Function with Diﬀerent Weights and Target Values, Total Quality Management, Vol. 9, No. 7, pp.645-643, 1998. [10] Matsuo, H., Suh, C. J. and Sullivan R. S., A Controlled Search Simulated Annealing Method for the Single Machine Weighted Tardiness Problem, Annals of Operations Research, Vol. 21, pp.85-108, 1989. [11] Nagar, A., Haddock, J. and Heragu, S., Multiple and Bi-criteria Scheduling: A Literature Review, European Journal of Operational Research, Vol. 81, pp.88-104, 1995. [12] Pi, W. and Low, C., Supplier Evaluation and Selection using Taguchi Loss Functions, International Journal Advanced Manufacturing Technology, Vol. 26, pp.155-160, 2005. [13] Pi, W. and Low, C., Supplier Evaluation and Selection via Taguchi Loss Functions and an AHP, International Journal Advanced Manufacturing Technology, Vol. 27, pp.625-630, 2006. [14] Chan, W. M. and Ibrahim, R. N., Evaluating the Quality Level of a Product with Multiple Quality Characteristics, Vol. 24, pp.738-742., 2004. [15] Roslund, J. L., Evaluating Management Objectives with the Quality Loss Function, Quality Progress, August, pp.45-49, 1989. [16] Sen, T. and White, C. S., A Note on Two Scheduling Problems on Single Machines with Dual Criteria, OMEGA, Vol. 20, No. 3, pp.404-406, 1991. [17] Smith, W. E., Various Optimizers for Single-Stage Production, Naval Research Logistics, Vol. 3, pp.59-66, 1956. [18] Snow. J. M., Rating Quality and Selecting Suppliers Using Taguchi Loss Functions, Naval Engineers Journal, January, pp.51-57, 1993. [19] Stevens, D. P. and Baker, R. C., A Generalized Loss Function for Process Optimization, Decision Science, Vol. 25, No. 1, pp.41-56, 1994. [20] Stewart, T. J., A Critical Survey on the Status of Multiple Criteria Decision Making Theory and Practice, OMEGA, Vol. 20, No. 5/6, pp.569-586, 1992. [21] Woolsey, R. E. D., Production Scheduling Quick and Dirty Methods For Parallel Machines, Produc- tion and Inventory Management Journal, Vol. 31, No. 3, pp.84-87, 1990. i i i i i “M19N43” — 2008/11/5 — 17:14 — page 600 — #12 i 600 International Journal of Information and Management Sciences, Vol. 19, No. 4, December, 2008 Author’s Information Bryan is currently an associate professor in the Department of Management and Marketing, Middle Tennessee State University, USA. He received his Ph.D. in Production/Operations from the University of Mississippi, USA. His research interests are quality, loss functions and employee selection. Department of Management and Management, P.O. Box X173, Middle Tennessee State University, Murfreesboro, Tennessee, 37132, U.S.A. E-mail: bkethley@mtsu.edu TEL : 615-898-5882 i i i