Using Taguchi Loss Functions to Develop a Single Objective by xld14276

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    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.




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       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.




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          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.




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    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




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          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.




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    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




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          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)




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    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.




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          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




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    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

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        on Computing, Vol. 6, No. 2, pp.154-160, 1994.




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           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
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    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




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