A fuzzy analysis on perishable-asset in earning management of mainland China airlines.

ABSTRACT

This paper is to study solution perishable-asset problem by fuzzy no-show and overbooking and improve their earnings. The research is using the triangular fuzzy numbers and fuzzy set theory to get the optimal decision. In addition, this paper is the first to use fuzzy concept to solve the no-show and overbooking problems in earning management. By utilizing the new decision method, the decision-maker can handle more information and improve planning and earning decision management with respect to no-show and overbooking utilizing fuzzy economic scenario. To efficiently handle the fuzziness of a decision variable with respect to the planning and determination of optimal earning management, the triangular fuzzy numbers are used to act as an evaluation tool. Some computational methods of fuzzy no-show and fuzzy overbooking are also proposed.

Keywords: Perishable-asset, Earnings management, Overbooking

1. Introduction

An airline operating is in an uncertain environment when the airplane was departure from airport. Thus the airlines use seat overbooking and no-show that leads to an uncertain income. Expected airlines overbooking determination frequently depends on past performance and forecasts of seat sales. The seat booking was uncertainty. Therefore, airlines are overbooking to reduce income uncertainty and losses. This research focuses on perishable-asset by earning and overbooking management of airlines under uncertain demand conditions. In this study, we use Fuzzy Set Theory (Zadeh, 1965) that is rational method to determine the effects of non-measurable hypothesis and to offer a rational model to reduce losses to the airlines. The extension to earning management method is demonstrated by using triangular fuzzy numbers. The fuzzy considerations have an advantage over other methods. The figure 1 is showed the logic about perishable-asset of seats.

FIGURE 1. THE LOGIC OF PERISHABLE-ASSET

[demand uncertain] [arrow right] [Perishable-asset] [right arrow] [Overbooking] [right arrow] [Earning increase]

2. PREVIOUS RESEARCH

When airlines operate in an uncertain environment, they frequently accept reservations for more than the number of seats available. Smith et al. (1992) found that sold-out airline flights still experience about 15% of unfilled seats during the actual flights in the US. This is the main reason for airlines overbooking in order to compensate for the unrecoverable revenue.

Another distinction between probability and fuzzy sets is that probability is the theory of random events, while fuzzy set theory, is not concerned with events at all. (Klir, Clair and Yuan 1997) Bodily and Therford (1995) studied perishable-asset revenue management in generic form and multiple-price yield management form with diversion. The study's purpose was to determine the size of the uncertainty of customers' showing-up to a flight, its influence on airline earning management and on the number of discount booking. Thus, once discount sales are curtailed, the airlines begin to book at full-price. They found that they had to reserve the same number of seats for discount customers as before. Furthermore, the effects of the bumping penalty offset the amount that one would overbook due to the uncertainty of the show up.

Suzuki (2002) seeks the optimal overbooking policies for US major airlines by considering denied-boarding passengers behavior after they are bumped. The results indicate that overbooking improves an airline's "current" revenue, but it reduces the airline's future revenues. Although, there is a significant negative overbooking effect, no airline should decrease the overbooking levels because the positive side of overbooking is much stronger and it more than offsets the negative side.

The fuzzy set analysis is based on Zadeh (1965) who introduced the subject. He introduced a fuzzy subset of a given set X as a function f. [right arrow] [0, 1] interpreting the value f(x) as the degree of membership of the element x [member of] X. If we consider the real unit interval [0, 1] as a set of true-values, it is apparent that fuzzy subsets are characteristic functions with respect to an infinitely many-valued logic.

Bodily and Pfeifer (1992) using traditional methodology of probability precision numbers, analyzed aviation market demand developed an overbooking model which, included earning management. They found that under uncertain conditions of no-show and overbooking, earning management are dilemmatic on aviation market. Decision makers frequently use linguistic terms in discussing the issues, for example: around 320 travels or more than 5% overbooking. The sense of uncertainty represented by fuzziness, however, is not the uncertainty of expectation. It is the uncertainty associated with the imprecision of a concept expressed by a linguistic term in ordinary language.

Although many papers are studying the problems of perishable-asset revenue management and overbooking, the main problem is the uncertainty of no-show to the degree that it effects airlines decision making with respect to overbooking numbers. The traditional analytical methods of no-show and earning could not explain the maximum and minimum earning. However, the fuzzy method solves earning problem, and assists the decision maker understand the interval of earning.

3. METHODS

First, we will define concepts and variables such as: fuzzy set theory, triangular fuzzy number, fuzzy acut, ranking of fuzzy numbers and earning management. Second, we transfer the earning management method onto a fuzzy model.

3.1 Fuzzy Set Theory

Let X be a space of points, with a generic element of X denoted by x. Thus, X={x}. A fuzzy set A in X is characterized by a membership function [f.sub.A](x) which is associated with each point in X a real number in the interval [0, 1], with the value of [f.sub.A](x) at x representing the "degree of membership" of x in A. Thus, the closer the value of [f.sub.A](x) to unity the higher is the degree of membership of x in A. When A is a set in the ordinary sense of the term, its membership function can take on only two values 0 and 1, with [f.sub.A](x)=1 or accordingly as x does or does not belong to A. Thus, in this case [f.sub.A](x) is reduced to the familiar characteristic function of a set A (Zadeh, 1965).

3.2 Triangular Fuzzy Number

A fuzzy number A (Dubois and Prade, 1978) in R (real line) is a triangular fuzzy number if its membership function [f.sub.A]: R [right arrow] [0, 1] is:

With -[infinity] < c [greater than or equal to] a [greater than or equal to] b < [infinity], the triangular fuzzy number A can be represented by (c, a, b), that is A =(c, a, b). Triangular fuzzy number A has max degree of membership on a. i.e. [f.sub.A](a)=1. In addition, c and b are the lower and upper bounds of the available area for the evaluation data. They are used to reflect the fuzziness of the evaluation data, the narrower the interval [c, b], the lower the fuzziness of the evaluation data.

3.3 The Algebraic Operation of Fuzzy Numbers based on The a-cut Concept

The [alpha]-cut of fuzzy number is defined as [A.sup.[alpha]] = {x [member of] X | [f.sub.A](x) [greater than or equal to] [alpha], 0 [less than or equal to] [alpha] [less than or equal to]1} = [[A.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]]. The A and B are positive fuzzy numbers, i.e., [A.sub.l.sup.[alpha]] > 0, [B.sub.l.sup.[alpha]] > 0 for all [alpha] [member of] [0,1]. Let [A.sup.[alpha]] = [Al[alpha], [A.sub.u.sup.[alpha]] and [B.sup.[alpha]] = [[B.sub.l.sup.[alpha]], [B.sub.u.sup.[alpha]]. According to extension principle (Zadeh, 1965) and vertex method (Dong and Shah, 1987), the algebraic operations of any two positive fuzzy numbers A and B can be expressed as:

1. Fuzzy addition: [(A [direct sum] B).sup.[alpha]] = [[A.sub.l.sup.[alpha]] + [B.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]]+ [B.sub.u.sup.[alpha]]

2. Fuzzy subtraction: [(A[??]B).sup.[alpha]] = [[A.sub.l.sup.[alpha]] - [B.sub.u.sup.[alpha]], [A.sub.u.sup.[alpha]] - [B.sub.l.sup.[alpha]]]

3. Fuzzy multiplication: [(A [cross product] B).sup.[alpha]] = [[A.sub.l.sup.[alpha]] [B.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]] [B.sub.u.sup.[alpha]]]

4. Fuzzy division: [(A[empty set]B).sup.[alpha]] = [[A.sub.l.sup.[alpha]]/ [B.sub.u.sup.[alpha]], [A.sub.u.sup.[alpha]]/[B.sub.l.sup.[alpha]]]

3.4 Ranking of Fuzzy Numbers

A fuzzy set can be expressed in terms of the concept of [alpha]-cut without resorting to the membership function (Terano, Asai and Sugeno, 1991). Thus, we use [alpha]-cut method to sort fuzzy numbers. Let [A.sub.1], [A.sub.2], ..., [A.sub.i], ..., [A.sub.n], be n fuzzy numbers, and the left and right membership function of fuzzy number [A.sub.i] are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the inverse functions of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Define the left integral value [I.sup.L]([A.sub.i]) and right integral value [I.sup.L]([A.sub.i]) of [A.sub.i] as (Liou and Wang, 1992; Yager, 1981):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [[alpha]].sub.j][member of][0,1], j=0,1, ..., k, and 0[][[alpha].sub.0] < [[alpha].sub.1] ... < [[alpha].sub.j] < ... [[alpha].sub.k]=1. then the left and right integral values can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where [DELTA][[alpha].sub.j] = [[alpha].sub.j] - [[alpha].sub.j-1]. The ranking value R([A.sub.i]) of fuzzy number [A.sub.i], is defined as R([A.sub.i])= [I.sup.L]([A.sub.i]) + [I.sup.R]([A.sub.i]). Define the ranking of the fuzzy numbers [A.sub.i] and [A.sub.j] based on the following rules:

[A.sub.i] > [A.sub.j] [??] R([A.sub.i]) > R([A.sub.j])

[A.sub.i] < [A.sub.j] [??] R([A.sub.i]) < R([A.sub.j])

[A.sub.i] = [A.sub.j] [??] R([A.sub.i]) = R([A.sub.j])

3.5 Earning Management

The main reason that airlines' overbook is to compensate for uncertainty that a customer would not show-up to a flight. On full flights, these unpredictability results in empty seats that could have been filled, and monetary rewards or compensation for customers. To reduce losses, airlines determine the overbooking ratio for each flight. Thus, the earning model utilized to obtain the optimal overbooking levels that maximizes an airline's cumulative passenger revenue can be determined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

There are carrier i at case t to be defined. [R.sub.it] is revenue. [P.sub.it] is the per-mile fare. [Q.sub.it] is the total passenger. And that [P.sub.it][MILE.sub.it] represents the airline income from sum of flight miles of per passenger. [OB.sub.it] is defined as the numbers of passengers overbooked. [COMP.sub.i] is the compensation that airline i pays per bumped passenger. [NS.sub.it] measures those who are no-show.

3.6 Fuzzy Earning Management

We are deep thinking them to draw figure 3 which represent no-show and fuzzy overbooking in the fuzzy earning management environment. Thus, ([c.sub.1], [a.sub.1], [b.sub.1]) is fuzzy no-show, ([c.sub.2], [a.sub.2], [b.sub.2]) is fuzzy overbooking, and ([c.sub.2], [a.sub.3], [b.sub.1]) represents the difference between no-show and overbooking. Furthermore, the fuzzy noshow is ([c.sub.1], [a.sub.1], [c.sub.2]) that the seats would become a perishable-asset when the airplane is departure. The fuzzy overbooking is ([b.sub.1], [a.sub.2], [b.sub.2]) which means that airlines have to compensate customers. Finally, there is an addition of an F before each variable of equation (3), which represents a fuzzy function. The equation is as following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[FIGURE 2 OMITTED]

4. AN EARNING MANAGEMENT CASE

4.1 Case Study in Mainland China

According the study presented by Smith et al. in 1992, for flights whose seats are completely sold out airlines still experience approximately 15% of seat unfilled during the actual flights. Thus, the fuzzy numbers we depend on 15%. We use Mainland China airlines to study. There are offer 200 seat for flight. The total seat fuzzy numbers (200, 200, 200), Thus, they accept fuzzy overbooking numbers are (30, 30, 30). Price discount is depended on which distributors that they had sold. Compensation is the same fare. The list is on the Table 1.

4.2 The Computation of Fuzzy Earning Management

We have scenarios of [alpha]-cut of management earning under difference scenarios. Thus, according to table 1 and the[alpha]-cut principle, we can calculate the fuzzy values of each case in the earning management matrix. We set six ranges to calculate the[alpha]a-cut values. Because the material too is huge, we only demonstrate case 19 to 23 on Table 2. The results are shown on Table 2.

After we obtain the [alpha]-cut value, we rank the fuzzy earning management values as shown in Table 3. The maximum number of ranking of earning management is 125,100 in case of 21, which can also be seen in figure 4. It is a trend of ranking of fuzzy management earning. Thus, the benefit interval is between 57,600 and 67,600 in the case of 21 (Table 4). This method assists management in making decisions. It does not only utilize precise/fuzzy numbers for decision-making, but it also provides for which interval the minimum and maximum under difference scenarios.

In the case of 21, the overbooking value is (30,30,30) and the no-show value is (100, 105, 110). This can be explaining that when the numbers of travelers is under (130, 135, 140) revenue will go down. Thus, the airlines can earn between $57,600 and $67,600 when the travelers are in this interval. In fact, we only offer a new method with respect to no-show and overbooking of management earning. This simple management-earning example does not include other costs. We are only focused on no-show and overbooking cost. Airlines can add other costs in the process of calculation.

5. CONCLUSION

Under the environment information is indefinite of demand and supply market, the information gained from uncertain demand is very important. Perishable-asset earnings management and decision-making are often uncertain and fuzzy. Under these situations, it is difficult to make good decision, especially, when the demand is in the peak time. This paper provides a solution method with respect to no-show and overbooking in perishable-asset management. The research results provides for that under uncertain conditions decision markers can better understand the interval of profit. The airlines companies can go farther by adding other cost in the process of calculation.

REFERENCE

Bodily, SE and Pfeifer, PE, "Overbooking decision rules", Omega, Vol. 20, 1992, 129-133.

Bodily, SE and Weatherford, LR, "Perishable-asset Revenue Management Generic and Multiple-price Yield Management with Diversion", Omega Int. J. Mgmt. Sic., Vol. 23, 1995, 173-185.

Dong, W. and Shah, H. C., "Vertex methods for computing functions of fuzzy variable", Fuzzy sets and Systems, Vol. 24, 1987, 65-78.

Dubois, D. and Prade, H., "Operations on fuzzy numbers", International Journal of System Science, Vol. 9, 1978, 613-626.

Klir, George J., Clair, Ute H. St. and Yuan, Bo, Fuzzy set theory foundations and applications, USA, 1997.

Liou, T. S. and Wang, M J. J., "Ranking fuzzy numbers with integral value", Fuzzy Sets and Systems, Vol. 50, 1992, 247-255.

Simon, J., "An almost practical solution to airline overbooking", Journal of Transport Economics and Policy, Vol. 2, 1968, 201-22.

Simon, J., "Airline overbooking: the state of the art a reply", Journal of Transport Economics and Policy, Vol. 6, 1972, 254-256.

Smith, C., Laeimkuhler, J. F. and Darrow, R. M., "Yield management at American airlines", Interfaces, Vol. 22, 1992, 8-31.

Suzuki, Yoshinori, "An empirical analysis of the optimal overbooking policies for US major airlines", Transportation Research Part E, Vol. 38, 2002, 135-149.

Terano, Toshiro, Asai, Kiyoji and Sugeno, Michio, Fuzzy systems theory and its applications, Academic Press Inc., Tokyo, Japan, 1992.

Yager, R. R., "A procedure for ordering fuzzy subsets of the unit interval", Information Science, Vol. 24, 1981, 143-101.

Zadeh, L. A., "Fuzzy sets", Information Control, Vol. 8, 1965, 338-353.

Zadeh, L. A., "The concept of a linguistic variable and its application to approximate reasoning", Part 1, 2 and 3, Information Science, Vol. 8, 1975-1976, 199-249, 301-357, 43-58.

Yu-Feng Lin, National Sun Yat-Sen University, TAIWAN

Dr. Yu-Feng Lin earned her Ph.D. at Sun Yat-Sen University, Institute of Mainland China Studies, Taiwan.

TABLE 1. DATA IN FUZZY MODEL

Cases t    Total seats    Total Overbooking  Total No-show
                                                persons

1        (200, 200, 200)     (30,30, 30)      (0, 5, 10)
2        (200, 200, 200)     (30,30, 30)      (5, 10, 15)
3        (200, 200, 200)     (30,30, 30)     (10, 15, 20)
4        (200, 200, 200)     (30,30, 30)     (15, 20, 25)
5        (200, 200, 200)     (30,30, 30)     (20, 25, 30)
6        (200, 200, 200)     (30,30, 30)     (25, 30, 35)

Cases t  Price of per mile        Miles         Compensates
              (units)

1            (4, 5, 6)      (4000, 4000, 4000)   (4, 5, 6)
2            (5, 6, 7)      (4000, 4000, 4000)   (5, 6, 7)
3            (6, 7, 8)      (4000, 4000, 4000)   (6, 7, 8)
4            (7, 8, 9)      (4000, 4000, 4000)   (7, 8, 9)
5           (8, 9, 10)      (4000, 4000, 4000)  (8, 9, 10)
6           (9, 10, 11)     (4000, 4000, 4000)  (9, 10, 11)

TABLE 2. THE [alpha]-CUT OF FUZZY REVENUE [FR.sub.IT]

Cases t  [alpha]  [alpha]-cut of fuzzy revenue [FR.sub.it]

            0                  157200, 67200]
           0.2                 [58180, 66180]
           0.4                 [59160, 65160]
19         0.6                 [60140, 64140]
           0.8                 [61120, 63120]
            1                  [62100, 621001
            0                  [57500, 67500]
           0.2                 [58480, 66480]
20         0.4                 [59460, 65460]
           0.6                 [60440, 64440]
           0.8                 [61420, 63420]
            1                  [62400, 62400]
            0                  [57600, 67600]
           0.2                 [58580, 66580]
21         0.4                 159560, 65560]
           0.6                 [60540, 64540]
           0.8                 [61520, 63520]
            1                  [62520, 62520]
            0                  [57500, 67500]
           0.2                 [58480, 66480]
22         0.4                 [59460, 65460]
           0.6                 [60440, 64440]
           0.8                 [61420, 63420]
            1                  f62400, 62400]
            0                  [57200, 67200]
           0.2                 [58180, 66180]
23         0.4                 [59160, 65160]
           0.6                 [60140, 64140]
           0.8                 161120, 63120]
            1                  [62100, 62100]

TABLE 3. RANKING VALUE OF FUZZY REVENUE FRIT

Cases t    [I.sup.L]      [I.sup.R]    R ([FR.sub.it])
         ([FR.sub.it])  ([FR.sub.it])

19          59,650         64,650          124,300
20          59,950         64,950          124,900
21          60,050         65,050          125,100
22          59,950         64,950          124,900
23          59,650         64,650          124,300

TABLE 4. FUZZY REVENUE [FR.sub.IT] AT T=19, 20, 21, 22, AND 23

Cases    Fuzzy revenue
t        ([FR.sub.it])

19       (57200, 62100,
            67200)
20       (57500, 62400,
            67500)
21       (57600, 62500,
            67600)
22       (57500, 62400,
            67500)
23       (57200, 62100,
            67200)