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Management of surgical waiting lists through a Possibilistic Linear Multiobjective Programming problem
Applied Mathematics and Computation 167 (2005) 477–495 www.elsevier.com/locate/amc
Management of surgical waiting lists through a Possibilistic Linear Multiobjective Programming problem Blanca Pe´rez Gladish *, Mar Arenas Parra, Amelia Bilbao Terol, Ma Victoria Rodrı´guez Urı´a Dpto. Economı´a Cuantitativa, Facultad de Ciencias Econo´micas y Empresariales de la Universidad de Oviedo, Avda. del Cristo s/n, Oviedo, Asturias (Espan˜a) C.P. 33006, Spain
Abstract This study attempts to apply a management science technique to improve the efficiency of Hospital Administration. We aim to design the performance of the surgical services at a Public Hospital that allows the Decision-Maker to plan surgical scheduling over one year in order to reduce waiting lists. Real decision problems usually involve several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory. Here we apply a Possibilistic Linear Multiobjective Programming method, developed by the authors, for solving a hospital management problem using a Fuzzy Compromise Programming approach. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy Sets; Compromise programming; Decision making; Surgical waiting lists
*
Corresponding author. E-mail addresses: [email protected] (B. Pe´rez Gladish), [email protected] (M. Arenas Parra), [email protected] (A. Bilbao Terol), [email protected] (Ma Victoria Rodrı´guez Urı´a). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.07.015
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1. Introduction Hospitals are considered public utility companies and are supposed to be non-profit institutions and they usually work in a political and highly complex environment. Managerial authority is shared between doctors and administrators and each of the two groups aims to formulate its own individual policies and objectives. As these policies and objectives may not coincide, and may even be in direct opposition to each other, giving rise to a situation that is certain to affect hospital performance and they must be taken into account when proposing any kind of result analysis. Once the quantitative target values for these objectives have been established, the main aim of any hospital administration is to improve the efficiency of hospital performance. Spanish public hospitals, as providers of health services, must draw up at the end of each year with financial authorities, a document called ‘‘Management Contract’’, which expresses the objectives to be reached over the following year and also aims at establishing quantitative target values for these objectives. One of the main objectives of the current Management Contracts is to reduce the waiting lists for surgical processes. The existence of surgical waiting lists is an important problem which infringes on citizenÕs health care rights. There are several causes for the existence of waiting lists but everyone agrees that the problem is essentially one of distribution and optimal design of resources. Thus, it seems necessary to apply modern management techniques to assure the efficient use of medical facilities and resources (see [1,2]). In this paper, a Possibilistic Multiobjective Linear Programming problem (FM-LP) is developed as an information system in order to analyze the internal coherency of the goals expressed by administrative authorities. Also, applying a Multicriteria Decision technique, Compromise Programming, we intend to design and manage, in an optimal way, the real performance of the surgical services of a medium-sized hospital of INSALUD (The Spanish Health Department), taking into account all the function constraints. This offers the decision centre a suitable methodology with which to analyze whether or not it is possible to improve the running of the services.
2. Description of the model In order to reach the previously mentioned objectives, hospitals are allowed to use several methods of operating scheduling: ordinary activity, within regular-operating hours, or extraordinary activity that would be developed overtime or through private hospital contracts. The model developed deals with
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all the services of the hospital. However, in order to keep the presentation simple, we focus on a specific service: General Surgery. 2.1. Program variables and data The variables considered are grouped into two main blocks, ordinary activity and extraordinary activity, and they are denoted according to the initials of the corresponding service. Each variable has two subscripts: the first one, i, denotes the specific process, while the second one denotes the month when the operation is carried out. An X precedes the extraordinary activity and an L precedes the state of the list (Tables 1 and 2). Planning the hold year in advance is possible but, usually, reality imposes its own rules and a permanent re-planning process is required. In this sense, it is necessary to carry out a monthly update, once the activity reports and new admissions and exclusions are known. For this reason, the model has been designed so that the starting point, or initial value of the state variables, as well as the corresponding parameters, can be related to this update at a specific moment of the process. In this paper the update was carried out at the end of the first quarter of the year, so the index j which corresponds to the months that follow the current one, will range from 4 to 12. We consider nine processes which represent about 45% of the total activity of the service under consideration (Table 3). The model includes data corresponding on the time spent on the operations and estimations on incoming and outgoing patients. Time spent on each process are uncertain data. The Decision Maker (DM) considers that the information provided by the average duration time obtained from the operating theatre reports of the preceding year, is not enough because the time employed for different cases of the same process are very different. For these reasons they are considered as imprecise or fuzzy data and they are
Table 1 Program decision variables Modality
Service
Process
Month
–/X
C
i
j
Table 2 Program state variables List
Service
Process
Month
L
C
i
j
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Table 3 Service: General Surgical Code
Process
Variable
241 278 454 455 550 553 565 574 685
Nodular goiter Morbid Obesity Varicose Veins Haemorrhoids Inguinal Hernias Incisional Hernias Anal Fissure/Anal Fistula Cholelithiasis Pilonidal Cyst
C01 C02 C03 C04 C05 C06 C07 C08 C09
µti 1
0
ti t i
BE
t i
RD
t i
WE
Fig. 1. Possibility distribution of the fuzzy duration data ~ti .
handled by triangular fuzzy numbers described by their possibility distributions estimated by the DM. 1 ~ti ¼ ðBEti ; RDti ; WEti Þ; where RDti is the average duration time, which has been determined using the mean data, plus 20 minutes, which is the time needed to prepare the theatre for the next operation; BEti and WEti are the best and the worst estimations given by the DM. In Fig. 1, l~ti , with 0 6 l~ti 6 1, represents the possibility degree of occurrence of parameter ~ti which represents the time spent on each process. Thus, the DM considers that the value with a higher possibility degree of occurrence, the central value of the fuzzy triangular number, is the real data RDti , which has l~ti ¼ 1. As well, we can observe that the DM believes best and worse estimations, WEti , BEti , have no possibility of occurring, l~ti ¼ 0, 1
Information about possibility distribution of fuzzy numbers can be found in [3].
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Table 4 Duration processes estimations Code
Process
Variable
BEti
RDti
WEti
241 278 454 455 550 553 565 574 685
Nodular goiter Morbid Obesity Varicose Veins Haemorrhoids Inguinal Hernias Incisional Hernias Anal Fissure/Anal Fistula Cholelithiasis Pilonidal Cyst
C01 C02 C03 C04 C05 C06 C07 C08 C09
128 132 115 63 97 114 53 115 54
151 155 135 74 114 134 62 135 63
182 186 162 89 137 161 75 162 76
representing lateral data in the fuzzy triangular number. Remaining values for the time spent on processes have intermediate possibility degrees of occurrence (Table 4). The number of monthly incoming/outgoing patients without an operation (admissions and exclusions respectively) has been estimated by the DM using e ij ) and excludata from the two preceding years. The number of admissions ( A e sions ð S ij Þ will also be considered as fuzzy triangular numbers, given their unknown character and will be expressed as follows: e ij ¼ ðBEA ; WRA ; WEA Þ; Admissions A ij ij ij Exclusions e S ij ¼ ðWESij ; WRSij ; BESij Þ; where WRAij and WRSij represent the worst historic real data for admissions and exclusions respectively; WEAij and WESij are the pessimistic estimations of the DM for the current year and BEAij and BESij are the optimistic ones. In Tables 5 and 6 we show the estimations for the current year of incoming and outgoing patients (admissions and exclusions, respectively), considered as imprecise data represented by fuzzy triangular numbers. 2.2. Constraints Four groups of constraints have been considered: 2.2.1. Evolution of waiting lists Evolution of the waiting lists is reflected by f iðjþ1Þ LC f ij þ AN f ij XC ij C ij ; LC
ð1Þ
f ij denotes the state of the waiting list for process i at the beginning of where LC f ij ¼ A e ij e e ij is the estimated number of admissions for month j, AN S ij where A process i during month j, and e S ij represents the estimated number of exclusions
482
Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
(7, 8, 10) (3, 3, 4) (22, 24, 31) (15, 17, 22) (49, 54, 70) (23, 26, 34) (16, 18, 23) (21, 23, 30) (24, 27, 35)
(5, 6, 8) (4, 4, 5) (23, 25, 33) (14, 15, 20) (54, 60, 78) (19, 21, 27) (14, 15, 20) (27, 30, 39) (29, 32, 42)
(4, 4, 5) (2, 2, 3) (20, 22, 29) (10, 11, 14) (51, 57, 74) (23, 25, 33) (14, 16, 21) (22, 24, 31) (23, 25, 33)
(5, 5, 7) (1, 1, 1) (15, 17, 22) (11, 12, 16) (42, 47, 61) (23, 25, 33) (11, 12, 16) (15, 17, 22) (17, 19, 25)
(5, 5, 7) (2, 2, 3) (11, 12, 16) (7, 8, 10) (27, 30, 39) (14, 15, 20) (9, 10, 13) (13, 14, 18) (13, 14, 18)
(14, 15, 20) (5, 5, 7) (19, 21, 27) (7, 8, 10) (46, 51, 66) (26, 29, 38) (8, 9, 12) (19, 21, 27) (17, 19, 25)
(7, 8 10) (4, 4, 5) (22, 24, 31) (12, 13, 17) (56, 62, 81) (31, 34, 44) (19, 21, 27) (23, 25, 33) (35, 39, 51)
(5 5 7) (2, 2, 3) (26, 29, 38) (16, 18, 23) (65, 72, 94) (32, 35, 46) (19, 21, 27) (21, 23, 30) (43, 48, 62)
(8, 9, 12) (5, 5, 7) (14, 15, 20) (13, 14, 18) (37, 41, 53) (18, 20, 26) (14, 16, 21) (18, 20, 26) (30, 33, 43)
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Table 5 Incoming estimations (admissions) for General Surgery
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Table 6 Outgoing estimations (exclusions) for General Surgery Code
Apr
241 278 454 455 550 553 565 574 685
(0, (1, (2, (0, (4, (1, (0, (1, (5,
0, 1, 2, 0, 4, 1, 0, 1, 5,
May 0) 1) 3) 0) 5) 1) 0) 1) 7)
(0, (0, (0, (1, (1, (0, (1, (0, (1,
0, 0, 0, 1, 1, 0, 1, 0, 1,
Jun 0) 0) 0) 1) 1) 0) 1) 0) 1)
(1, (0, (0, (0, (0, (1, (0, (5, (1,
Jul
1, 0, 0, 0, 0, 1, 0, 6, 1,
1) 0) 0) 0) 0) 1) 0) 8) 1)
(0, (0, (0, (0, (1, (0, (0, (1, (0,
Aug 0, 0, 0, 0, 1, 0, 0, 1, 0,
0) 0) 0) 0) 1) 0) 0) 1) 0)
(0, (0, (0, (0, (0, (2, (0, (0, (1,
0, 0, 0, 0, 0, 2, 0, 0, 1,
Sep 0) 0) 0) 0) 0) 3) 0) 0) 1)
(0, (0, (1, (1, (5, (5, (0, (4, (2,
0, 0, 1, 1, 5, 5, 0, 4, 2,
Oct 0) 0) 1) 1) 7) 7) 0) 5) 3)
(0, (0, (1, (0, (4, (3, (0, (2, (2,
0, 0, 1, 0, 4, 3, 0, 2, 2,
Nov 0) 0) 1) 0) 5) 4) 0) 3) 3)
(0, (0, (0, (2, (5, (1, (0, (2, (2,
Dec
0, 0, 0, 2, 5, 1, 0, 2, 2,
0) 0) 0) 3) 7) 1) 0) 3) 3)
(0, (0, (0, (1, (0, (1, (1, (0, (1,
0, 0, 0, 1, 0, 1, 1, 0, 1,
0) 0) 0) 1) 0) 1) 1) 0) 1)
without surgical operation for process i, for each month j; XCij, Cij are respectively the obtained extraordinary and ordinary optimal surgical activity. The state of the waiting lists at the beginning of the scheduling period is known, as it is the residual list of the the previous scheduled period (Table 7). 2.2.2. Monthly operating theatre availability The available operating theatre times are computed considering that the maximum daily activity is six and a half hours. Moreover, it is assumed that the operating theatre sessions are assigned in advance to each service through internal agreements, and cannot be changed at this stage of the process. Thus, the constraint corresponding to month j takes the following form: 9 X
~ti C ij 6 CQj ;
ð2Þ
i¼1
where the right-hand side CQj represents the total operating theatre time available for the General Surgery service in month j, and can be found in Table 8. 2.2.3. Upper bounds on the stay on a waiting list The main priority with respect to the surgical activity for the planning year is that no patient should stay on the waiting list for more than 6 months. With this aim, it must be assured that the sum of the ordinary and extraordinary operations performed between April and the month k be greater than the number of patients that would be 6 months or longer on the waiting list for each process i in month k: Table 7 Initial state of the waiting list
Number of patients
241
278
454
455
550
553
565
574
685
17
13
51
21
118
58
28
59
49
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Table 8 Total monthly available theatre time for General Surgery
Minutes
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
3955
4704
4976
3316
2997
2997
5183
4815
4385
9 X k X ðC ij þ XC ij Þ P sik : i¼1
ð3Þ
j¼4
Above equations set lower bounds on the global activity level of each process, thereby establishing the maximum stay requirements. These lower bounds must be fixed for each process and month. We obtain the minimum bounds, sik, from the state of the waiting list at the beginning of the scheduled year (Table 8), accumulating these values month by month. Parameters sik by process and month are displayed in Table 9. 2.2.4. Bounds on extraordinary operations DM know at the beginning of the year that it is not going to be possible to carry out in the hospital all the surgical activity, so they concert with Health Authorities for some processes to be transferred to other centres at least in a certain number. In the General Surgery service, these constraints correspond to Inguinal Hernias, which is the surgical process with the highest waiting list: XC 5j P 1;
j ¼ 4; . . . ; 12:
ð4Þ
Also, there are several surgical processes that are preferred not to be transferred to other centres due to their surgical complexity. This is the case of Nodular goiter, Morbid Obesity and Cholelithiasis:
Table 9 Parameters sik corresponding to General Surgery Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
25 8 29 17 78 55 19 41 25
28 10 52 33 139 84 35 61 48
36 15 67 46 179 104 46 80 69
42 17 86 66 214 124 60 89 114
57 17 105 88 281 164 83 110 162
63 17 151 97 325 203 94 119 224
67 20 175 112 358 243 110 129 249
75 24 186 118 417 261 126 157 261
75 26 223 133 505 301 150 175 281
B. Pe´rez Gladish et al. / Appl. Math. Comput. 167 (2005) 477–495 12 X
XC 1j ¼ 0;
j¼4
12 X
XC 2j ¼ 0;
j¼4
12 X
XC 8j ¼ 0:
485
ð5Þ
j¼4
No integrity conditions are imposed in the model, given that the estimated times per process are average times, and, thus, only approximate planning is required. 2.3. Objectives and formulation of the problem To determine the number of processes which it is necessary to carry out for each month two criteria have been considered: performance capability of the hospital and extraordinary activity, which means operations in the hospital taking extra time, or referrals to other centres. With regard to the first criteria the objective is to maximise hospital performance capability, while the objective with regard to the second criteria is to fix the minimum extraordinary activity that allows us to reduce the residual waiting list. The first objective function carries out the maximum total employment of operational theatres with ordinary activity per month and process throughout the year: Fe 1 ¼ Max
9 X 12 X i¼1
~ti C ij :
ð6Þ
j¼4
The second objective represents the minimization of the total minutes of extraordinary activity we have to carry out if we want to maintain waiting lists under 6 months: Fe 2 ¼ Min
9 X 12 X i¼1
~ti XC ij :
ð7Þ
j¼4
3. Methodology The above problem describes a real decision situation. It is a Multiobjective Possibilistic Linear Programming problem (FP-MOLP) in which all the parameters are fuzzy and they are represented by fuzzy numbers described by their possibility distribution estimated by the analyst from the information supplied by the DM [4]. The uncertain and/or imprecise nature of the problemÕs parameters involves two main problems: feasibility and optimality. Feasibility may be handled by comparing fuzzy numbers. In this paper we use a fuzzy relationship to compare
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486
fuzzy numbers [5] that verifies suitable properties and that, besides, is computationally efficient to solve linear problems because it preserves its linearity. Optimality is handled through Compromise Programming (CP). CP is a well-known Multiple Criteria Decision Making approach developed by Yu [6] and Zeleny [7]. The basic idea in CP is the identification of an ideal solution as a point where each attribute under consideration achieves its optimum value. Zeleny states that alternatives that are closer to the ideal are preferred to those that are farther from it because being as close as possible to the perceived ideal is the rationale of human choice. We shall consider the following Multiobjective Possibilistic Programming problem: min ~z ¼ ð~z1 ; ~z2 ; . . . ; ~zk Þ ¼ ð~c1 x; ~c2 x; . . . ; ~ck xÞ FP-MOLP ( ) ~ bi ; i ¼ 1; . . . ; m ai x 6 ~ ~ e ; s:t: x 2 vð A; bÞ ¼ xP0 where xt = (x1, x2, . . . , xn) is the crisp decision vector, ~ct ¼ ð~c1 ; ~c2 ; . . . ; ~ck Þ are the e ¼ ½~aij fuzzy parameters of the k considered objectives, A mn is the fuzzy techt ~ ~ ~ ~ nological matrix and b ¼ ðb1 ; b2 ; . . . ; bm Þ are also fuzzy parameters. The uncertain and/or imprecise nature of the technological matrix and of the resource vector which defines the set of constraints of the model leads us to compare fuzzy numbers. In this work we compare fuzzy numbers handle them through their expected intervals as defined by Heilpern [8] and we use the fuzzy relationship defined by [5] which leads us to the concept of b-feasibility of a decision vector. A decision vector x 2 Rn , is said to be b-feasible for the problem FP-MOLP if x verifies the constraints at least in a degree b. That is: ~ ai x6b ~ bi ;
i ¼ 1; . . . ; m:
ð8Þ
In accordance with the above considerations we shall solve the FP-MOLP through a family of b-FP-MOLP problems, where 0 6 b 6 1: min ~z ¼ ð~z1 ; ~z2 ; . . . ; ~zk Þ ¼ ð~c1 x; ~c2 x; . . . ; ~ck xÞ s:t:
b-FP-MOLP
9 ~ ~ ð1 bÞE~a1i þ bE~a2i x 6 bEb1i þ ð1 bÞEb2i ; > = ¼ vðbÞ: i ¼ 1; . . . ; m > ; xP0
~i ¼ ð~ai1 ; ~ai2 ; . . . ; ~ain Þ is a vector The expected interval of the fuzzy vector a whose components are the expected intervals of each fuzzy number of vector ~ ai , that is: ai1 Þ; EIð~ ai2 Þ; . . . ; EIð~ ain ÞÞ: EIð~ ai Þ ¼ ðEIð~
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487
In order to apply the CP approach to solve the problem, we need to obtain the fuzzy ideal solution of the b-FP-MOLP problem. For this we use the solving method proposed by Arenas et al. [9]. In the CP framework, once the fuzzy ideal solution is found values of the decision variables, which determine a fuzzy solution, as accurately as possible to the fuzzy ideal solution. 2 We will solve the problem FP-MOLP by handling the fuzzy objectives, ~z ¼ ~cx, and the b-fuzzy ideal solution, ~z ðbÞ, through their expected intervals. 3 Therefore, the problem now is: ~ EIð~zr ðbÞÞ; Find a x 2 vðbÞ such that : EIð~cr xÞ ! EIð~zr ðbÞÞ
½E~1cr x ; E~2cr x
r ¼ 1; . . . ; k:
ð9Þ
~z ðbÞ ~z ðbÞ ½E1r ; E2r .
where EIð~cr xÞ ¼ and ¼ It should be considered desirable to obtain a fuzzy objective vector with less amplitude than the b-fuzzy ideal solution, i.e., such that it should verify the following condition: ~z ðbÞ
E~2cr x E~1cr x 6 E2r
~z ðbÞ
E1r
ð10Þ
:
From here, we assert that ~ EIð~zr ðbÞÞ; EIð~cr xÞ !
r ¼ 1; . . . ; k if and only if Dr ! 0;
where
n o ~z ðbÞ ~z ðbÞ Dr ¼ max E~1cr x E1r ; E2r E~2cr x ;
r ¼ 1; . . . ; k;
is the discrepancy between the rth fuzzy objective ~cr x and the r-th componet of the b-fuzzy ideal solution ~zr ðbÞ [10]. We shall solve now a new crisp CP problem where the objective is to minimize the discrepancy between the b-fuzzy ideal solution and fuzzy objectives. Therefore, the ideal solution is the null vector and we define a b-compromise solution of the FP-MOLP as a decision vector x* that is a compromise solution to the problem: min
ðD1 ; D2 ; . . . ; Dk Þ
s:t:
x 2 vðbÞ:
ð11Þ
As problem (11) is crisp, the compromise programming approach to solve it is based on the Lp family of distances: !1p k X p p w rD r Min Lp ¼ Min ð12Þ r¼1 s:t:
2 3
x 2 vðbÞ;
We shall denote the relation of ‘‘accurate’’ as ð!Þ. ~ More details about these results can be found in [10].
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488
where wr P 0. wr can be regarded as a normalizing coefficient and also as a weight that measures the relative importance of the discrepancy between the rth fuzzy objective and its fuzzy ideal value and may be established by the DM in an interactive process with the analyst. The most commonly obtained compromise solutions are for metrics p = 1 and p = 1 because for other metrics non-linear mathematical programming algorithms are needed. Also, under certain conditions (see [11]) they are the bounds of the whole compromise set.
4. Resolution of the problem We have solved the b-parametric Multiobjective Programming (13), with which we determine the b-efficient solutions, in an interactive way with the DM. In absence of preferential subjective weights for both objectives are assigned equal values (i.e., W1 = W2 = 1). As a first step we ask the DM to fix the feasibility degree he/she wants to assume. Once the DM has fixed the initial feasibility degree, the b-fuzzy ideal solution is obtained through its a-cuts. Table 10 contains the possibility distri butions of the ideal point ð Fe 1 ; Fe 2 Þ consisting of the optimal value of each of the considered objectives, maximum internal capability and minimum external activity, for each feasibility degree b fixed by the DM. As we can see, as DM establishes higher feasibility degrees optimal solution worsens, that is we have lower levels of internal activity and higher of external surgical activity. ! 9 X 12 9 X 12 X X ~ti C ij ; ~ti XC ij Min ð Fe 1 ; Fe 2 Þ ¼ i¼1
j¼4
i¼1
j¼4
s:t: e AN bÞE2 ij
þ LC iðjþ1Þ 6 LC ij þ ð1 h i 9 P ~ ~ ð1 bÞE1ti þ bEt2i C ik 6 CQk
e AN bE1 ij
i¼1 9 P k P
C ij þ XC ij P sik
9 > XC ij C ij > > > > > > > > > > > > > > > > > > =
i¼1 j¼4
XC 5j P 1 12 P j¼4
XC 1j ¼ 0;
12 P j¼4
XC 2j ¼ 0;
12 P
XC 8j ¼ 0
j¼4
i ¼ 1; . . . ; 9; j ¼ 1; . . . ; 12; k ¼ 4; . . . ; 12
> > > > > > > > > > > > > > > > > > > ;
ð13Þ vðbÞ:
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489
Table 10 a-Cuts of each b-fuzzy ideal solution a
Fe 1
Fe 2
b = 0.6 0 0.2 0.4 0.6 0.8 1
[30854, 43544] [31930, 42148] [33001, 40631] [34131, 39160] [35160, 37657] [36244, 36244]
[80325, 113380] [82895, 109478] [85828, 105792] [88572, 101899] [91601, 98198] [94312, 94312]
b = 0.8 0 0.2 0.4 0.6 0.8 1
[29838, 42107] [30877, 40737] [31897, 39288] [32960, 37845] [33982, 36403] [35034, 35034]
[81341, 114816] [83945, 110901] [86932, 107135] [89732, 103211] [92780, 99452] [95522, 95522]
b=1 0 0.2 0.4 0.6 0.8 1
[28775, 40617] [29781, 39330] [30776, 37901] [31817, 36535] [32785, 35109] [33811, 33811]
[82403, 116316] [85041, 112294] [88053, 108532] [90874, 104521] [93978, 100746] [96746, 96746]
Due to the level of conflict between the considered objectives no solution generated by the single optimization of any criterion seems acceptable in practice. For instance, maximization of internal activity implies very higher levels of external activity to be done at the beginning of the plannified year, in order to maintain waiting lists into the desired levels (see Table 11). Once fuzzy ideal solution is obtained the b-compromise fuzzy solutions L1 and L1 are determined for each feasibility degree b fixed by the DM. Table 11 External activity for Fe 1 with possibility degree a = 1 and b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
0 0 74 39 338 183 93 0 122
0 0 26 15 1 0 0 0 0
0 0 23 12 1 0 0 0 0
0 0 3 1 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
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490
Let us consider r = 1, 2 objectives. The following notation will be used: n o ~z ðbÞ ~z ðbÞ Dr ¼ max E~1cr x E1r ; E2r E~2cr x ;
DT ¼ max Dr ;
h 9 X 12 iX e e ~ ~ EIð~c1 xÞ ¼ EIð Fe 1 Þ ¼ E1F 1 ; E2F 1 ¼ E1ti ; Et2i C ij ; i¼1
e e EIð~c2 xÞ ¼ EIðF~ 2 Þ ¼ E1F 2 ; E2F 2
j¼4
h
9 X 12 iX ~ ~ XC ij ; ¼ Et1i ; E2ti
eF r eF r e EIð~zr Þ ¼ EIð F r Þ ¼ E1 ; E2 :
i¼1
j¼4
In Table 12 we show DM the expected intervals of the fuzzy compromise solutions for the fixed feasibility levels. Once the DM has selected the risk-level he/she wants to tolerate, he/she will be able to choose the most suitable solution. We can observe that when the risk-level increases the obtained compromise solution for this risk-level worsens due to the fact that the feasible set is smaller. Problem (A.1) (see Appendix A) provides the solution of maximum weighted aggregate achievement for both objectives: internal and external activity. However this solution can extremely biased towards the achievement of one of the objectives. For contrary, problem (A.2) provides the most balanced solution between the achievement of both objectives. Therefore it seems interesting to present the two solutions to the DM for assessments since they can offer very different surgical scheduling planning. For each risk-level b we offer DM two choices either the maximum efficient solution (compromise solution L1) or the maximum balanced solution (compromise solution L1). In both cases the total ordinary activity will be almost the same because of
Table 12 b-compromise fuzzy solutions Fe 1
Fe 2
EIð Fe 1 Þ
EIð Fe 2 Þ
b = 0.6 L1 (30837, 36214, 43542) L1 (30840, 36230, 43540)
(80341, 94342, 113381) (80339, 94326, 113384)
[33526, 39878] [33535, 39885]
[87342, 103862] [87333, 103855]
b = 0.8 (29784, 34976, 42053) L1 L1 (29820, 35018, 42107)
(81395, 95580, 114870) (81358, 95539, 114816)
[32380, 38515] [32419, 38563]
[88488, 105225] [88449, 105178]
b=1 L1 L1
(82407, 96750, 116321) (82403, 96746, 116319)
[31289, 37204] [31293, 37207]
[89579, 106536] [89575, 106533]
(28771, 33806, 40602) (28775, 33810, 40604)
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491
Table 13 Compromise solution L1: internal activity for b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
5 2 0 0 0 0 8 16 0
3 1 0 0 0 0 0 28 0
13 5 0 0 0 0 3 13 0
0 0 0 0 0 0 0 23 0
7 11 0 0 0 0 0 0 0
0 0 0 0 0 0 0 17 13
12 6 0 0 0 0 18 8 0
0 1 0 0 0 0 22 22 0
0 2 3 0 0 0 16 18 0
Table 14 Compromise solution L1: internal activity for b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
5 2 0 0 0 0 2 19 0
16 12 0 0 0 0 3 0 0
0 0 0 0 0 0 17 27 0
0 0 0 0 0 0 0 23 0
2 11 0 0 0 0 16 11 0
5 9 0 0 0 0 0 5 0
12 0 0 0 0 0 12 17 0
0 4 0 0 0 0 18 21 0
0 2 0 0 0 0 22 18 0
25 number of patients
241
20
278 454
15
455 550
10
553 565
5
574 685
0
685
May Jun
Jul
Aug Sep
Oct
Nov Dec Jan
Fig. 2. Evolution of the waiting lists. 0.8-compromise solution L1.
the maximum performance capability of the hospital, but there are differences in the scheduling (see Tables 13 and 14). Finally, we can complete the information provided to the DM with the expected monthly evolution of the waiting lists (see Figs. 2 and 3). The information provided by the results of the set out problem can be very useful for the DM in order to determine whether is possible or not to reach the
492
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number of patients
25 20 15 10 5 0 May Jun
Jul
Aug Sep Oct Nov Dec Jan
241 278 454 455 550 553 565 574 685 685
Fig. 3. Evolution of the waiting lists. 0.8-compromise solution L1.
objectives regarding waiting lists fixed by the Health Authorities at the beginning of the year, with the given conditions of the hospital. In this sense, the DM will have quantitative data to sing the conditions established by Health Authorities in the management contract.
5. Conclusions In this work we have planified in an optimal way the real performance of a surgical service at a medium sized public hospital, in order to give DM quantitative data for the analysing of the internal coherency of the goals expressed by the Spanish Health Service in relation to the maximum stay on a waiting list. The imprecise nature of the modelÕs data led us to consider them as fuzzy numbers. Estimations about time spent on the surgical processes and about incoming and outcoming patients were made by the DM in terms of fuzzy logic due to the impossibility of obtaining time series for these data. For these reasons we have solved the problem through a Possibilistic Multiobjective Linear Programming approach. Applying Compromise Programming to the resolution of the model we offer the DM a set of compromise solutions which depend on the feasible-level he/she wants to support. These solutions imply maximum efficiency on the one hand (solution L1), and on the other maximum equilibrium between objectives (solution L1), so depending on DM preferences he/she can choose between them. It is also important to remark that the proposed model allows not only to determine best-compromise surgical scheduling but also allows to understand the underlying conflict and interaction between the different criteria involved in the DM process. For all this reasons the analytical framework introduced in this paper can be considered a sound basis for establishing decision support systems for determining surgical plans in order to reduce waiting lists, within a sustainable multiple criteria perspective.
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493
Appendix A Determination of b-compromise solution L1 Min s:t:
D1 þ D2 þ d1 þ d2 9 X 12 X i¼1
eF ~ Et1i C ij E1 1 D1 6 Md1 ;
j¼4
9 X 12 X eF ~ E1 1 Et1i C ij D1 6 Mð1 d1 Þ;
i¼1 9 X
12 X
i¼1
j¼4
j¼4
eF ~ Et1i XC ij E1 2 D2 6 Md2 ;
9 X 12 X eF ~ E1 2 Et1i XC ij D2 6 Mð1 d2 Þ;
i¼1
j¼4
9 X 12 X eF ~ E2 1 Et2i C ij D1 6 Md1 ;
i¼1 9 X 12 X i¼1
eF ~ Et2i C ij E2 1 D1 6 Mð1 d1 Þ;
j¼4
eF 2
E2
9 X 12 X i¼1
9 X 12 X i¼1
j¼4
ðA:1Þ ~ Et2i XC ij
D2 6 Md2 ;
j¼4
eF ~ Et2i XC ij E2 2 D2 6 Mð1 d2 Þ;
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij 6 E2 1 E1 1 þ Md1 ; i¼1
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij þ Mð1 d1 Þ P E2 1 E1 1 ; i¼1
h ~ ~ E2ti Et1i
9 iX i¼1
j¼4 12 X
e eF F XC ij 6 E2 2 E1 2 þ Md2 ;
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i XC ij þ Mð1 d2 Þ P E2 2 E1 2 ; i¼1
dp 2 f0; 1g;
j¼4
p ¼ 1; 2;
and
vðbÞ:
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494
Determination of b-compromise solution L1 Min s:t:
DT þ d1 þ d2 9 X 12 X i¼1
eF ~ Et1i C ij E1 1 DT 6 Md1 ;
j¼4
9 X 12 X eF ~ E1 1 Et1i C ij DT 6 Mð1 d1 Þ;
i¼1 9 X 12 X i¼1
j¼4
eF ~ Et1i XC ij E1 2 DT 6 Md2 ;
j¼4
9 X 12 X eF ~ E1 2 Et1i XC ij DT 6 Mð1 d2 Þ;
i¼1
j¼4
9 X 12 X eF ~ E2 1 Et2i C ij DT 6 Md1 ;
i¼1 9 X
12 X
i¼1
j¼4
eF 2
E2
eF ~ Et2i C ij E2 1 DT 6 Mð1 d1 Þ;
9 X 12 X i¼1
9 X 12 X i¼1
j¼4
ðA:2Þ ~ Et2i XC ij
DT 6 Md2 ;
j¼4
eF ~ Et2i XC ij E2 2 DT 6 Mð1 d2 Þ;
j¼4
9 X 12 h iX eF 1 eF 1 ~ti ~ti C ij 6 E2 E1 þ Md1 ; E2 E1 i¼1
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij þ Mð1 d1 Þ P E2 1 E1 1 ; i¼1
j¼4
9 X 12 h iX e eF ~ ~ F E2ti Et1i XC ij 6 E2 2 E1 2 þ Md2 ; i¼1
j¼4
9 X 12 h iX eF 2 eF 2 ~ti ~ti E2 E1 XC ij þ Mð1 d2 Þ P E2 E1 ; i¼1
dp 2 f0; 1g
j¼4
p ¼ 1; 2 and
vðbÞ.
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References [1] M. Arenas, A. Bilbao, M. Jime´nez, M.V. Rodrı´guez, E. Cerda´, Management of surgical waiting lists in public hospitals, in: Y.Y. Haimes, J. Steuer (Eds.), Research and Practice in Multiple Criteria Decision Making, Springer-Verlag, 2000. [2] M. Arenas, A. Bilbao, R. Caballero, T. Go´mez, M.V. Rodrı´guez, F. Ruı´z, Analysis via goal programming of the minimum achievable stay in surgical waiting lists, Journal of the Operational Research Society 53 (2002) 387–396. [3] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [4] H. Tanaka, Fuzzy data analysis by possibilistic linear models, Fuzzy Sets and Systems 24 (1987) 363–375. [5] M. Jime´nez, Ranking fuzzy numbers through the comparison of its expected intervals, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 4 (4) (1996) 379–388. [6] P.L. Yu, A class of solutions for group decision problems, Management Science 19 (1973) 936– 946. [7] M. Zeleny, Compromise programming, in: J.L. Cochrane, M. Zeleny (Eds.), Multiple Criteria Decision Making, University of South Carolina Press, Columbia, 1973. [8] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems 47 (1992) 81–86. [9] M. Arenas, A. Bilbao, M. Jime´nez, M.V. Rodrı´guez, A Theory of possibilistic approach to the solution of a fuzzy linear programming, in: J. Giro´n (Ed.), Applied Decision Analysis, Kluwer Academic Publishers, 1998. [10] M. Arenas, A. Bilbao, B. Pe´rez, M.V. Rodrı´guez, Solving a multiobjective possibilistic problem through compromise programming, European Journal of Operational Research (avalaible online on 12th March 2004). [11] E. Blasco, E. Cuchillo-Iba´n˜ez, M.A. Moro´n, C. Romero, On the monotonicity of the compromise set in multicriteria problems, Journal of Optimization Theory and its Applications 102 (1) (1999) 69–82.
Management of surgical waiting lists through a Possibilistic Linear Multiobjective Programming problem Blanca Pe´rez Gladish *, Mar Arenas Parra, Amelia Bilbao Terol, Ma Victoria Rodrı´guez Urı´a Dpto. Economı´a Cuantitativa, Facultad de Ciencias Econo´micas y Empresariales de la Universidad de Oviedo, Avda. del Cristo s/n, Oviedo, Asturias (Espan˜a) C.P. 33006, Spain
Abstract This study attempts to apply a management science technique to improve the efficiency of Hospital Administration. We aim to design the performance of the surgical services at a Public Hospital that allows the Decision-Maker to plan surgical scheduling over one year in order to reduce waiting lists. Real decision problems usually involve several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory. Here we apply a Possibilistic Linear Multiobjective Programming method, developed by the authors, for solving a hospital management problem using a Fuzzy Compromise Programming approach. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy Sets; Compromise programming; Decision making; Surgical waiting lists
*
Corresponding author. E-mail addresses: [email protected] (B. Pe´rez Gladish), [email protected] (M. Arenas Parra), [email protected] (A. Bilbao Terol), [email protected] (Ma Victoria Rodrı´guez Urı´a). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.07.015
478
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1. Introduction Hospitals are considered public utility companies and are supposed to be non-profit institutions and they usually work in a political and highly complex environment. Managerial authority is shared between doctors and administrators and each of the two groups aims to formulate its own individual policies and objectives. As these policies and objectives may not coincide, and may even be in direct opposition to each other, giving rise to a situation that is certain to affect hospital performance and they must be taken into account when proposing any kind of result analysis. Once the quantitative target values for these objectives have been established, the main aim of any hospital administration is to improve the efficiency of hospital performance. Spanish public hospitals, as providers of health services, must draw up at the end of each year with financial authorities, a document called ‘‘Management Contract’’, which expresses the objectives to be reached over the following year and also aims at establishing quantitative target values for these objectives. One of the main objectives of the current Management Contracts is to reduce the waiting lists for surgical processes. The existence of surgical waiting lists is an important problem which infringes on citizenÕs health care rights. There are several causes for the existence of waiting lists but everyone agrees that the problem is essentially one of distribution and optimal design of resources. Thus, it seems necessary to apply modern management techniques to assure the efficient use of medical facilities and resources (see [1,2]). In this paper, a Possibilistic Multiobjective Linear Programming problem (FM-LP) is developed as an information system in order to analyze the internal coherency of the goals expressed by administrative authorities. Also, applying a Multicriteria Decision technique, Compromise Programming, we intend to design and manage, in an optimal way, the real performance of the surgical services of a medium-sized hospital of INSALUD (The Spanish Health Department), taking into account all the function constraints. This offers the decision centre a suitable methodology with which to analyze whether or not it is possible to improve the running of the services.
2. Description of the model In order to reach the previously mentioned objectives, hospitals are allowed to use several methods of operating scheduling: ordinary activity, within regular-operating hours, or extraordinary activity that would be developed overtime or through private hospital contracts. The model developed deals with
B. Pe´rez Gladish et al. / Appl. Math. Comput. 167 (2005) 477–495
479
all the services of the hospital. However, in order to keep the presentation simple, we focus on a specific service: General Surgery. 2.1. Program variables and data The variables considered are grouped into two main blocks, ordinary activity and extraordinary activity, and they are denoted according to the initials of the corresponding service. Each variable has two subscripts: the first one, i, denotes the specific process, while the second one denotes the month when the operation is carried out. An X precedes the extraordinary activity and an L precedes the state of the list (Tables 1 and 2). Planning the hold year in advance is possible but, usually, reality imposes its own rules and a permanent re-planning process is required. In this sense, it is necessary to carry out a monthly update, once the activity reports and new admissions and exclusions are known. For this reason, the model has been designed so that the starting point, or initial value of the state variables, as well as the corresponding parameters, can be related to this update at a specific moment of the process. In this paper the update was carried out at the end of the first quarter of the year, so the index j which corresponds to the months that follow the current one, will range from 4 to 12. We consider nine processes which represent about 45% of the total activity of the service under consideration (Table 3). The model includes data corresponding on the time spent on the operations and estimations on incoming and outgoing patients. Time spent on each process are uncertain data. The Decision Maker (DM) considers that the information provided by the average duration time obtained from the operating theatre reports of the preceding year, is not enough because the time employed for different cases of the same process are very different. For these reasons they are considered as imprecise or fuzzy data and they are
Table 1 Program decision variables Modality
Service
Process
Month
–/X
C
i
j
Table 2 Program state variables List
Service
Process
Month
L
C
i
j
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480
Table 3 Service: General Surgical Code
Process
Variable
241 278 454 455 550 553 565 574 685
Nodular goiter Morbid Obesity Varicose Veins Haemorrhoids Inguinal Hernias Incisional Hernias Anal Fissure/Anal Fistula Cholelithiasis Pilonidal Cyst
C01 C02 C03 C04 C05 C06 C07 C08 C09
µti 1
0
ti t i
BE
t i
RD
t i
WE
Fig. 1. Possibility distribution of the fuzzy duration data ~ti .
handled by triangular fuzzy numbers described by their possibility distributions estimated by the DM. 1 ~ti ¼ ðBEti ; RDti ; WEti Þ; where RDti is the average duration time, which has been determined using the mean data, plus 20 minutes, which is the time needed to prepare the theatre for the next operation; BEti and WEti are the best and the worst estimations given by the DM. In Fig. 1, l~ti , with 0 6 l~ti 6 1, represents the possibility degree of occurrence of parameter ~ti which represents the time spent on each process. Thus, the DM considers that the value with a higher possibility degree of occurrence, the central value of the fuzzy triangular number, is the real data RDti , which has l~ti ¼ 1. As well, we can observe that the DM believes best and worse estimations, WEti , BEti , have no possibility of occurring, l~ti ¼ 0, 1
Information about possibility distribution of fuzzy numbers can be found in [3].
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481
Table 4 Duration processes estimations Code
Process
Variable
BEti
RDti
WEti
241 278 454 455 550 553 565 574 685
Nodular goiter Morbid Obesity Varicose Veins Haemorrhoids Inguinal Hernias Incisional Hernias Anal Fissure/Anal Fistula Cholelithiasis Pilonidal Cyst
C01 C02 C03 C04 C05 C06 C07 C08 C09
128 132 115 63 97 114 53 115 54
151 155 135 74 114 134 62 135 63
182 186 162 89 137 161 75 162 76
representing lateral data in the fuzzy triangular number. Remaining values for the time spent on processes have intermediate possibility degrees of occurrence (Table 4). The number of monthly incoming/outgoing patients without an operation (admissions and exclusions respectively) has been estimated by the DM using e ij ) and excludata from the two preceding years. The number of admissions ( A e sions ð S ij Þ will also be considered as fuzzy triangular numbers, given their unknown character and will be expressed as follows: e ij ¼ ðBEA ; WRA ; WEA Þ; Admissions A ij ij ij Exclusions e S ij ¼ ðWESij ; WRSij ; BESij Þ; where WRAij and WRSij represent the worst historic real data for admissions and exclusions respectively; WEAij and WESij are the pessimistic estimations of the DM for the current year and BEAij and BESij are the optimistic ones. In Tables 5 and 6 we show the estimations for the current year of incoming and outgoing patients (admissions and exclusions, respectively), considered as imprecise data represented by fuzzy triangular numbers. 2.2. Constraints Four groups of constraints have been considered: 2.2.1. Evolution of waiting lists Evolution of the waiting lists is reflected by f iðjþ1Þ LC f ij þ AN f ij XC ij C ij ; LC
ð1Þ
f ij denotes the state of the waiting list for process i at the beginning of where LC f ij ¼ A e ij e e ij is the estimated number of admissions for month j, AN S ij where A process i during month j, and e S ij represents the estimated number of exclusions
482
Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
(7, 8, 10) (3, 3, 4) (22, 24, 31) (15, 17, 22) (49, 54, 70) (23, 26, 34) (16, 18, 23) (21, 23, 30) (24, 27, 35)
(5, 6, 8) (4, 4, 5) (23, 25, 33) (14, 15, 20) (54, 60, 78) (19, 21, 27) (14, 15, 20) (27, 30, 39) (29, 32, 42)
(4, 4, 5) (2, 2, 3) (20, 22, 29) (10, 11, 14) (51, 57, 74) (23, 25, 33) (14, 16, 21) (22, 24, 31) (23, 25, 33)
(5, 5, 7) (1, 1, 1) (15, 17, 22) (11, 12, 16) (42, 47, 61) (23, 25, 33) (11, 12, 16) (15, 17, 22) (17, 19, 25)
(5, 5, 7) (2, 2, 3) (11, 12, 16) (7, 8, 10) (27, 30, 39) (14, 15, 20) (9, 10, 13) (13, 14, 18) (13, 14, 18)
(14, 15, 20) (5, 5, 7) (19, 21, 27) (7, 8, 10) (46, 51, 66) (26, 29, 38) (8, 9, 12) (19, 21, 27) (17, 19, 25)
(7, 8 10) (4, 4, 5) (22, 24, 31) (12, 13, 17) (56, 62, 81) (31, 34, 44) (19, 21, 27) (23, 25, 33) (35, 39, 51)
(5 5 7) (2, 2, 3) (26, 29, 38) (16, 18, 23) (65, 72, 94) (32, 35, 46) (19, 21, 27) (21, 23, 30) (43, 48, 62)
(8, 9, 12) (5, 5, 7) (14, 15, 20) (13, 14, 18) (37, 41, 53) (18, 20, 26) (14, 16, 21) (18, 20, 26) (30, 33, 43)
B. Pe´rez Gladish et al. / Appl. Math. Comput. 167 (2005) 477–495
Table 5 Incoming estimations (admissions) for General Surgery
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483
Table 6 Outgoing estimations (exclusions) for General Surgery Code
Apr
241 278 454 455 550 553 565 574 685
(0, (1, (2, (0, (4, (1, (0, (1, (5,
0, 1, 2, 0, 4, 1, 0, 1, 5,
May 0) 1) 3) 0) 5) 1) 0) 1) 7)
(0, (0, (0, (1, (1, (0, (1, (0, (1,
0, 0, 0, 1, 1, 0, 1, 0, 1,
Jun 0) 0) 0) 1) 1) 0) 1) 0) 1)
(1, (0, (0, (0, (0, (1, (0, (5, (1,
Jul
1, 0, 0, 0, 0, 1, 0, 6, 1,
1) 0) 0) 0) 0) 1) 0) 8) 1)
(0, (0, (0, (0, (1, (0, (0, (1, (0,
Aug 0, 0, 0, 0, 1, 0, 0, 1, 0,
0) 0) 0) 0) 1) 0) 0) 1) 0)
(0, (0, (0, (0, (0, (2, (0, (0, (1,
0, 0, 0, 0, 0, 2, 0, 0, 1,
Sep 0) 0) 0) 0) 0) 3) 0) 0) 1)
(0, (0, (1, (1, (5, (5, (0, (4, (2,
0, 0, 1, 1, 5, 5, 0, 4, 2,
Oct 0) 0) 1) 1) 7) 7) 0) 5) 3)
(0, (0, (1, (0, (4, (3, (0, (2, (2,
0, 0, 1, 0, 4, 3, 0, 2, 2,
Nov 0) 0) 1) 0) 5) 4) 0) 3) 3)
(0, (0, (0, (2, (5, (1, (0, (2, (2,
Dec
0, 0, 0, 2, 5, 1, 0, 2, 2,
0) 0) 0) 3) 7) 1) 0) 3) 3)
(0, (0, (0, (1, (0, (1, (1, (0, (1,
0, 0, 0, 1, 0, 1, 1, 0, 1,
0) 0) 0) 1) 0) 1) 1) 0) 1)
without surgical operation for process i, for each month j; XCij, Cij are respectively the obtained extraordinary and ordinary optimal surgical activity. The state of the waiting lists at the beginning of the scheduling period is known, as it is the residual list of the the previous scheduled period (Table 7). 2.2.2. Monthly operating theatre availability The available operating theatre times are computed considering that the maximum daily activity is six and a half hours. Moreover, it is assumed that the operating theatre sessions are assigned in advance to each service through internal agreements, and cannot be changed at this stage of the process. Thus, the constraint corresponding to month j takes the following form: 9 X
~ti C ij 6 CQj ;
ð2Þ
i¼1
where the right-hand side CQj represents the total operating theatre time available for the General Surgery service in month j, and can be found in Table 8. 2.2.3. Upper bounds on the stay on a waiting list The main priority with respect to the surgical activity for the planning year is that no patient should stay on the waiting list for more than 6 months. With this aim, it must be assured that the sum of the ordinary and extraordinary operations performed between April and the month k be greater than the number of patients that would be 6 months or longer on the waiting list for each process i in month k: Table 7 Initial state of the waiting list
Number of patients
241
278
454
455
550
553
565
574
685
17
13
51
21
118
58
28
59
49
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484
Table 8 Total monthly available theatre time for General Surgery
Minutes
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
3955
4704
4976
3316
2997
2997
5183
4815
4385
9 X k X ðC ij þ XC ij Þ P sik : i¼1
ð3Þ
j¼4
Above equations set lower bounds on the global activity level of each process, thereby establishing the maximum stay requirements. These lower bounds must be fixed for each process and month. We obtain the minimum bounds, sik, from the state of the waiting list at the beginning of the scheduled year (Table 8), accumulating these values month by month. Parameters sik by process and month are displayed in Table 9. 2.2.4. Bounds on extraordinary operations DM know at the beginning of the year that it is not going to be possible to carry out in the hospital all the surgical activity, so they concert with Health Authorities for some processes to be transferred to other centres at least in a certain number. In the General Surgery service, these constraints correspond to Inguinal Hernias, which is the surgical process with the highest waiting list: XC 5j P 1;
j ¼ 4; . . . ; 12:
ð4Þ
Also, there are several surgical processes that are preferred not to be transferred to other centres due to their surgical complexity. This is the case of Nodular goiter, Morbid Obesity and Cholelithiasis:
Table 9 Parameters sik corresponding to General Surgery Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
25 8 29 17 78 55 19 41 25
28 10 52 33 139 84 35 61 48
36 15 67 46 179 104 46 80 69
42 17 86 66 214 124 60 89 114
57 17 105 88 281 164 83 110 162
63 17 151 97 325 203 94 119 224
67 20 175 112 358 243 110 129 249
75 24 186 118 417 261 126 157 261
75 26 223 133 505 301 150 175 281
B. Pe´rez Gladish et al. / Appl. Math. Comput. 167 (2005) 477–495 12 X
XC 1j ¼ 0;
j¼4
12 X
XC 2j ¼ 0;
j¼4
12 X
XC 8j ¼ 0:
485
ð5Þ
j¼4
No integrity conditions are imposed in the model, given that the estimated times per process are average times, and, thus, only approximate planning is required. 2.3. Objectives and formulation of the problem To determine the number of processes which it is necessary to carry out for each month two criteria have been considered: performance capability of the hospital and extraordinary activity, which means operations in the hospital taking extra time, or referrals to other centres. With regard to the first criteria the objective is to maximise hospital performance capability, while the objective with regard to the second criteria is to fix the minimum extraordinary activity that allows us to reduce the residual waiting list. The first objective function carries out the maximum total employment of operational theatres with ordinary activity per month and process throughout the year: Fe 1 ¼ Max
9 X 12 X i¼1
~ti C ij :
ð6Þ
j¼4
The second objective represents the minimization of the total minutes of extraordinary activity we have to carry out if we want to maintain waiting lists under 6 months: Fe 2 ¼ Min
9 X 12 X i¼1
~ti XC ij :
ð7Þ
j¼4
3. Methodology The above problem describes a real decision situation. It is a Multiobjective Possibilistic Linear Programming problem (FP-MOLP) in which all the parameters are fuzzy and they are represented by fuzzy numbers described by their possibility distribution estimated by the analyst from the information supplied by the DM [4]. The uncertain and/or imprecise nature of the problemÕs parameters involves two main problems: feasibility and optimality. Feasibility may be handled by comparing fuzzy numbers. In this paper we use a fuzzy relationship to compare
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fuzzy numbers [5] that verifies suitable properties and that, besides, is computationally efficient to solve linear problems because it preserves its linearity. Optimality is handled through Compromise Programming (CP). CP is a well-known Multiple Criteria Decision Making approach developed by Yu [6] and Zeleny [7]. The basic idea in CP is the identification of an ideal solution as a point where each attribute under consideration achieves its optimum value. Zeleny states that alternatives that are closer to the ideal are preferred to those that are farther from it because being as close as possible to the perceived ideal is the rationale of human choice. We shall consider the following Multiobjective Possibilistic Programming problem: min ~z ¼ ð~z1 ; ~z2 ; . . . ; ~zk Þ ¼ ð~c1 x; ~c2 x; . . . ; ~ck xÞ FP-MOLP ( ) ~ bi ; i ¼ 1; . . . ; m ai x 6 ~ ~ e ; s:t: x 2 vð A; bÞ ¼ xP0 where xt = (x1, x2, . . . , xn) is the crisp decision vector, ~ct ¼ ð~c1 ; ~c2 ; . . . ; ~ck Þ are the e ¼ ½~aij fuzzy parameters of the k considered objectives, A mn is the fuzzy techt ~ ~ ~ ~ nological matrix and b ¼ ðb1 ; b2 ; . . . ; bm Þ are also fuzzy parameters. The uncertain and/or imprecise nature of the technological matrix and of the resource vector which defines the set of constraints of the model leads us to compare fuzzy numbers. In this work we compare fuzzy numbers handle them through their expected intervals as defined by Heilpern [8] and we use the fuzzy relationship defined by [5] which leads us to the concept of b-feasibility of a decision vector. A decision vector x 2 Rn , is said to be b-feasible for the problem FP-MOLP if x verifies the constraints at least in a degree b. That is: ~ ai x6b ~ bi ;
i ¼ 1; . . . ; m:
ð8Þ
In accordance with the above considerations we shall solve the FP-MOLP through a family of b-FP-MOLP problems, where 0 6 b 6 1: min ~z ¼ ð~z1 ; ~z2 ; . . . ; ~zk Þ ¼ ð~c1 x; ~c2 x; . . . ; ~ck xÞ s:t:
b-FP-MOLP
9 ~ ~ ð1 bÞE~a1i þ bE~a2i x 6 bEb1i þ ð1 bÞEb2i ; > = ¼ vðbÞ: i ¼ 1; . . . ; m > ; xP0
~i ¼ ð~ai1 ; ~ai2 ; . . . ; ~ain Þ is a vector The expected interval of the fuzzy vector a whose components are the expected intervals of each fuzzy number of vector ~ ai , that is: ai1 Þ; EIð~ ai2 Þ; . . . ; EIð~ ain ÞÞ: EIð~ ai Þ ¼ ðEIð~
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In order to apply the CP approach to solve the problem, we need to obtain the fuzzy ideal solution of the b-FP-MOLP problem. For this we use the solving method proposed by Arenas et al. [9]. In the CP framework, once the fuzzy ideal solution is found values of the decision variables, which determine a fuzzy solution, as accurately as possible to the fuzzy ideal solution. 2 We will solve the problem FP-MOLP by handling the fuzzy objectives, ~z ¼ ~cx, and the b-fuzzy ideal solution, ~z ðbÞ, through their expected intervals. 3 Therefore, the problem now is: ~ EIð~zr ðbÞÞ; Find a x 2 vðbÞ such that : EIð~cr xÞ ! EIð~zr ðbÞÞ
½E~1cr x ; E~2cr x
r ¼ 1; . . . ; k:
ð9Þ
~z ðbÞ ~z ðbÞ ½E1r ; E2r .
where EIð~cr xÞ ¼ and ¼ It should be considered desirable to obtain a fuzzy objective vector with less amplitude than the b-fuzzy ideal solution, i.e., such that it should verify the following condition: ~z ðbÞ
E~2cr x E~1cr x 6 E2r
~z ðbÞ
E1r
ð10Þ
:
From here, we assert that ~ EIð~zr ðbÞÞ; EIð~cr xÞ !
r ¼ 1; . . . ; k if and only if Dr ! 0;
where
n o ~z ðbÞ ~z ðbÞ Dr ¼ max E~1cr x E1r ; E2r E~2cr x ;
r ¼ 1; . . . ; k;
is the discrepancy between the rth fuzzy objective ~cr x and the r-th componet of the b-fuzzy ideal solution ~zr ðbÞ [10]. We shall solve now a new crisp CP problem where the objective is to minimize the discrepancy between the b-fuzzy ideal solution and fuzzy objectives. Therefore, the ideal solution is the null vector and we define a b-compromise solution of the FP-MOLP as a decision vector x* that is a compromise solution to the problem: min
ðD1 ; D2 ; . . . ; Dk Þ
s:t:
x 2 vðbÞ:
ð11Þ
As problem (11) is crisp, the compromise programming approach to solve it is based on the Lp family of distances: !1p k X p p w rD r Min Lp ¼ Min ð12Þ r¼1 s:t:
2 3
x 2 vðbÞ;
We shall denote the relation of ‘‘accurate’’ as ð!Þ. ~ More details about these results can be found in [10].
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where wr P 0. wr can be regarded as a normalizing coefficient and also as a weight that measures the relative importance of the discrepancy between the rth fuzzy objective and its fuzzy ideal value and may be established by the DM in an interactive process with the analyst. The most commonly obtained compromise solutions are for metrics p = 1 and p = 1 because for other metrics non-linear mathematical programming algorithms are needed. Also, under certain conditions (see [11]) they are the bounds of the whole compromise set.
4. Resolution of the problem We have solved the b-parametric Multiobjective Programming (13), with which we determine the b-efficient solutions, in an interactive way with the DM. In absence of preferential subjective weights for both objectives are assigned equal values (i.e., W1 = W2 = 1). As a first step we ask the DM to fix the feasibility degree he/she wants to assume. Once the DM has fixed the initial feasibility degree, the b-fuzzy ideal solution is obtained through its a-cuts. Table 10 contains the possibility distri butions of the ideal point ð Fe 1 ; Fe 2 Þ consisting of the optimal value of each of the considered objectives, maximum internal capability and minimum external activity, for each feasibility degree b fixed by the DM. As we can see, as DM establishes higher feasibility degrees optimal solution worsens, that is we have lower levels of internal activity and higher of external surgical activity. ! 9 X 12 9 X 12 X X ~ti C ij ; ~ti XC ij Min ð Fe 1 ; Fe 2 Þ ¼ i¼1
j¼4
i¼1
j¼4
s:t: e AN bÞE2 ij
þ LC iðjþ1Þ 6 LC ij þ ð1 h i 9 P ~ ~ ð1 bÞE1ti þ bEt2i C ik 6 CQk
e AN bE1 ij
i¼1 9 P k P
C ij þ XC ij P sik
9 > XC ij C ij > > > > > > > > > > > > > > > > > > =
i¼1 j¼4
XC 5j P 1 12 P j¼4
XC 1j ¼ 0;
12 P j¼4
XC 2j ¼ 0;
12 P
XC 8j ¼ 0
j¼4
i ¼ 1; . . . ; 9; j ¼ 1; . . . ; 12; k ¼ 4; . . . ; 12
> > > > > > > > > > > > > > > > > > > ;
ð13Þ vðbÞ:
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Table 10 a-Cuts of each b-fuzzy ideal solution a
Fe 1
Fe 2
b = 0.6 0 0.2 0.4 0.6 0.8 1
[30854, 43544] [31930, 42148] [33001, 40631] [34131, 39160] [35160, 37657] [36244, 36244]
[80325, 113380] [82895, 109478] [85828, 105792] [88572, 101899] [91601, 98198] [94312, 94312]
b = 0.8 0 0.2 0.4 0.6 0.8 1
[29838, 42107] [30877, 40737] [31897, 39288] [32960, 37845] [33982, 36403] [35034, 35034]
[81341, 114816] [83945, 110901] [86932, 107135] [89732, 103211] [92780, 99452] [95522, 95522]
b=1 0 0.2 0.4 0.6 0.8 1
[28775, 40617] [29781, 39330] [30776, 37901] [31817, 36535] [32785, 35109] [33811, 33811]
[82403, 116316] [85041, 112294] [88053, 108532] [90874, 104521] [93978, 100746] [96746, 96746]
Due to the level of conflict between the considered objectives no solution generated by the single optimization of any criterion seems acceptable in practice. For instance, maximization of internal activity implies very higher levels of external activity to be done at the beginning of the plannified year, in order to maintain waiting lists into the desired levels (see Table 11). Once fuzzy ideal solution is obtained the b-compromise fuzzy solutions L1 and L1 are determined for each feasibility degree b fixed by the DM. Table 11 External activity for Fe 1 with possibility degree a = 1 and b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
0 0 74 39 338 183 93 0 122
0 0 26 15 1 0 0 0 0
0 0 23 12 1 0 0 0 0
0 0 3 1 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
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490
Let us consider r = 1, 2 objectives. The following notation will be used: n o ~z ðbÞ ~z ðbÞ Dr ¼ max E~1cr x E1r ; E2r E~2cr x ;
DT ¼ max Dr ;
h 9 X 12 iX e e ~ ~ EIð~c1 xÞ ¼ EIð Fe 1 Þ ¼ E1F 1 ; E2F 1 ¼ E1ti ; Et2i C ij ; i¼1
e e EIð~c2 xÞ ¼ EIðF~ 2 Þ ¼ E1F 2 ; E2F 2
j¼4
h
9 X 12 iX ~ ~ XC ij ; ¼ Et1i ; E2ti
eF r eF r e EIð~zr Þ ¼ EIð F r Þ ¼ E1 ; E2 :
i¼1
j¼4
In Table 12 we show DM the expected intervals of the fuzzy compromise solutions for the fixed feasibility levels. Once the DM has selected the risk-level he/she wants to tolerate, he/she will be able to choose the most suitable solution. We can observe that when the risk-level increases the obtained compromise solution for this risk-level worsens due to the fact that the feasible set is smaller. Problem (A.1) (see Appendix A) provides the solution of maximum weighted aggregate achievement for both objectives: internal and external activity. However this solution can extremely biased towards the achievement of one of the objectives. For contrary, problem (A.2) provides the most balanced solution between the achievement of both objectives. Therefore it seems interesting to present the two solutions to the DM for assessments since they can offer very different surgical scheduling planning. For each risk-level b we offer DM two choices either the maximum efficient solution (compromise solution L1) or the maximum balanced solution (compromise solution L1). In both cases the total ordinary activity will be almost the same because of
Table 12 b-compromise fuzzy solutions Fe 1
Fe 2
EIð Fe 1 Þ
EIð Fe 2 Þ
b = 0.6 L1 (30837, 36214, 43542) L1 (30840, 36230, 43540)
(80341, 94342, 113381) (80339, 94326, 113384)
[33526, 39878] [33535, 39885]
[87342, 103862] [87333, 103855]
b = 0.8 (29784, 34976, 42053) L1 L1 (29820, 35018, 42107)
(81395, 95580, 114870) (81358, 95539, 114816)
[32380, 38515] [32419, 38563]
[88488, 105225] [88449, 105178]
b=1 L1 L1
(82407, 96750, 116321) (82403, 96746, 116319)
[31289, 37204] [31293, 37207]
[89579, 106536] [89575, 106533]
(28771, 33806, 40602) (28775, 33810, 40604)
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Table 13 Compromise solution L1: internal activity for b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
5 2 0 0 0 0 8 16 0
3 1 0 0 0 0 0 28 0
13 5 0 0 0 0 3 13 0
0 0 0 0 0 0 0 23 0
7 11 0 0 0 0 0 0 0
0 0 0 0 0 0 0 17 13
12 6 0 0 0 0 18 8 0
0 1 0 0 0 0 22 22 0
0 2 3 0 0 0 16 18 0
Table 14 Compromise solution L1: internal activity for b = 0.8 Code
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
241 278 454 455 550 553 565 574 685
5 2 0 0 0 0 2 19 0
16 12 0 0 0 0 3 0 0
0 0 0 0 0 0 17 27 0
0 0 0 0 0 0 0 23 0
2 11 0 0 0 0 16 11 0
5 9 0 0 0 0 0 5 0
12 0 0 0 0 0 12 17 0
0 4 0 0 0 0 18 21 0
0 2 0 0 0 0 22 18 0
25 number of patients
241
20
278 454
15
455 550
10
553 565
5
574 685
0
685
May Jun
Jul
Aug Sep
Oct
Nov Dec Jan
Fig. 2. Evolution of the waiting lists. 0.8-compromise solution L1.
the maximum performance capability of the hospital, but there are differences in the scheduling (see Tables 13 and 14). Finally, we can complete the information provided to the DM with the expected monthly evolution of the waiting lists (see Figs. 2 and 3). The information provided by the results of the set out problem can be very useful for the DM in order to determine whether is possible or not to reach the
492
B. Pe´rez Gladish et al. / Appl. Math. Comput. 167 (2005) 477–495
number of patients
25 20 15 10 5 0 May Jun
Jul
Aug Sep Oct Nov Dec Jan
241 278 454 455 550 553 565 574 685 685
Fig. 3. Evolution of the waiting lists. 0.8-compromise solution L1.
objectives regarding waiting lists fixed by the Health Authorities at the beginning of the year, with the given conditions of the hospital. In this sense, the DM will have quantitative data to sing the conditions established by Health Authorities in the management contract.
5. Conclusions In this work we have planified in an optimal way the real performance of a surgical service at a medium sized public hospital, in order to give DM quantitative data for the analysing of the internal coherency of the goals expressed by the Spanish Health Service in relation to the maximum stay on a waiting list. The imprecise nature of the modelÕs data led us to consider them as fuzzy numbers. Estimations about time spent on the surgical processes and about incoming and outcoming patients were made by the DM in terms of fuzzy logic due to the impossibility of obtaining time series for these data. For these reasons we have solved the problem through a Possibilistic Multiobjective Linear Programming approach. Applying Compromise Programming to the resolution of the model we offer the DM a set of compromise solutions which depend on the feasible-level he/she wants to support. These solutions imply maximum efficiency on the one hand (solution L1), and on the other maximum equilibrium between objectives (solution L1), so depending on DM preferences he/she can choose between them. It is also important to remark that the proposed model allows not only to determine best-compromise surgical scheduling but also allows to understand the underlying conflict and interaction between the different criteria involved in the DM process. For all this reasons the analytical framework introduced in this paper can be considered a sound basis for establishing decision support systems for determining surgical plans in order to reduce waiting lists, within a sustainable multiple criteria perspective.
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Appendix A Determination of b-compromise solution L1 Min s:t:
D1 þ D2 þ d1 þ d2 9 X 12 X i¼1
eF ~ Et1i C ij E1 1 D1 6 Md1 ;
j¼4
9 X 12 X eF ~ E1 1 Et1i C ij D1 6 Mð1 d1 Þ;
i¼1 9 X
12 X
i¼1
j¼4
j¼4
eF ~ Et1i XC ij E1 2 D2 6 Md2 ;
9 X 12 X eF ~ E1 2 Et1i XC ij D2 6 Mð1 d2 Þ;
i¼1
j¼4
9 X 12 X eF ~ E2 1 Et2i C ij D1 6 Md1 ;
i¼1 9 X 12 X i¼1
eF ~ Et2i C ij E2 1 D1 6 Mð1 d1 Þ;
j¼4
eF 2
E2
9 X 12 X i¼1
9 X 12 X i¼1
j¼4
ðA:1Þ ~ Et2i XC ij
D2 6 Md2 ;
j¼4
eF ~ Et2i XC ij E2 2 D2 6 Mð1 d2 Þ;
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij 6 E2 1 E1 1 þ Md1 ; i¼1
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij þ Mð1 d1 Þ P E2 1 E1 1 ; i¼1
h ~ ~ E2ti Et1i
9 iX i¼1
j¼4 12 X
e eF F XC ij 6 E2 2 E1 2 þ Md2 ;
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i XC ij þ Mð1 d2 Þ P E2 2 E1 2 ; i¼1
dp 2 f0; 1g;
j¼4
p ¼ 1; 2;
and
vðbÞ:
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Determination of b-compromise solution L1 Min s:t:
DT þ d1 þ d2 9 X 12 X i¼1
eF ~ Et1i C ij E1 1 DT 6 Md1 ;
j¼4
9 X 12 X eF ~ E1 1 Et1i C ij DT 6 Mð1 d1 Þ;
i¼1 9 X 12 X i¼1
j¼4
eF ~ Et1i XC ij E1 2 DT 6 Md2 ;
j¼4
9 X 12 X eF ~ E1 2 Et1i XC ij DT 6 Mð1 d2 Þ;
i¼1
j¼4
9 X 12 X eF ~ E2 1 Et2i C ij DT 6 Md1 ;
i¼1 9 X
12 X
i¼1
j¼4
eF 2
E2
eF ~ Et2i C ij E2 1 DT 6 Mð1 d1 Þ;
9 X 12 X i¼1
9 X 12 X i¼1
j¼4
ðA:2Þ ~ Et2i XC ij
DT 6 Md2 ;
j¼4
eF ~ Et2i XC ij E2 2 DT 6 Mð1 d2 Þ;
j¼4
9 X 12 h iX eF 1 eF 1 ~ti ~ti C ij 6 E2 E1 þ Md1 ; E2 E1 i¼1
j¼4
9 X 12 h iX eF eF ~ ~ E2ti Et1i C ij þ Mð1 d1 Þ P E2 1 E1 1 ; i¼1
j¼4
9 X 12 h iX e eF ~ ~ F E2ti Et1i XC ij 6 E2 2 E1 2 þ Md2 ; i¼1
j¼4
9 X 12 h iX eF 2 eF 2 ~ti ~ti E2 E1 XC ij þ Mð1 d2 Þ P E2 E1 ; i¼1
dp 2 f0; 1g
j¼4
p ¼ 1; 2 and
vðbÞ.
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