Uncertain Models for Bed Allocation

Transcription

1 Global Journal of Health Scence Vol., No. ; October 00 Uncertan Models for Bed Allocaton Lng Gao (Correspondng author) College of Scence, Guln Unversty of Technology Box 733, Guln 54004, Chna Tel: E-mal: rng04@yahoo.com.cn Lang Ln College of Scence, Guln Unversty of Technology Guln 54004, Chna E-mal: lnlang6666@6.com Lanlong Gao College of Scence, Guln Unversty of Technology Guln 54004, Chna E-mal: gaolanlong@yahoo.com.cn The research s fnanced by Guangx Natonal Scence Foundaton of Chna (Gu No. 0836) Abstract The purpose of ths paper s to develop a methodology for modelng sckbed allocaton problems wth uncertan length of stay for each patent. Two uncertan bed allocaton models are presented. Hybrd ntellgent algorthm s employed for solvng these models. Fnally, numercal example s provded to demonstrate the feasblty of the proposed algorthm. Keywords: Uncertan varable, Sckbed, Allocaton, Uncertan programmng. Introducton Queue s frequently encountered n daly lfe. Queueng theory s the mathematcal study of watng lnes (or queues) and generally consdered a branch of operatons research because the results are often used when makng busness decsons about the resources needed to provde servce. It s applcable n a wde varety of stuatons that may be encountered n busness, commerce, ndustry, publc servce, engneerng and healthcare. In supermarkets and n banks queues form when there are nsuffcent server unts to meet the demand for servce. Smlarly, n hosptals, the queues form when there are nsuffcent beds avalable to admt ll people. In supermarkets and n banks customers can go elsewhere. But sck people have no alternatve optons: they ust have to wat. When the lack of sckbeds occurs patents wat to be admtted. The bed crss was consdered as a queung system and dscrete event smulaton was employed to evaluate the model numercally by C. Vaslaks and E. El-Darz. Other researches have exsted n healthcare servce. A methodology that uses system smulaton combned wth optmzaton to determne the optmal number of doctors, lab techncans and nurses requred to maxmze patent throughput and to reduce patent tme n the system subect to budget restrctons was presented by Mohamed and Talal. Real-lfe decsons are usually made n the state of uncertanty. By uncertan programmng we mean the optmzaton theory n uncertan envronments. In many cases, fuzzness and randomness smultaneously appear n a system. In order to descrbe these phenomena, a fuzzy random varable was ntroduced by Kwakernak as a random element takng fuzzy varable values. A random fuzzy varable was proposed by Lu as a fuzzy element takng random varable values. More generally, a hybrd varable was ntroduced by Lu as a measurable functon from a chance space to the set of real numbers. Fuzzy random varable and random fuzzy varable are nstances of hybrd varable. In order to 3 ISSN E-ISSN

2 Global Journal of Health Scence Vol., No. ; October 00 measure hybrd events, a concept of chance measure was ntroduced by L and Lu. After that, a general framework of hybrd programmng was proposed by Lu. Uncertanty theory was founded by Lu n 007 as a branch of mathematcs based on normalty, monotoncty, selfdualty, and countable subaddtvty axoms. Uncertan programmng was presented by Lu n 008 as the optmzaton theory n uncertan envronments. It has been appled n system relablty desgn, proect schedulng problem, vehcle routng problem, faclty locaton problem, and machne schedulng problem. In ths paper, we wll ntroduce two uncertan models for bed allocaton problem. The rest of the paper s organzed as follows: n Secton, we wll brefly revew the concepts of uncertanty. Then we descrbe the assumptons and notatons for uncertan bed allocaton models n Secton 3. After that, two uncertan models for bed allocaton are formulated n Secton 4. In Secton 5, a hybrd ntellgent algorthm desgned for solvng the models s descrbed. In Secton 6, a numercal example s gven to show the effectveness of the proposed algorthm. Fnally, we dscuss the conclusons and provde future drecton for research.. Prelmnares Let be a nonempty set, and let L be a σ-algebra over. Each element L s called an event. M s a set functon whch follows the four axoms gven by Lu: Axom. (Normalty) M. Axom. (Monotoncty) M M whenever. C Axom 3. (Self-Dualty) M M for any event. Axom 4. (Countable Subaddtvty) For every countable sequence of events M M., we have Then M s called an uncertan measure, the trplet, LM, s called an uncertanty space. A measurable functon defned on nto s called an uncertan varable.. Identfcaton Functon A random varable may be characterzed by a probablty densty functon, and a fuzzy varable may be descrbed by a membershp functon. Lu ntroduce an dentfcaton functon to characterze an uncertan varable. Defnton (Lu) An uncertan varable s sad to have a frst dentfcaton functon f () ( x) s a nonnegatve functon on such that sup ( x) ( y) ; () for any set B of real numbers, we have sup ( x), f sup ( x) 0.5 xb xb M B sup ( x), f sup ( x) 0.5 C xb xb x y Example : By a trapezodal uncertan varable we mean the uncertan varable fully determned by the quadruplet ( abcd ; ; ; ) of crsp numbers wth a bc d, whose frst dentfcaton functon s ( x a) ( b a), f a x b; 0.5, f b x c; ( x) ( x d) ( c d), f c x d; 0, otherwse.. Expected Value Expected value s the average value of uncertan varable n the sense of uncertan measure, and represents the sze of uncertan varable. Defnton (Lu) Let be an uncertan varable. Then the expected value of s defned by Publshed by Canadan Center of Scence and Educaton 33

3 Global Journal of Health Scence Vol., No. ; October 00 E[ ] M { r} dr M { r} dr 0 provded that at least one of the two ntegrals s fnte. 3. Assumptons and notatons Bed crss sometmes appears n hosptals. The crss s usually attrbuted to factors such as the bad weather, nfluenza, older people, geratrcans, and lack of cash or nurse shortages. As mentoned above, n hosptals, the queues form when there are nsuffcent beds avalable to admt ll people, and sck people have to wat untl free bed s avalable. When a patent arrves, he/she enter a queung system. We wll dvde the servce system nto three phases ncludng watng sckbed, preoperatve preparaton and postoperatve recovery. After that, the patent leaves the system. For optmze the beds allocaton durng the gusty busy perod, we descrbe assumptons and notatons as follows: 3. Assumptons ) treatment receved across dfferent sckbeds s dentcal; ) the total account of sckbeds s fxed and each bed s occuped by one patent; 3) the out-patents arrvals randomly and ndependently of each other; 4) LOS(length of stay) for each patent s consdered to be a uncertan varable. 3. Notatons n : the number of out-patents need to be hosptalzed durng a perod of t days; m : the number of fxed sckbeds;,,, m : the sckbeds; m : the number of patents cared for n sckbed ;,,, m : the patents cared for n sckbed ; k(, ): the patent cared for n sckbed ; x : the decson vector, where x (,), (, ),, (, ), (,), (, ),, (, ),, (,), (, ),, (, ) k k k m k k k m k k k m and the sequence (, ),,, n ; Dk(, ) : the arrvng tme of patent k(, ),,,, m;,,, m ; Tk(, ) : the LOS(length of stay) of patent k(, ), 0,,,, m ;,,, m k s the rearrangement of, and Tk(,0) s the occupaton tme of ntal patent n sckbed, so patent k(, ) wll leave the hosptal at the tme Tk(, 0 ) ; 0 0 : the uncertan vector where T x. Thus the watng tme of patent (, ) k s Tk Dk 0 (, 0) (, ) We denote the total watng tmes and the longest stayng tme n hosptal of all the n patents by f ( x, ) and (, ) f x, respectvely. Then we have f m m x T k D k (, ) (, ) (, ) f (, ) max m x 0Tk (, ) m 4. Uncertan Bed Allocaton Model Wth assumpton of the uncertan, we ntroduce two uncertan models for bed allocaton., 34 ISSN E-ISSN

4 Global Journal of Health Scence Vol., No. ; October Expected Tme Goal Model In order to balance the above conflctng obectves f ( x, ) and f ( x, ), we may have the followng target levels and prorty structure: At the frst prorty level, the expected total watng tme Thus we have a goal constrant n whch d 0 wll be mnmzed. E f ( x, ) b d, E f ( x, ) should not exceed the target value b. At the second prorty level, the expected longest stayng tme n hosptal E f ( x, ) should not exceed the target value b. That s, we have a goal constrant E f ( x, ) b d, n whch d 0 wll be mnmzed. Then we have the followng expected tme goal programmng model for the bed allocaton problem: lexmn d 0, d 0 st.. Ef( x, ) b d Ef( x, ) b d k(, ) n,,, m,,,, m, ntegers k(, ) k(, ),f (, ) (, ) where lexmn represents lexcographcally mnmzng the obectve vector. 4. Chance-Constraned Goal Programmng We assume the followng prorty structure. At the frst prorty level, the total watng tme f (, ) x should not exceed the target value b wth confdence level. Thus we have a goal constrant n whch d 0 wll be mnmzed. M f ( x, ) b d At the second prorty level, the longest stayng tme n hosptal f (, ) x should not exceed the target value b wth confdence level. Thus we have a goal constrant M f ( x, ) b d n whch d 0 wll be mnmzed. Then we have the followng chance-constraned goal programmng model for the bed allocaton problem: lexmn d 0, d 0 st.. M f( x, ) b d M f( x, ) b d k(, ) n,,, m,,,, m, ntegers k(, ) k(, ), f (, ) (, ) Publshed by Canadan Center of Scence and Educaton 35

5 Global Journal of Health Scence Vol., No. ; October Intellgent Algorthm Based on Uncertan Smulaton In order to solve the models, we ntegrate uncertan smulaton, neural network and GA to produce a hybrd ntellgent algorthm. Step. Generate tranng nput-output data for uncertan functons lke U : x E[ f( x, )] U : x M f ( x, ) 0 for any gven decson vector x. Step. Tran a neural network to approxmate the uncertan functons by the generated tranng data. Step 3. Intalze pop_sze chromosomes whose feasblty may be checked by the traned neural network. Step 4. Update the chromosomes by crossover and mutaton operatons and the traned neural network may be employed to check the feasblty of offsprng. Step 5. Calculate the obectve value for all chromosomes by the traned neural network. Step 6. Compute the ftness of each chromosome by rank-based evaluaton functon based on the obectve values. Step 7. Select the chromosome by rank-based evaluaton functon based on the obectve values. Here, we adopt the followng rank-based evaluaton functon Eval( V ) a( a),,,, pop _ sze where parameter a (0,), means the best ndvdual, and pop_ sze the worst ndvdual. Step 8. Repeat the fourth to seventh steps a gven number of cycles. Step 9. Report the best chromosome as the optmal soluton. 6. Numercal Experments In ths secton, we gve a smple example to llustrate the effectveness of the proposed hybrd ntellgent algorthms employng MATLAB. Here the parameters are set as follows: the populaton sze s 00, the probablty of crossover P c s 0.6, the probablty of mutaton P m s 0.3, and the parameter a n the rank-based evaluaton functon s Example. Let us consder 0 patents arrve n one day and 3 sckbeds are avalable. The estmated lengths of stay for each patent are trapezodal uncertan varables as table. At the frst prorty level, the expected total watng tme have a goal constrant n whch d wll be mnmzed. E f ( x, ) 0, d E f (, ) x should be as lttle as possble. Then we At the second prorty level, the expected longest stayng tme n hosptal target value8. That s, we have a goal constrant d E f ( x, ) 8, d E f (, ) x should not exceed the n whch wll be mnmzed. Then we have the followng expected tme goal programmng model for the bed allocaton problem: 36 ISSN E-ISSN

6 Global Journal of Health Scence Vol., No. ; October 00 lexmn d 0, d 0 st.. Ef( x, ) d 0 Ef( x, ) d 8 k(, ) n,, 3,,,, m3, ntegers k(, ) k(, ),f (, ) (, ) A run of the hybrd ntellgent algorthm (3000 cycles n fuzzy smulaton, 500 generaton n GA) shows that the optmal allocaton s Sckbed : 45 3; Sckbed : 69 7; Sckbed 3: Conclusons In ths paper, wth the ntroduced uncertan varable, the bed allocaton problem n hosptal s dscussed under uncertan envronment; thereby two uncertan models are presented. Ths model effectvely reduce the patents tme n queue, thus can mprove the publc satsfacton to the health servce. The realzaton of ths method wll open up new research feld of the applcaton of uncertanty theory. In the future research, dynamc bed allocaton s concerned and we wll mprove the effcency of algorthm. References C. Vaslaks and E. El-Darz. (00). A smulaton study of the wnter bed crss. Health Care Management Scence, Vol.4, Kwakernaak H. (978). Fuzzy random varablesłi. defntons and theorems. Informaton Scences, Vol.5, -9. Mohamed A. Ahmed, Talal M. Alkhams. (009). Smulaton optmzaton for an emergency department healthcare unt n Kuwat. European Journal of Operatonal Research, Vol.98, 3, Lu B. (00). Theory and Practce of Uncertan Programmng. Hedelberg: Physca-Verlag. Lu B. (006). A survey of credblty theory. Fuzzy Optmzaton and Decson Makng, Vol.5, No.4, Lu B. (007). Uncertanty Theory. (nd ed.). Berln: Sprnger-Verlag. Lu B. (009). Theory and Practce of Uncertan Programmng. (nd ed.). Berln: Sprnger-Verlag. Lu B. (00). Theory and Practce of Uncertan Programmng. (3rd ed.). [Onlne] Avalable: Lu B. (00). Uncertanty Theory. (3rd ed.). [Onlne] Avalable: (March 0, 00). L X and Lu B. (009). Chance measure for hybrd events wth fuzzness and randomness. Soft Computng, Vol.3, No., Zxong Peng. (00). Uncertan Smulaton. [Onlne] Avalable: (March 0, 00). Table. The estmated length of stay for each patent Patent Length of Stay (6,9,,4) (4,7,9,) (3,6,8,) (,4,6,9) (,3,5,8) Patent Length of Stay (0,3,5,8) (,3,5,7) (0,,3,4) (0,,3,4) (0,,3,4) Publshed by Canadan Center of Scence and Educaton 37