眼科病床的合理安排-2009年全国大学生数学建模竞赛B题

amao

医院就医排队是大家都非常熟悉的现象，它以这样或那样的形式出现在我们面前，例如，患者到门诊就诊、到收费处划价、到药房取药、到注射室打针、等待住院等，往往需要排队等待接受某种服务。
我们考虑某医院眼科病床的合理安排的数学建模问题。
该医院眼科门诊每天开放，住院部共有病床79张。该医院眼科手术主要分四大类：白内障、视网膜疾病、青光眼和外伤。附录中给出了2008年7月13日至2008年9月11日这段时间里各类病人的情况。
白内障手术较简单，而且没有急症。目前该院是每周一、三做白内障手术，此类病人的术前准备时间只需1、2天。做两只眼的病人比做一只眼的要多一些，大约占到60%。如果要做双眼是周一先做一只，周三再做另一只。
外伤疾病通常属于急症，病床有空时立即安排住院，住院后第二天便会安排手术。
其他眼科疾病比较复杂，有各种不同情况，但大致住院以后2-3天内就可以接受手术，主要是术后的观察时间较长。这类疾病手术时间可根据需要安排，一般不安排在周一、周三。由于急症数量较少，建模时这些眼科疾病可不考虑急症。
该医院眼科手术条件比较充分，在考虑病床安排时可不考虑手术条件的限制，但考虑到手术医生的安排问题，通常情况下白内障手术与其他眼科手术（急症除外）不安排在同一天做。当前该住院部对全体非急症病人是按照FCFS（First come, First serve）规则安排住院，但等待住院病人队列却越来越长，医院方面希望你们能通过数学建模来帮助解决该住院部的病床合理安排问题，以提高对医院资源的有效利用。
问题一：试分析确定合理的评价指标体系，用以评价该问题的病床安排模型的优劣。
问题二：试就该住院部当前的情况，建立合理的病床安排模型，以根据已知的第二天拟出院病人数来确定第二天应该安排哪些病人住院。并对你们的模型利用问题一中的指标体系作出评价。
问题三：作为病人，自然希望尽早知道自己大约何时能住院。能否根据当时住院病人及等待住院病人的统计情况，在病人门诊时即告知其大致入住时间区间。

问题四：若该住院部周六、周日不安排手术，请你们重新回答问题二，医院的手术时间安排是否应作出相应调整?
问题五：有人从便于管理的角度提出建议，在一般情形下，医院病床安排可采取使各类病人占用病床的比例大致固定的方案，试就此方案，建立使得所有病人在系统内的平均逗留时间（含等待入院及住院时间）最短的病床比例分配模型。


附录略，见附件。

amao

The inpatient department of CREGOT Ophthalmology Hospital has seventy-nine hospital beds. Treatment and surgery services for cataract, retinal diseases, glaucoma and ocular trauma are offered. Related Information from 13 July 2008 to 11 September 2008 is given in the appendix.

The cataract surgery is relatively easy, and there is no emergency. According to current schedule, the cataract surgery service is on Monday and Wednesday. Surgery preparation needs one or two days. Patients who have two-eye cataract account for 60% of the whole number of cataract patients. The first eye surgery is on Monday, and the second is on Wednesday for them.

The ocular traumas patients usually need emergency treatment. They are hospitalized as soon as possible. The surgery will be on the following day when they are hospitalized.

For retinal diseases and glaucoma patients, usually, preparation needs two or three days before surgery when they are hospitalized, and their postoperative period will be longer. Generally, this type of surgery is not on Monday or Wednesday.

Considering the workload of oculists, CREGOT does not put the cataract surgery and other surgery (except for an emergency treatment) on the same day. The patients(except for emergency treatment) queue up for beds available under the FCFS(First Come, First Serve) principle, while the queue becomes longer and longer. CREGOT hope you can help him to improve the utilization of their resources with mathematical modeling.

Your tasks are as follows:

Task 1: To developed a reasonable evaluation index system to evaluate the merits and demerits of the beds assignment model.

Task 2: Build a model to determine which patients should be hospitalized based on the number of patients who will be discharged on the following day, and evaluate your model using the index system developed in Task 1.

Task 3: Outpatients surely want to know when they can be hospitalized as early as possible. Can you give them a prediction of hospitalized interval based on the statistic data in the appendix?

Task 4: If the inpatient department does not arrange surgery on Saturday and Sunday, redo Task 2. Discuss whether the schedule should be changed.

Task 5: In order to simplify the scheme, it is suggested that the beds of various types of patients should be assigned by using fixed proportion principle. Develop a fixed proportion principle model to minimize the average time of stay in this system (include waiting time for beds available and hospital stays).