The Optimal Number of Hospital Beds Under Uncertainty: A Costs Management Approach

Document Type: Original Manuscript


1, Department of Industrial Engineering, Yazd University, Yazd, Iran

2 Faculty member, Department of Industrial Engineering, University of Torbat Heydarieh, Torbat Heydarieh, Iran.

3 Faculty member, Department of Industrial Engineering, quchan university advanced technologyUniversity, quchan, Iran



Equipping hospital beds uses a great deal of a hospital''''s resources. Therefore, it is essential to consider the hospital beds'''' efficiency. To increase its efficiency, a fuzzy unrestricted model for managing hospital expenses is presented in this paper. The lack of beds in hospitals leads to patients’ admission loss and consecutively profit loss. On the other hand, increasing the bed count leads to an increase in equipment expenses. Therefore, in order to determine optimal bed capacity, it is of utmost importance to consider these two costs simultaneously. In our paper, hospital admission system is modeled with a multi-server queuing system (M/M/K). Therefore, to calculate the total cost function, limiting probabilities of multi-server queueing model is used. Furthermore, due to uncertain nature of parameters, such as interest rate and hospitalization profit in various future time periods, these uncertainties are covered by fuzzy logic. Finally, to determine the optimal bed count, Lee and Li''''s fuzzy ranking method is used. This model is implemented ona case study. Its goal is to determine the optimal bed count for emergency unit of Razi hospital in Torbat Heydarieh. Considering the high capability of Markovian chains in modeling different circumstances and the various queueing models, the proposed model can be extended for various hospital units.

Graphical Abstract

The Optimal Number of Hospital Beds Under Uncertainty: A Costs Management Approach


  • Unrestricted fuzzy non-linear model. This model is used in addition to fuzzy ranking. Together, this approach reduces execution time. Furthermore, compared to the corresponding restricted fuzzy non-linear model, this approach is understood more easily by humans.
  • The use of Markovian chains and queueing theory. The high flexibility of Markov chains in modeling would lead to a suitable model for various hospitals and conditions. Therefore, the resulting model will have a high level of flexibility.
  • Integration of fuzzy logic and queuing theory. This integration leads to the coverage of uncertain conditions which is an inherent condition of servicing environments and especially hospitals.
  • A novel cost function with the consideration of time value of money


Main Subjects

Asady, B. (2010). The revised method of ranking LR fuzzy number based on deviation degree. Expert Systems with Applications, 37(7), 5056-5060.

Asady, B, & Zendehnam, A. (2007). Ranking fuzzy numbers by distance minimization. Applied Mathematical Modelling, 31(11), 2589-2598.

Bazzoli, Gloria J, Brewster, Linda R, Liu, Gigi, & Kuo, Sylvia. (2003). Does US hospital capacity need to be expanded? Health Affairs, 22(6), 40-54.

Ben Bachouch, Rym, Guinet, Alain, & Hajri-Gabouj, Sonia. (2012). An integer linear model for hospital bed planning. International Journal of Production Economics, 140(2), 833-843. doi:

Cheng, Ching-Hsue. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy sets and systems, 95(3), 307-317.

Chu, Ta-Chung, & Tsao, Chung-Tsen. (2002). Ranking fuzzy numbers with an area between the centroid point and original point. Computers & Mathematics with Applications, 43(1), 111-117.

Cochran, Jeffery K., & Roche, Kevin T. (2009). A multi-class queuing network analysis methodology for improving hospital emergency department performance. Computers & Operations Research, 36(5), 1497-1512. doi:

Coile, Russell C, & Association, American Hospital. (2002). Futurescan 2002: A Forecast of Healthcare Trends, 2002-2006. Chicago: Health Administration Press Chicago.

Crandall, Richard E., & Markland, Robert E. (1996). DEMAND MANAGEMENT-TODAY''''S CHALLENGE FOR SERVICE INDUSTRIES. Production and Operations Management, 5(2), 106-120. doi: 10.1111/j.1937-5956.1996.tb00389.x

Garg, Lalit, McClean, Sally, Meenan, Brian, & Millard, Peter. (2010). A non-homogeneous discrete time Markov model for admission scheduling and resource planning in a cost or capacity constrained healthcare system. Health Care Management Science, 13(2), 155-169. doi: 10.1007/s10729-009-9120-0

Gelenbe, Erol, Pujolle, Guy, & Nelson, JCC. (1987). Introduction to queueing networks (2th ed.): Wiley Chichester.

Getz, Donald. (1983). Capacity to absorb tourism: Concepts and implications for strategic planning. Annals of Tourism Research, 10(2), 239-263. doi:

Gong, Yue-jiao, Zhang, Jun, & Fan, Zhun. (2010). A multi-objective comprehensive learning particle swarm optimization with a binary search-based representation scheme for bed allocation problem in general hospital. Paper presented at the Systems Man and Cybernetics (SMC), Istanbul.

Gorunescu, Florin, McClean, SallyI, & Millard, PeterH. (2002). Using a Queueing Model to Help Plan Bed Allocation in a Department of Geriatric Medicine. Health Care Management Science, 5(4), 307-312. doi: 10.1023/A:1020342509099

Gu, Zheng. (2003). Analysis of Las Vegas strip casino hotel capacity: an inventory model for optimization. Tourism Management, 24(3), 309-314.

Hershey, John C, Weiss, Elliott N, & Cohen, Morris A. (1981). A stochastic service network model with application to hospital facilities. Operations Research, 29(1), 1-22.

Hwang, Johye, Gao, Long, & Jang, Wooseung. (2010). Joint demand and capacity management in a restaurant system. European Journal of Operational Research, 207(1), 465-472. doi:

Kao, Edward PC, & Tung, Grace G. (1981). Bed allocation in a public health care delivery system. Management Science, 27(5), 507-520.

Klassen, Kenneth J, & Rohleder, Thomas R. (2002). Demand and capacity management decisions in services: how they impact on one another. International Journal of Operations & Production Management, 22(5), 527-548.

Kokangul, Ali. (2008). A combination of deterministic and stochastic approaches to optimize bed capacity in a hospital unit. Computer methods and programs in biomedicine, 90(1), 56-65.

Lapierre, Sophie D., Goldsman, David, Cochran, Roger, & DuBow, Janice. (1999). Bed allocation techniques based on census data. Socio-Economic Planning Sciences, 33(1), 25-38. doi:

Lee, Donald KK, & Zenios, Stefanos A. (2009). Optimal capacity overbooking for the regular treatment of chronic conditions. Operations research, 57(4), 852-865.

Lee, E. S., & Li, R. J. (1988). Comparison of fuzzy numbers based on the probability measure of fuzzy events. Computers & Mathematics with Applications, 15(10), 887-896. doi:

Li, Ling, & Benton, W. C. (2003). Hospital capacity management decisions: Emphasis on cost control and quality enhancement. European Journal of Operational Research, 146(3), 596-614. doi:

Li, X., Beullens, P., Jones, D., & Tamiz, M. (2008). An integrated queuing and multi-objective bed allocation model with application to a hospital in China. J Oper Res Soc, 60(3), 330-338.

McManus, Michael L, Long, Michael C, Cooper, Abbot, & Litvak, Eugene. (2004). Queuing theory accurately models the need for critical care resources. Anesthesiology, 100(5), 1271-1276.

Milne, Eugene, & Whitty, Paula. (1995). Calculation of the need for paediatric intensive care beds. Archives of disease in childhood, 73(6), 505-507.

Nejad, Ali Mahmodi, & Mashinchi, Mashaallah. (2011). Ranking fuzzy numbers based on the areas on the left and the right sides of fuzzy number. Computers & Mathematics with Applications, 61(2), 431-442.

Pullman, Madeleine, & Rodgers, Svetlana. (2010). Capacity management for hospitality and tourism: A review of current approaches. International Journal of Hospitality Management, 29(1), 177-187.

Taylor, Gordon D. (1980). How to match plant with demand: a matrix for marketing. International Journal of Tourism Management, 1(1), 56-60.

Walczak, Steven, Pofahl, Walter E., & Scorpio, Ronald J. (2003). A decision support tool for allocating hospital bed resources and determining required acuity of care. Decision Support Systems, 34(4), 445-456. doi:

Wang, Yu-Jie, & Lee, Hsuan-Shih. (2008). The revised method of ranking fuzzy numbers with an area between the centroid and original points. Computers & Mathematics with Applications, 55(9), 2033-2042.

White, John A, & Francis, Richard L. (1971). Normative models for some warehouse sizing problems. AIIE Transactions, 3(3), 185-190.