Solving surgical cases assignment problem by a branch-and-price approach

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Abstract

In this paper, we study a surgical cases assignment problem (SCAP) of assigning a set of surgical cases to several multifunctional operating rooms with an objective of minimizing total operating cost. Firstly, we formulate this problem as an integer problem and then reformulate the integer program by using Dantzig–Wolf decomposition as a set partitioning problem. Based on this set partitioning formulation, a so-called branch-and-price exact solution algorithm, combining Branch-and-Bound procedure with column generation (CG) method, is designed for the proposed problem where each node is the linear relaxation problem of a set partitioning problem. This linear relaxation problem is solved by a CG approach in which each column represents a plan for one operating room and is generated by solving a sub-problem (SP) of single operating room planning problem. The computational results indicate that the decomposition approach is promising and capable of solving large problems.

Introduction

Operating theatre, consisting of operating rooms and recovery rooms, is the kernel in a hospital. With almost 10–15% of the intended budget (Clergue, 1999 and Macario et al., 1995), costs of the operating theatre are the most significant portion of those of hospital sectors. As a result of centrally constrained budgets, resource-allocation problems, in which such limited resources as surgeons, nurses and beds should be allocated appropriately to services by minimizing the operating cost and maximizing operating theatre's utilization, are known as the common issues in health-care planning problems.

Although the operating theatre management is similar to the production management problem if the surgical cases are regarded as options and the operating theatre as production center, there is much difference between them considering the components of the system, the objective of the management problem, the necessary constraints and so on:

  • In an industrial system, normally the production process is pre-determined while in a hospital system, all services offered to a patient might vary according to his health situation and preference. What's more, more urgent cases occur in a hospital system than those of an industrial one.

  • In an industrial system, one machine is normally occupied by one technician while the surgical team is composed of different specialists (such as a surgeon, nurses and an anaethesist), so it is much harder to coordinate the activities concerned.

  • Considering the objective of those two management problems, we can also find the difference: the operating theatre management problem normally aims at minimizing the operating cost while the objective of production management problem is normally to maximize the factory's profit.

  • By comparing the main constraints of both systems, we find that the requests of the patients (for example, the operating deadline) should be absolutely satisfied, and it is much more difficult for a manager to assign human resources in a hospital because the surgical team consists various roles and their capacity and requests should also be met while these are not necessarily done in the factory.

  • Another difference between these two systems is based on the difficulty of estimating the operating time. In fact, it is extremely difficult to determine the operating time of a surgical case because it depends on many parameters (patient's age, surgeon's experience, etc.) while normally the operating time of a product can be well evaluated according to the previous experience.

  • In an industrial system, normally the semi-finished products can be temporarily stored when there is no machine available in the precedent stage and working process of the current stage will not be interrupted; however, if there is no recovery bed available in the recovery room of the hospital when the operation of a patient is finished, this patient has to remain in the operating room where he is treated and begin to regain consciousness.

In conclusion, the operating theatre management problem is much more complex than a typical production planning one though there are also many things in common.

In this paper, we will focus on the surgical cases assignment problems (SCAP), one part of the operating theatre planning problem, which belongs to a primary category of operating theatre planning problem. This kind of problem aims at assigning a set of surgical cases to operating theatres during one planning period (1 day, 1 week or 1 month) by minimizing total operating cost with given resources’ constraints. As a popular issue, it is absolutely attractive to researchers, such as Ozkarahan (1995), Vissers (1998), Lovejoy and Li (2002), Jebali et al.(2003), Dexter et al.(1999), Blake and Donald (2002), Chaabane et al.(2003), Guinet and Chaabane (2003), and Weinbroum et al.(2003).

We assume, in this paper, that the operating time of a surgical case is estimated in advance and used as a determined data in the mathematical model.

We aim to assign Ncase surgical cases to several multifunctional operating rooms during 1 week (5 days). Each surgical case is identified by its operating time and deadline. A surgical case can be assigned to an operating room for a day whenever the total operating time of assigned surgical cases is less than or equal to this operating room's maximal opening duration in this day (ordinary opening time plus its maximal additional time). The operation date has to be fixed before its deadline. Each operating room can have different opening periods. The objective is to minimize total operating cost (unexploited time's cost or overtime cost).

As for this assignment problem, some precedent work has been done by Fei et al. (2004). In that paper, we constructed a heuristic procedure, based on column generation (CG) and dynamic programming, to get the solution to this problem. We obtained approximate solutions, so in this paper we will try to solve this problem for the exact solution by a so-called branch-and-price procedure here. Furthermore, in this paper we will loosen the hypothesis that all operating rooms are identical.

This paper is organized as follows. Firstly, a general integer programming (GIP) formulation is described for the problem concerned. Then, framework of applied branch-and-price procedure is described. After that the decomposition approach is given, and the CG method assisted with dynamic program method is presented to solve linear relaxation of the set partitioning problem. In Section 4, we will present the CG procedure and its key points, and in Section 5 the detail of the branch-and-price procedure are presented for the exact optimal solution of the general integer problem. Afterwards we will show the experimental results and finally give out conclusions and perspectives.

Section snippets

General integer programming

In SCAP, the objective is to find a minimum cost for assigning Ncase surgical cases to a set of available operating rooms during 1 week considering restrictions on each operating room's capacity and patients’ deadline. Due to the complexity of the operating room planning and scheduling problems, some simplifications have been made as follows:

  • All the operating rooms are multifunctional; i.e., each operating room can operated any assigned surgical case.

  • Human and instruments resources are always

Framework of branch-and-price procedure

As presented in Barnhart et al. (1998), the successful solution of large-scale mixed integer programming, such as SCAP, requires formulations whose linear programming (LP) relaxations make a good approximation of feasible solutions. Based on this idea, a great deal of attention has been paid to the Branch-and-Cut and Branch-and-Price approaches. The philosophies of Branch-and-Cut and branch-and-price are similar: both of them are generations of Branch-and-Bound approach, where pricing and

Set partitioning general problem (GP) corresponding to general integer problem (GIP)

Utilizing Dantzig and Wolfe (1960), we decompose this GIP problem into a set partitioning master problem (MP) consisting of constraints (1), (2) and a set of SPs with feasible region defined by (3) with Cj defined by (5). All decision variables in the MP and SPs are binary integers defined by (4).


Notes

    aij

    1 if surgical case i is assigned to plan j; 0 otherwise;

    bjd

    1 if plan j is scheduled on day d; 0 otherwise;

    ekj

    1 if operating room k is used by plan j; 0 otherwise.

    A plan

    one possible assignment

Framework of heuristic procedure based on column generation

As mentioned in Section 4, in the proposed branch-and-price procedure, the linear relaxation of each node is solved by a CG procedure. In addition, the first node is solved by an HPBCG for an approximate solution, which is set as the first UB in the branch-and-price procedure. In this section, we will describe the framework of the heuristic procedure HPBCG where CG procedure is one of its parts.

Selection strategy of node and branching variable

In this section, we will give the details of the branch-and-price procedure, whose framework was already described in Section 3, in order to solve the proposed SCAP, especially its branching criteria.

As is known, usually two classes of decision need to be made throughout a Branch-and-Bound procedure. One, called node selection, is to select an active node in the Branch-and- Bound tree (node list) to be explored; the other, called branching variable selection, is to select a fractional variable

Experimental results

Requirements for the numerical experiments are as follows:

  • Hardware for running the algorithm: IBM ThinkPad T23 (CPU: PM 1.6 GHz, Memory: 256 MB).

  • Software development environment: Microsoft VC++ 6.0

  • Software support: COIN, a LP solver, downloadable at IBM's website (http://www.coin-or.org/index.html).

Conclusion and perspective

In this paper, we have developed a decomposition-based branch-and-price algorithm to optimally solve a SCAP with an objective of minimizing total unexploited and overtime operating cost. At the end of this paper, we tested problems with a size up to 160 surgical cases. The results show our algorithm works quite well for tested problems.

However, it is the fact that our proposed problem has taken into account too many strict hypotheses that make our results far from practical application. Thus,

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This research is funded by “Inter-university Attraction Poles Programme – Belgian State – Belgian Science Policy”.

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