Integral resource capacity planning for inpatient care services based on bed census predictions by hour

Abstract

The design and operations of inpatient care facilities are typically largely historically shaped. A better match with the changing environment is often possible, and even inevitable due to the pressure on hospital budgets. Effectively organizing inpatient care requires simultaneous consideration of several interrelated planning issues. Also, coordination with upstream departments like the operating theatre and the emergency department is much-needed. We present a generic analytical approach to predict bed census on nursing wards by hour, as a function of the Master Surgical Schedule and arrival patterns of emergency patients. Along these predictions, insight is gained on the impact of strategic (ie, case mix, care unit size, care unit partitioning), tactical (ie, allocation of operating room time, misplacement rules), and operational decisions (ie, time of admission/discharge). The method is used in the Academic Medical Center Amsterdam as a decision-support tool in a complete redesign of the inpatient care operations.

Appendices

Appendix A

In the appendix, the derivations are presented that were omitted in the main article for reasons of readability. Note that the exposition is such that it is supplementary to the main text, and is therefore not intended to be comprehensible in isolation.

Demand predictions for elective patients

Single surgery block

To calculate we first determine the admission process under a given number of performed surgeries y. Define a_{n, t}^j(x|y) as the probability that x patients are admitted until time t on day n, given that y admissions take place in total. Then:

where v_{n, t}^j is the probability for a type j patient to be admitted in time t, given that he/she will be admitted at day n and is not yet admitted before t:

Finally,

To calculate d_{n, t}^j(x), we first determine d _n ^j(x), for day 0 the probability that x patients are present at the start of the discharge process (t=ϑ _j ) and for days n>0 the probability that x patients are present at the start of the day:

where s _n ^j is the probability that a type j patient who is still present at the begin of day n is discharged on day n:

Starting from d _n ^j(x), we determine the day process:

where z_n, t^j is the probability of a type j patient to be discharged during time interval [t, t+1) on day n, given this patient is still present at time t:

Single MSS cycle

We determine the overall probability distribution of the number of patients in recovery resulting from a single MSS, using discrete convolutions. If specialty j is assigned to OR block b_{i, s}, then the distribution for the number of recovering patients of block b_i,s present at time t on day m(m∈{0, 1, 2, …, S, S+1, S+2, …}) is given by:

where 0 means and all other probabilities are 0. Then, H_{m, t} is computed by:

Steady state

Since the cyclic structure of the MSS implies that the recovery of patients receiving surgery during one cycle may overlap with patients from the next cycle, the distributions H_{m, t} have to be overlapped in the correct manner. H_s, t^SS can be computed as follows:

where M=max{m|∃t, x with H_{m, t}(x)>0}.

Appendix B

Demand predictions for acute patient types

Single patient type

For patient type j=(p, r, θ), the admission process is determined by a non-homogeneous Poisson process:

To calculate , we first determine , for day 0 the probability that x patients are present at the start of the discharge process (t=θ+1) and for days n>0 the probability that x patients are present at the start of the day:

where is the probability that a type j patient who is still present at the begin of day n is discharged during day n:

Starting from we determine the day process:

where is the probability of a type j patient to be discharged during time interval [t, t+1) on day n, given this patient is still present at time t:

Single cycle

To determine the overall probability distribution of the number of patients in recovery resulting from a single AAC, define as the probability distribution of the number of recovering patients of type j present at time interval t on day w(w∈{0, 1, 2, …, R, R+1, R+2, …}). The distribution is given by:

Then, G_{w, t} is computed by:

Steady state

G_{r, t}^SS can be computed as follows:

where W=max{r|∃t, x with G_{r, t}(x)>0}.

Appendix C

Performance indicators

In this appendix, the derivation of Z_{q, t}^k(x _k |n) is presented. To this end, let us first introduce the concept cohort. A cohort is a group of patients originating from a single instance of an OR block (electives) or admission time interval (acute patients). Then,

where Ω is the total number of cohorts, ω the number of cohorts that do generate arrivals during time interval [t, t+1) on day q, and the permutation σ is such that the patient types σ(1), …, σ(ω) are the types that can generate those arrivals. Further, for notational convenience we introduce the function f_{q, t}ⁱ as f_{q, t}ⁱ=h_{q, t}ⁱ for the elective patients, and for acute patient types. Also, we introduce α_{q, t}^j as α_{q, t}^j=α_{q, t}^j for the elective patient types and for the acute patient types. It remains to define the probability that for an arriving cohort, from the y _j patients present in total, n _j arrivals occur during time interval [t, t+1):

where for elective patient types v_{n, t}^j=(w_{n, t}^je _n ^j)/e _n ^j∑_k=0^tw_{n, k}^j+e₋₁^j·1_(n=0) and for acute patient types v_{n, t}^j=1.