Column Generation Approach for Operating Rooms Planning Mehdi LAMIRI, Xiaolan XIE and ZHANG Shuguang...

Column Generation Approach for Operating Rooms Planning

Mehdi LAMIRI, Xiaolan XIE and ZHANG Shuguang

Industrial Engineering and Computer Sciences Division

Engineering and Health Division

ORAHS 06, Wroclaw - 2 -

Outline

• Motivation & Problem description

• Problem modelling

• A column generation approach

• Computational results

• Conclusions and perspectives


Problem description: Motivations

Operating rooms represent one of the most expensive sectors of the hospital

Involves coordination of large number of resources

Must deal with random demand for emergent surgery and unplanned activities

Planning and scheduling operating rooms’ has become one of the major priorities of hospitals for reducing cost and improving service quality


Problem description

How to plan elective cases when the operating rooms capacity is shared between two patients classes : elective and emergent patients

Elective patients :

Electives cases can be delayed and planned for future dates

Emergent patients :

Emergent cases arrive randomly and have to be performed in the day of arrival


Outline







Model: Operating rooms capacities

We consider the planning of a set of elective surgery cases over an horizon of H periods (days)

In each period there are S operating rooms

For each OR-day (s, t) we have a regular capacity Tts

Exceeding the regular capacity generates overtime costs (COts)

1 2 H1 2 H

OR 1

OR S

… OR 1

OR S

… OR 1

OR S

…

… …

T11 T1S T21

T2S

THSTH1


Model: Emergent patients

In this work, the OR capacity needed for emergency cases in OR-day (s, t) is assumed to be a random variable ( wt s ) based on:

- The distribution of the number of emergent patients in a given period estimated using information systems and / or by operating rooms’ manager

- The distribution of the OR time needed for emergency surgeries estimated from the historical data


Model: Elective cases

At the beginning of the horizon, there are N requests for elective surgery

A plan that specifies the subset of elective cases to be performed in each OR-day under the consideration of uncertain demand for emergency surgery

T1,2

T1,1

OR-day (1, 1)

OR-day (S, H)

TH,S

OR-day (2, 1)

Case 5

Case 12

Case 10

Case1


Model: Elective cases

Each elective case i ( 1…N ) has the following characteristics :

Operating Room Time needed for performing the case i : ( pi )

Estimated using information systems and/or surgeons’ expertises

A release period (Bi)

It represents hospitalisation date, date of medial test delivery

A set of costs CEits ( t = Bi …H, H+1 )

The CEits represents the cost of performing elective case i in period t in OR s

CEi,H+1 : cost of not performing case i in the current plan


Model: Elective cases related costs

The cost structure is fairly general. It can represent many situations :

Hospitalization costs / Penalties for waiting time

Patient’s or surgeon’s preferences

Eventual deadlines

ORs availabilities


Model : Example of planning

T12

T1,1

Case 5

Case 1

Case 10

Case 16

Case 7Case 9

Case 12

Case 21

OR-day (1, 1)

OR-day (S, H)

Case 2

Case 16

Case 14Case 20Case 1

Case 25

Case 33

Case 2Case 32

Case 27

The plan must minimizes the sum of elective patient related costs and the expected overtime costs

Overtime costs

Cases related costs


Mathematical Model

Unplanned activities time

Planned activities time

Regular capacity

(P) *J = Min

1 1

1 1 1i i

N M H M H

its its ts ts

i s t B s t B

J X CE X CO O

(1)

Subject to:

1

ts

N

ts W ts i its ts

i

O E W p X T

(2)

1

1

1i

M H

it

s t B

X

(3)

Xits {0, 1} (4)

overtime

Patient related cost Overtime cost

Decision:

- Assign case i to OR-day (s, t) , Xits = 1

- Reject case i from plan, Xi,H+1,s = 1


Outline







Plan for one OR-day

A “plan” is a possible assignment of patients to a particular OR-day

p : plan for a particular OR-day is defined as follows

aip = 1 if case i is in plan p

btsp = 1 if plan p is assigned to OR-day (s, t)

Cost of the plan :

, ,p ip tsp its tsp ts ts i ip ts

i t s t s i

C a b CE b CO E W p a T

Costs related to patients

assigned to the palnOvertime cost in the OR-day

related to the plan


( ) *p p

p

J Min J Y const c Y

1

1

0 1

{ },

ip pp

tsp pp

p

a Y , i

b Y , t ,s

Y , p

Subject to:

Column formulation for the planning problem

Each OR-day receives at most one plan

Each patient is assigned at most to one selected plan

: set of all possible plans

Yp = 1, if plan p is selected and Yp = 0, otherwise

Master problem


( ) *p p

p

J Min J Y const c Y

1

1

0 1

{ },

ip pp

tsp pp

p

a Y , i

b Y , t ,s

Y , p

Subject to:

Column formulation for the planning problem

Each OR-day receives at most one plan

Each patient is assigned at most to one selected plan

The master problem is an integer linear programming problem, whereas the initial formulation has a nonlinear objective:

The nonlinear quantities (expected overtime costs) are now imbedded into the columns costs

The master problem has a huge number of variables (columns)

Master problem


Master Problem

Linear master problem (LMP)

Optimal solution of the LMP

Near-optimal solution

Solution Methodology

Solve by Column Generati

on

Construct a

“good” feasible solution

Relax the integrality constraint

s


Linear Master problem (LMP)

The Linear Master Problem (LMP) is the same as the master problem MP except that the integrity of Yp is relaxed.

Problem LMP provides a lower bound of the master problem and hence a lower bound of the original problem.

Problem LMP can be solved by the column generation technique


Solving the linear master problem

simplex multipliers

i , t s

reduced cost < 0add new column

Y

N STOP

*

p pp

const c Y

1

1

0

*

*

ip pp

tsp pp

*p

a Y , i

b Y , t ,s

Y p

,

st

min

Reduced Linear Master Problem

over Ω* Ω

,p i ip ts tsp

i t s

c a b min

Pricing problem

minimizes reduced cost

0,ip tsp ia b t B

,

1tspt s

b , 0,1ip tspa b

st


The pricing problem

The pricing problem can be decomposed into H×S subproblems

One sub-problem for each OR-day

min ( ) its i ip ts ts i ip ts tsi i

CE a CO E W p a T

0,1 ,ipa i Subject to:

Simplex multipliers


The pricing problem


CE a CO E W p a T



The pricing problem

The pricing sub-problem is a stochastic knapsack problem:

The capacity of the sack is a random variable

There is a penalty cost if the capacity is exceeded


CE a CO E W p a T


Dynamic programming method


Master Problem

Linear master problem (LMP)

Optimal solution of the LMP

Near-optimal solution

Solution Methodology

Solve by Column Generati

on

Construct a

“good” feasible solution

Relax the integrality constraint

s


Constructing a near optimal solution

Step 1: Determine the corresponding patient assignment matrix (Xits) from the solution (Yp) of the Linear Master Problem.

Step 2: Derive a feasible solution starting from (Xits)

Step 3: Improve the solution obtained in Step 2


Derive a feasible solution

Method I : Solving the integer master problem MP by restricting to generated columns

Method II : Complete Reassignment

Fix assignments of cases in plans with Yp = 1

Reassign myopically but optimally all other cases one by one by taking into account scheduled cases.

Method III : Progressive reassignment

Reassign each case to one OR-day by taking into account the current assignment (Xits) of all other cases, fractional or not.


Improvement of a feasible schedule

Heuristic 1 : Local optimization of elective cases.

Reassign at each iteration the case that leads to the largest improvement

Heuristic 2 : Pair-wise exchange of elective cases

Exchange the assignment of a couple of cases that leads to the largest improvement

Heuristic 3 : Period-based re-optimization

Re-optimize the planning of all cases assigned to a given OR-day (s, t) and all rejected cases.


Overview of the optimization methods

Method LMP Deriving a feasible solution Improving solutions

M1 CPLEX IP M2 Complete reassign Local opt M3 Progressive reassign Local opt M4 Progressive reassign Local opt, Period-based M5 Progressive reassign Exchange, Period-based M6 Progressive reassign Exchange, Local opt, Period-based M7

column generation

CPLEX LP

+ Dynamic

Programming Progressive reassign Local opt, Period-based, Exchange


Outline







Computational experiments

Problem instances generation

Number of periods : H = 5

Number of operating rooms: S = 3, 6, 9, 12

OR-day’s regular capacity : Tts = 8 hours

Capacity needed for emergency cases : Wts is exponentially distributed with a mean of 3 hours

Overtime cost : COts = 500 € / hour

Duration of elective surgeries : pi are randomly generated from the interval [0.5, 3 hours]

Release periods : Bi are randomly generated from the set {1…H}


Computational experiments

Problem instances generation

Patients related costs : CEits = (t- Bi) x c

c is set equal to 150 € (hospitalisation cost)

Case 1: Identical ORs

Case 2: Non-Identical ORs

ORs are equally allocated to 3 specialties, and an extra charge of 100€ is added for cases assigned to another speciality’s ORs

The number of elective cases is determined such that the workload of ORs due to elective cases is 85% of the regular capacity of the entire planning horizon.


Computational experiments: Gap

Duality GapM1 M2 M3 M4 M5 M6 M7

　R=0　　

Nb Rooms=3(60 cases)

9.95% 3.35% 0.86% 0.82% 0.59% 0.58% 0.50%

Nb Rooms=6(119.3 cases)

10.59% 2.22% 1.01% 0.94% 0.64% 0.62% 0.53%


--- 1.94% 0.83% 0.74% 0.45% 0.44% 0.31%


--- 2.30% 1.02% 0.93% 0.60% 0.59% 0.47%

　R=100　　


0.15% 0.26% 0.14% 0.14% 0.17% 0.08% 0.05%


0.27% 2.39% 0.40% 0.40% 0.28% 0.24% 0.16%


1.54% 1.75% 0.64% 0.60% 0.31% 0.30% 0.22%


--- 1.85% 0.64% 0.62% 0.31% 0.31% 0.19%

Case R = 0: Identical ORs

Case R = 100: extra charge of 100 € for assigning cases to another specialty's ORs

Results based on 10 randomly generated instances: the average Gap


Computation TimeM1 M2 M3 M4 M5 M6 M7

　R=0　　


49,3 8,7 8,2 8,6 8,2 8,8 8,7


>5000 53,3 51,7 52,79 53,5 53,4 54


--- 184 182,1 183,6 182,3 184 186,2


--- 436,7 438,1 433,6 435 433,8 438,8

　R=100　　


9,6 9,8 9,3 10,8 9,5 9,9 9,9


120 71,7 70,6 77,8 72,7 72,3 72,1


>5000 245,7 244,9 259,4 251 246,7 247,5


--- 569,5 572,2 593,5 584,1 570,2 590,1

Computational experiments: computation time

• Over 65% of the computation for CGP is spent on pricing problems.

Results based on 10 randomly generated instances: the average Computation time (second)


Computation results

The lower bound of the Column generation is very tight

Solving the integer master problem with generated columns can be very poor and it is very time consuming

Progressive reassignment outperforms the complete reassignment as progressive reassignment preserves the solution structure of the column generation solution


Outline







Model extension: Overtime capacity and under utilization cost

We introduce an additional penalty cost when the overtime capacity is exceeded

Operating Room related cost

regular capacity

overtime capacity

OR workload

under use

overtime cost overtime

capacity exceeded


Conclusions and perspectives

The proposed model can represent many real world constraints

Column generation is an efficient technique for providing provably good solutions in reasonable time for large problem.

Future work

Make the stochastic model realistic enough to take into account random operating times, ...

Take into account other criteria such as reliability of OR plans

Develop exact algorithms able to solve problems with large size

Test with field data

Column Generation Approach for Operating Rooms Planning Mehdi LAMIRI, Xiaolan XIE and ZHANG Shuguang...

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