Linear Programming Formulation of the Elevator Trip

Origin-destination Matrix Estimation Problem

Juha-Matti Kuusinen, Mirko Ruokokoski, Janne Sorsa and Marja-Liisa Siikonen

KONE Technology, KONE Corporation, POB 7, 02151 Espoo, Finland

Keywords:

Elevator trafﬁc, Origin-destination Matrix, Linear Programming, Branch-and-bound.

Abstract:

Elevator group control dispatches elevators to passengers’ calls in a dynamic environment where new calls

constantly emerge. At the moment of making a dispatching decision, it is not known when and at which ﬂoors

new passengers will register new calls, what is the number of passengers waiting behind these and existing

calls, and what are their destinations. Robust dispatching decisions require that future passenger trafﬁc is

forecast based on the realized passenger ﬂow in a building. The problem is that this ﬂow cannot be directly

measured. It can, however, be estimated by ﬁnding the passenger counts for the origins and destinations of

every elevator trip occurring in a building. An elevator trip consists of successive stops in one direction of

travel with passengers inside the elevator. We formulate the elevator trip origin-destination matrix estimation

problem as a minimum cost network ﬂow problem. We also present a branch-and-bound algorithm for ﬁnding

all solutions to the problem and study its performance based on numerical experiments.

1 INTRODUCTION

Assume that when an elevator stops, the passengers

inside the elevator car ﬁrst alight and then the possi-

bly waiting passengers board the elevator. Then, an

elevator trip to up or down direction starts at a stop

where passengers board an empty elevator and ends

to a stop where the elevator becomes empty again.

At each stop, the elevator group control registers the

number of boarding and alighting passengers, and the

calls given by the boarding passengers.

Our goal is to estimate elevator trip origin-

destination (OD) matrices, i.e., the passenger counts

for the OD pairs of the elevator trips. The elevator

trip OD matrices estimated for a given time interval

are combined to form a building OD matrix that de-

scribes the trafﬁc ﬂow in the whole building during

that interval. The building OD matrices form trafﬁc

statistics that are used to forecast future trafﬁc. The

required information is: what is the number of pas-

sengers waiting behind each existing and new call at

the time of serving the call, when and at which ﬂoors

new calls are likely to occur, and what are the desti-

nations of the passengers. These forecasts are needed

in elevator dispatching to make robust call allocation

decisions in constantly changing trafﬁc conditions.

An elevator trip is analogous to a single transit

route such as a bus line where there is at most one

route connecting any OD pair, and usually counts of

boarding and alighting passengers are collected from

all stops on the route (Nguyen, 1984). The passenger

counts for the OD pairs of the route are typically esti-

mated by minimizing the distance to a reference OD

matrix with respect to a suitable distance metric while

also producing the observed boarding and alighting

counts. Popular OD matrix estimation methods are

iterative proportional ﬁtting which is known also as

the bi-proportional or Bregman’s balancing method

(Lamond and Stewart, 1981; Bell, 1983; Nguyen,

1984; Ben-Akiva et al., 1985), maximum entropy and

minimum information (Zuylen and Willumsen, 1980;

Nguyen, 1984; Ben-Akiva et al., 1985), maximum

likelihood and generalized least squares (Ben-Akiva

et al., 1985; Cascetta and Nguyen, 1988), Bayesian

inference (Maher, 1983; Cascetta and Nguyen, 1988;

Li, 2009), recursive methods (Tsygalnitsky, 1977;

Furth and Navick, 1992; Li and Cassidy, 2007) and bi-

level programming formulations (Fisk, 1988; Lund-

gren and Peterson, 2008).

An elevator trip OD matrix could in principle be

estimated with these methods but some basic differ-

ences between an elevator trip and a single transit

route prevent their use. A single transit route is typ-

ically deﬁned in advance and is not often changed.

Hence, usually a single OD matrix is estimated for

a certain time period using the corresponding refer-

150

Kuusinen J., Ruokokoski M., Sorsa J. and Siikonen M..

Linear Programming Formulation of the Elevator Trip Origin-destination Matrix Estimation Problem.

DOI: 10.5220/0004338502980303

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 298-303

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

ence OD matrix and boarding and alighting counts

observed during that period. An elevator trip is re-

quest driven, which means that each elevator trip has

its own set of OD pairs deﬁned by the calls given by

the boarding passengers. Hence, we need to estimate

a separate OD matrix for each elevator trip. A natural

requirement is then that the estimated OD passenger

counts are integer-valued. The above methods do not

in general meet this condition. In addition, when el-

evators are installed in a new building, the reference

matrix is not available and the elevator trip OD ma-

trices have to estimated based only on the observed

boarding and alighting counts.

We model the elevator trip OD matrix estimation

problem as a minimum cost network ﬂow problem.

This model can be used to solve the problem when the

measured boarding and alighting counts are consis-

tent taking into account also lower bounds on the OD

passenger counts. When the counts are consistent, the

result of minimizing the distance between the board-

ing and alighting counts produced by the estimated el-

evator trip OD matrix and the observed boarding and

alighting counts does not depend on the distance met-

ric. The minimum cost network ﬂow formulation is

appealing since it is a linear program which can be

efﬁciently solved with the network simplex algorithm

(Bertsimas and Tsitsiklis, 1997). The elevator trip OD

matrix estimation problem is either over-, exactly- or

under-determined, i.e., the number of OD pairs is ei-

ther less than, equal to or greater than the number

of boarding and alighting counts. In the ﬁrst case,

the problem has a unique solution and in the last two

cases it has more than one solution.

We present a simple branch-and-bound algorithm

to ﬁnd all solutions to the problem. The algorithm is

motivated by the one-three algorithm (Danna et al.,

2007). We want to ﬁnd all solutions and then select

one among them randomly because without any ad-

ditional information it is impossible to know which

of the solutions correspond to the true OD passen-

ger counts, i.e., to the realized passenger trafﬁc. In

the long term, the random selection results to traf-

ﬁc statistics, i.e., building OD matrices, that are not

affected by the algorithm used in solving the prob-

lem, and thus, model better the possible realizations

of the trafﬁc. This is desirable when the statistics are

used to make passenger trafﬁc forecasts for elevator

dispatching. An existing method to estimate an ele-

vator trip OD matrix from inconsistent boarding and

alighting passenger counts is based on solving a suc-

cession of positive inverse problems (Yoneda, 2007).

This method, however, cannot be used to ﬁnd all so-

lutions.

In a real time elevator group control, there is usu-

ally only a very short time for solving the elevator trip

OD matrix estimation problem. Therefore, we study

the execution time of the branch-and-boundalgorithm

in ﬁnding all solutions to four typical example prob-

lems.

2 FORMULATION

We deﬁne an elevator trip as a directed network of

nodes N = {1,2,... ,m}, and arcs A. The nodes

correspond to the ﬂoors of a building. Let b

and

denote the measured number of passengers who

board and alight the elevator at node i, respectively.

Furthermore, let N

⊂ {1,2,...,m − 1} and N

−

⊂

{2,3,..., m} denote the set of pickup and delivery

nodes, respectively. Node i ∈ N

if b

≥ 1, and node

j ∈ N

−

, if a

≥ 1. The nodes i ∈ N

and j ∈ N

−

i + 1 ≤ j ≤ m, deﬁne the OD pair (i, j) if either the

passengers who board the elevator at node i register a

delivery request to node j, or there is another node k,

k ∈ {1,2,..., i− 1}, where a delivery request is regis-

tered to node j. Let A denote the set of arcs deﬁned by

the OD pairs (i, j), i ∈ N

, j ∈ N

−

. Finally, deﬁne x

as the unobservable number of passengers from the

origin node i to the destination node j, i.e., the OD

passenger count along the arc (i, j) ∈ A.

Figure 1 presents an elevator trip with ﬁve nodes

and OD pairs. This example corresponds to a situa-

tion where the passengers who board the elevator at

the ﬁrst node register delivery requests to nodes 3 and

4, and the passengers who board the elevator at the

second node register a new delivery request to node 5.

In general, a boarding passenger cannot register a de-

livery request to a destination where a request already

exists but it is of course possible that the passenger

travels to this destination. Hence, even if the passen-

gers who board the elevator at the second node cannot

ble that some of them travel to these destinations. In

the example, a dashed arc describes this situation, and

thus, arcs (2,3) and (2,4) are dashed. It can also be as-

sumed that passengers who board an elevator cannot

travel to destinations deﬁned by delivery requests reg-

istered at later nodes. This is why there is no arc (1,5)

in the example.

1 2 3 4 5

Figure 1: Example elevator trip.

The OD passenger counts are related to the mea-

LinearProgrammingFormulationoftheElevatorTripOrigin-destinationMatrixEstimationProblem

151

sured boarding and alighting counts through the so

called ﬂow conservation constraints, i.e., the total

ﬂow out of the node i, i ∈ N

, equals b

, and the total

ﬂow into the node j, j ∈ N

−

, equals a

∑

j|(i, j)∈A

= b

∀i ∈ N

, (1)

∑

i|(i, j)∈A

= a

∀ j ∈ N

−

. (2)

We assume that if there is a delivery request from

an origin node to a destination node, then at least one

passenger must be assigned to that OD pair. Hence,

we obtain the following lower bounds for the OD pas-

senger counts:



1, if i → j

0, otherwise

, (3)

where i → j signiﬁes that a delivery request is regis-

tered from node i to node j.

The lower bounds could be further tightened. For

example, in Figure 1, the lower bound for the OD

passenger count x

would be a

. We assume, how-

ever, that the tightening of the bounds is done in the

solver before solving the problem. We also assume

that the measured boarding and alighting counts are

consistent, i.e.,

∑

i∈N

∑

i∈N

−

, and this holds

also when the lower bounds are subtracted from the

counts.

2.1 Properties

For every elevator trip with two or more nodes, m ≥ 2,

the number of arcs n satisﬁes the following bounds:

m− 1 ≤ n ≤ m(m− 1)/2. (4)

The lower bound is the minimum number of edges

and the upper bound is the maximum number of edges

in any undirected graph. Since the elevator does

not change direction during a single trip, the graph

describing the elevator trip is acyclic, and thus, the

bounds for undirected graphs are valid.

The coefﬁcient matrix corresponding to the con-

straints (1) and (2) is the node-incidence matrix of the

elevator trip. This matrix contains p = |N

| + |N

−

rows and n columns, and its rank is p − 1 (Bazaraa

et al., 2009). Note that any node on the elevator

trip that is both a pickup and a delivery node can

be split in two successive nodes so that the result-

ing delivery node precedes the pickup node. Then,

p = m and equation (4) implies that n ≥ p− 1 always.

This means that when the node-arc incidence matrix

is over-determined, n < p, then n = p−1 and the ma-

trix has full rank equal to p− 1. In this case there is a

unique OD passenger count vector satisfying the con-

straints (1)-(3). When the matrix is exactly- or under-

determined, n ≥ p, it is also rank deﬁcient and there

are always more than one OD passenger count vec-

tor satisfying the constraints. A unique solution can

be obtained by formulating the problem of ﬁnding the

elevator trip OD passenger counts as a minimum cost

network ﬂow problem.

2.2 Minimum Cost Network Flow

Formulation

To model the elevator trip OD matrix estimation prob-

lem as a minimum cost network ﬂow problem, we

augment the directed network describing the eleva-

tor trip as is done in the simplex method for network

ﬂow problems to ﬁnd an initial basic feasible solution

(Bazaraa et al., 2009). We ﬁrst add to the network

an additional node m + 1, N = N ∪ {m + 1}. Then,

for every pickup node i ∈ N

we deﬁne the OD pair

(i,m + 1) and for every delivery node j ∈ N

−

the OD

pair (m+ 1, j), and add the corresponding new arcs to

the set A. The lower bound for each of these arcs is

zero. Figure 2 presents the elevator trip of Figure 1

with these additions. A dotted arc corresponds to an

OD pair incident to or from the additional node.

1 2 3 4 5 6

Figure 2: Augmented example elevator trip.

We set the cost of each arc (i, j), c

, not incident

to or from the additional node m + 1 equal to zero,

and the cost of each arc incident to or from the addi-

tional node equal to one. The elevator trip OD matrix

estimation problem can now be mathematically for-

mulated as a minimum cost network ﬂow problem as

follows:

Minimize

∑

i∈N

im+1

∑

j∈N

−

m+1j

subject to

∑

j|(i, j)∈A

= b

∀i ∈ N

(5)

∑

i|(i, j)∈A

= a

∀ j ∈ N

−

(6)

∑

i|(i,m+1)∈A

im+1

∑

j|(m+1, j)∈A

m+1j

= 0 (7)

≥ l

∀(i, j) ∈ A.

Note that the optimal value of the objective function

is zero, i.e., the OD passenger counts of the arcs in-

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

152

cident to or from the additional node are zero. Fur-

thermore, because the measured boarding and alight-

ing counts b

and a

are always integer-valued and

the node-incidence matrix deﬁned by the constraints

(5)-(7) is always totally unimodular, all solutions to

the problem are always integer-valued (Bazaraa et al.,

2009).

3 BRANCH-AND-BOUND

ALGORITHM

All solutions to the elevator trip OD matrix estima-

tion problem can be found with a branch-and-bound

algorithm presented in Algorithm 1. The algorithm is

similar to the second phase of the one-tree algorithm

(Danna et al., 2007). One difference between the al-

gorithms is that our algorithm can be used only when

the solutions to the linear program (LP) are integer-

valued. A second difference is that we accept only

optimal solutions, i.e., whose objective value is zero,

whereas the one-tree algorithm also accepts solutions

whose objectivevalue does not differ more than a pre-

deﬁned percentage of the optimal value. Third, in our

algorithm every node always stores the solution of the

LP deﬁned by that node. Hence, when we select a

new node from the set N, the solution of the LP at

that node is readily available, and we proceed to test

whether all variables at the node are ﬁxed by the local

bounds of the node. If there is a variable that is not

ﬁxed by the local bounds, we branch with respect to

one such variable, otherwise the node is fathomed.

When branching with respect to variable i at node

n, we form the child nodes n

and n

so that they do

not contain the value of the variable i at node n, x

(n).

To ﬁnd also other optimal solutions where the valueof

the variable i is equal to x

(n), we set the local bounds

of this variable at node n equal to x

(n) and add it

back to the set of nodes N. When node n is next time

selected from the set N, branching at this node is done

with respect to some other variable than i, or the node

is fathomed. Furthermore, the child nodes and the

corresponding solutions are added to the sets N and S

only if the solutions are optimal.

It is not always necessary to create two child

nodes. For example, when the value of the variable

i at node n is equal to its lower bound, x

(n) = l

(n),

we create only the right children n

= {x

≥ x

(n)+1}

and ﬁx the upper bound of variable i at node n equal

to its lower bound, i.e., to its value, u

(n) = x

(n). In

addition, our branching strategy does not produce du-

plicate solutions. There are also other possibly more

efﬁcient algorithms than branch-and-boundto ﬁnd in-

teger solutions to network ﬂow problems (Baldoni-

Algorithm 1: Branch-and-bound algorithm.

Set of nodes: N ← {rootnode}

Solve LP at node rootnode.

Set of solutions: S ← {x(rootnode)}

while N 6=

0 do

Select node n from N.

Set of nodes: N ← N \ {n}

if there exists a variable i that is not ﬁxed by local

bounds of node n: l

(n) < u

(n) then

Create child nodes and ﬁx bounds of node n:

= {x

≤ x

(n) − 1}, n

= {x

≥ x

(n) + 1}

and l

(n) = u

(n) = x

(n)

Set of nodes: N ← N ∪ {n}

for j = 1 → 2 do

Solve LP at node n

if objective z(n

) = 0 then

Set of solutions: S ← S∪ {x(n

)}

Set of nodes: N ← N ∪ {n

}

end if

end for

end if

end while

Silva et al., 2003) but these are not considered in this

paper.

4 NUMERICAL EXPERIMENTS

To study the performanceof the branch-and-boundal-

gorithm, we formed four example problems. These

problems were selected based on simulations ran

with the Building Trafﬁc Simulator (BTS), and they

are typical for the elevator trip OD matrix estima-

tion problem (Siikonen et al., 2001; Kuusinen et al.,

2012).

The branch-and-bound algorithm was imple-

mented with MATLAB 7.10.0 (R2010a) on a Pentium

2.66 GHz machine with 3.0 GB memory and Win-

dows XP. The LP at each node was solved using the

Cplex Class API provided in the CPLEX Optimiza-

tion Studio V12.2 for MATLAB. CPLEX was chosen

since it is somewhat the academic and industry stan-

dard for linear optimization problems.

The Cplex class provides methods for the manip-

ulation of the model, which can be used to increase

solving speed. One important manipulation is the se-

lection of the algorithm used by the Cplex instance.

When the LP is solved for the ﬁrst time at the root

node of the branch-and-bound tree, the algorithm of

choice is the network simplex algorithm since it tends

to be faster than a general simplex algorithm for net-

work ﬂow problems (Bertsimas and Tsitsiklis, 1997).

LinearProgrammingFormulationoftheElevatorTripOrigin-destinationMatrixEstimationProblem

153

Another advantage of the Cplex class is that after

solving a LP, the solution and the corresponding basis

can be obtained from the Cplex instance. Every node

stores this basis which is used as a starting basis at its

child nodes together with the dual simplex algorithm.

This is a typical approach in branch-and-bound algo-

rithms where the subproblem at each node is a LP and

can be solved using a simplex method.

4.1 Example Problems

Problem 1, an over-determined problem, excluding

the additional node and arcs, consists of the follow-

ing nodes, arcs, and boarding and alighting passenger

count vectors:

= {1,2,3}, N

−

= {3,4},

A = {(1,3),(2, 3),(2,4),(3,4)},

b = [10,11,11], a = [11,21].

The lower bound of an arc written in bold equals one,

and zero otherwise. Hence, an arc written in bold

means that the passengers who board the elevator at

the origin of the OD pair register a delivery request to

the destination of the OD pair. Otherwise, the deliv-

ery request to the destination is registered at an ori-

gin before the origin of the OD pair. Furthermore, in

this problem there is no arc (1,4) because the deliv-

ery request to node 4 is registered by the passengers

who board the elevator at node 2, and thus, the pas-

sengers who board the elevator at node 1 cannot be

traveling to node 4. The boarding and alighting pas-

senger count vectors are formed with respect to the

order of the nodes in the sets N

and N

−

. For ex-

ample, b

= 10, b

= 11, b

= 11, and a

= 11 and

= 21 for Problem 1.

Problem 2, an exactly-determined problem is de-

ﬁned by:

= {1,2,5}, N

−

= {3,4,6},

A = {(1, 3),(1, 4),(2,3),(2, 4),(2,6),(5,6)},

b = [10,11,10], a = [5,10,16],

and Problem 3, an under-determined problem by:

= {1,2,3,4}, N

−

= {5,6,7},

A = {(1,5),(1, 6),(1, 7),(2,5),(2, 6),(2,7),

(3,5),(3, 6),(3,7), (4,5),(4, 6),(4,7)},

b = [6,5,5,5], a = [7, 7,7].

Finally, Problem 4, also an under-determinedproblem

is deﬁned by:

= {1,2,5,6}, N

−

= {2,.. .,16},

A = {(1,2),... ,(1,6),(1, 8),.. .,(1,16),

(2,3),... ,(2,6), (2,7),(2, 8),...,(2, 16),

(5,6),... ,(5,16),(6,7),. ..,(6,16)},

b = [21,1,2,1],

a = [5,1,2,2,2,1,1, 1,2,1,1,2, 2,1,1].

4.2 Numerical Results

The performance of the branch-and-bound algorithm

is studied based on the execution time in seconds. To

account for the machine dependent variations in the

execution time, we solved the ﬁrst, second and fourth

problem 500 times, and the third problem 10 times,

and computed the average execution time of these

runs. The third problem was solved only 10 times be-

cause of a much longer execution time. Table 1 shows

the results, namely, the average execution time in sec-

onds and the number of solutions for each problem.

Table 1: Results of the numerical experiments.

Problem Execution time No. of solutions

1 0.0075 1

2 0.22 5

3 68.6737 2016

4 1.0513 9

It can be seen that as the number of solutions in-

creases, the execution time increases. The shorter the

execution time, the more information about the pas-

senger trafﬁc the elevator group control has at the

moment of making a dispatching decision. Further-

more, the group control has to make a decision typ-

ically in less than 0.5 seconds. Hence, we conclude

that in a real time application, the minimum cost net-

work ﬂow formulation and the branch-and-bound al-

gorithm can be used to ﬁnd all solutions to problems

corresponding to Problems 1 and 2, i.e., over- and

exactly-determined problems. Fortunately, most real

problems are either over- or exactly-determined (Ku-

usinen et al., 2012).

5 CONCLUSIONS

In this paper, we studied the problem of ﬁnding the

origin-destination (OD) passenger counts for the OD

pairs of an elevator trip. An elevator trip is formally

deﬁned as successive stops in one direction of travel

with passengers inside the elevator. We formulated

the problem as a minimum cost network ﬂow prob-

lem. This formulation can be used to estimate the OD

passenger counts for elevator trips where the observed

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

154

number of boarding and alighting passengers are con-

sistent. The elevator trip OD matrices estimated for

a given time interval are combined to form a build-

ing OD matrix that describes the trafﬁc ﬂow between

every pair of ﬂoors in the building during that inter-

val. These matrices form trafﬁc statistics that are used

to forecast future trafﬁc. The elevator group control

uses these forecasts to make robust call allocation de-

cisions in constantly changing trafﬁc conditions.

The coefﬁcient matrix associated with the so

called ﬂow conservation constraints is either over-,

exactly- or under-determined. In the last two cases the

problem has always more than one solution. We pre-

sented a simple branch-and-bound algorithm to ﬁnd

all solutions to the problem. When all solutions are

available and one is selected randomly, the long-term

trafﬁc statistics, i.e., building OD matrices, are not

affected by the algorithm used to solve the problem,

and thus, model better the possible realizations of the

passenger trafﬁc. This is desirable when the statistics

are the basis of the passenger trafﬁc forecasts used

in elevator dispatching. To assess the performance of

the algorithm, we studied its execution time in solv-

ing four example problems. Based on the results, the

formulation and algorithm can be used in a real time

elevator group control application to solve over- and

exactly-determined problems. Fortunately, most real

problems correspond to these problems.

As the results suggest, the execution time is not

acceptable for under-determinedelevator trip OD ma-

trix estimation problems. Hence, an ongoing research

is to ﬁnd more efﬁcient ways to solve the problem.

We are also studying methods that can be used to esti-

mate the elevator trip OD matrices in the presence of

inconsistent boarding and alighting counts. A future

challenge is to ﬁnd out what method to use to make

forecasts based on the collected trafﬁc statistics, and

detect whether the forecasts made for a given time in-

terval are adequate, and thus, can be used by the ele-

vator group control application.

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