A Fuzzy Chance-constraint Programming Model for a Home Health

Care Routing Problem with Fuzzy Demand

Yong Shi, Touﬁk Boudouh, Olivier Grunder

IRTES EA 7274, Universit

e de Bourgogne Franche-Comt

e, UTBM, Rue Thierry Mieg, 90000 Belfort cedex, France

Keywords:

Home Health Care, Fuzzy Chance Constraint Programming, Hybrid Genetic Algorithm, Stochastic

Simulation.

Abstract:

Home Health Care (HHC) companies are widespread in European countries, and aim to serve patients at home

to help them recover from illness and injury in a personal environment. Since transportation costs constitute

one of the largest forms of expenditure in the Home Health Care industry, it is of great signiﬁcance to research

the optimization of the Home Health Care logistics. This paper considers the Home Health Care Routing

Problem with Fuzzy Demand, which comes from the logistics practice of the home health care company. A

fuzzy chance constraint programming model is proposed based on the fuzzy credibility theory, the hybrid

genetic algorithm and stochastic simulation method are integrated to solve the proposed model. Firstly the

uncertain constraints have been reduced to the deterministic ones, experimental results for the benchmark test

problem show the good efﬁciency of the proposed algorithm. Then the proposed hybrid algorithm has been

applied to solve the fuzzy model, the inﬂuence of the parameters to the objective function has been discussed.

This research will help HHC companies to make appropriate decisions when arranging their vehicle routes.

1 INTRODUCTION

Home health care (HHC) is a growing medical ser-

vice in France and other developed countries. This

service is provided by the Home Health Care compa-

nies, which aim to serve the patients at home to help

them recover from illness or injury in a personal envi-

ronment(Liu et al., 2014). Each day, a HHC company

carries out various logistics activities including the

delivery of drugs or medical instruments from the a

pharmacy to patients, and pickup the biological sam-

ples from patients’ home to the laboratory (Liu et al.,

2013). A large number of patients distributed in a

town or village, a certain quantity of the drugs needed

according to the recovery degree of them. For a HHC

company, the transportation cost is one of the most

important spendings in the company activities, so it is

of great signiﬁcance to optimize the vehicle routing

problem in home health care companies.

According to a survey (Mankowska et al., 2014;

Harris, 2015) of the home health care companies, the

main operational process of the HHC can be summa-

rized as 3 steps.

(1) The HHC company collects the information from

the patients, this information may include: the

name, address, sex, type of the illness, symptom

and other related information;

(2) The HHC company plan to arrange the visited

routes and assign nurses according to the infor-

mation collected;

(3) The nurses are scheduled to visit the patients.

Each nurse is assigned to a planned route, and

he/she has to carry out all of the service-related

activities for the route. This nurse will drive the

vehicle to visit the patients one by one according

to the designed route. In case of a lack of drugs,

the nurse has to go back to the depot, load more

drugs into the vehicle and continue to attend to

the remaining patients until all the patients are at-

tended to.

It is easy to ﬁnd that the home health care routing

optimization problem is closely related the Capaci-

tated Vehicle Routing Problem (CVRP) which is one

of the most classical combinatorial optimization prob-

lems (Eksioglu et al., 2009). CVRP is a basic model

in supply chain, and it has been applied into many

ﬁled. However, our problem is neither like the classi-

cal VRP nor like the Open Vehicle Routing problem

(OVRP); all the variations of VRP can be seen in the

literature (Pillac et al., 2013; Toth and Vigo, 2014). In

the classical VRP, each vehicle needs to return to the

depot again, while in the OVRP, each vehicle does not

return to the depot after servicing the last customer on

Shi Y., Boudouh T. and Grunder O.

A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand.

DOI: 10.5220/0006194703690376

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 369-376

ISBN: 978-989-758-218-9

369

a route, but may end at a different location.

Moreover, demand is one of the most important

parameters in supply chain optimization. However

in previous studies, most researchers have focused on

the deterministic demand. While in the real world, it

is usually very hard to determine the precise demands

of customers and thus they are estimated from histor-

ical data. Given this aspect of VRPs, a consideration

of stochastic vehicle routing problems (SVRP) and

fuzzy vehicle routing problems (FVRP) may be use-

ful (Wen and Iwamura, 2008). In the Vehicle Rout-

ing Problem with Stochastic Demand (Bianchi et al.,

2006), the demand is a stochastic variable, which is

decided by probability distribution parameters, but

the parameters are often obtained from the histori-

cal data. However sometimes, we could not obtain

enough historical data for the new customers or pa-

tients. Although stochastic models can cater for a

variety of cases, they are not sufﬁcient to describe

many other situations, where the probability distribu-

tion of customers demands may be unknown or par-

tially known (Wen and Iwamura, 2008). On the con-

trary, fuzzy language exists everywhere in the health

care domain, such as a doctor may inform us “small

penicillin”, or “you have a little fever ”. So, it is very

appropriate to describe the non-deterministic demand

in HHC domain using fuzzy variables.

This paper contributes to the home health care

routing optimization with fuzzy demand in the fol-

lowing aspects: (1) a fuzzy chance constraint pro-

graming model is proposed based on the fuzzy

credibility theory; (2) hybrid genetic algorithm and

stochastic simulation are integrated to solve the pro-

posed model; (3) Some experiments are carried out on

the deterministic model to validate the efﬁciency of

the proposed method, and then the heuristic method

has been employed to solve the fuzzy model. The

rest of this paper is organized as follows: in second

section some necessary theory of credibility is intro-

duced, then the home health care problem with fuzzy

demand model is constructed, and a hybrid heuristic

algorithm is proposed to solve the model, at last some

experiments are presented to illustrate the algorithm.

2 SUPPLY CHAIN MODELING

2.1 The Description of Fuzzy Demand

Constraints

Liu (Liu et al., 2003) recently developed credibility

theory, which can be used to measure the chance of

that a fuzzy chance occurs. The law of credibility in

the theory of fuzzy sets plays a role similar to that

played by the law of probability in measurement the-

ory of the ordinary sets. In this section we will ﬁrst

introduce the credibility theory (Liu et al., 2003), be-

cause it is crucial to describe the fuzzy demand.

In the deterministic VRP, it is straightforward to

describe the capacity constraints: the total demand of

the whole route should not exceed the vehicle capac-

ity. However, in the VRPFD, the capacity constraints

become more complex than the deterministic ones for

the uncertain demand. Now, we have to consider the

relationship between the fuzzy demand and the capac-

ity of the vehicle (Mousavi and Niaki, 2013).

For a vehicle, after serving the j th patient, the

remaining capacity is changed and it becomes a fuzzy

variable named

, where

= q −

∑

i=1

= (q −

∑

i=1

3,i

, q −

∑

i=1

2,i

, q −

∑

i=1

1,i

)

= (Q

1, j

, Q

2, j

, Q

3, j

)

In the deterministic model, if the remaining ca-

pacity of the vehicle is greater than a customer’s de-

mand, this vehicle has the chance to serve this cus-

tomer. However, facing with a fuzzy variable of de-

mand and remaining capacity, how can we make a de-

cision that whether the vehicle should continue visit-

ing the ( j + 1)th patient or go to the lab directly? It

should be compared with the demand of the ( j + 1)

th patient, of course, which is also a fuzzy variable.

Based the credibility we can derive equation (1) and

equation (2) as follows:

Cr = Cr{

j+1

≤

}

= Cr{(d

1, j+1

− Q

3, j

, d

2, j+1

− Q

2,k

, d

3,k+1

− Q

1, j

) ≤ 0};

(1)

Cr = Cr{

j+1

≤

}











0, d

1, j+1

≥ Q

3, j

− d

1, j+1

2(Q

3, j

− d

1, j+1

+ d

2, j+1

− Q

2, j

)

, d

1, j+1

≤ Q

3, j

, d

2, j+1

≥ Q

2, j

3, j+1

−Q

1, j

− 2(d

2, j+1

− Q

2, j

)

2(Q

2, j

− d

2, j+1

+ d

3, j+1

− Q

1, j

)

, d

2, j+1

≤ p

2, j

, d

3, j+1

≥ Q

1, j

1 d

3, j+1

≤ Q

1, j

(2)

There is no doubt that if the quantity of remaining

drugs is very high, and the demand of the next pa-

tient is very low, then the chance of the vehicle of be-

ing able to provide the next patient’s service becomes

greater (Cao and Lai, 2010).

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

370

We will describe the preference index by Cr,

which denotes the magnitude of our preference to

drive the vehicle to the next patient after it has served

the current patient according to formulation (2). Note

that Cr ∈[0, 1]. When Cr = 0, we declare that the ve-

hicle does not have the capacity to serve the next pa-

tient and it should terminate service at the current pa-

tient and return to the depot to replenish drugs. When

Cr = 1, we can be completely sure that the vehicle

should serve the next patient. However, in most cases,

Cr is neither 0 nor 1, but Cr ∈ (0, 1).

To describe Cr in a convenient way, let us in-

troduce the dispatcher preference index DPI, where

DPI ∈ [0, 1]. Note that DPI expresses the dispatcher’s

attitude toward risk. When the dispatcher is not a risk-

averse, he/she will choose lower values of parameter

DPI, which indicates that the dispatcher prefers to use

the vehicle available capacity as much as possible, al-

though there is an increase in the number of cases in

which the vehicle arrives at the next customer’s home

and is not able to carry out planned service due to

small available capacity. On the other hand, when

the dispatcher is a risk-averse, he will choose greater

DPI, this may result in a less complete utilization of

vehicle capacity along the planned routes and less ad-

ditional distance to cover due to failures

2.2 Mathematical Model

In order to describe the supply chain by formulation,

we give the assumption as follows.

(1)The vehicles are homogeneous.

(2)Each nurse responds for one route. In the process

of delivery, if the remaining drugs is not enough for

the patients, she must drive back and ﬁll up the drugs,

return to this patient, then she will continue visiting

the remaining patients until all the patients she re-

sponds are served.

(3)Each vehicle starts from the depot, then visits and

attends to patients, and terminates the journey at the

laboratory.

(4)we assume that the drugs have their own volume,

so the vehicle capacity must be taken into considera-

tion. However, the samples are vials of blood, or tem-

perature record sheets, which could be assumed to be

negligible and will not be considered with respect to

the capacity of the vehicle.

(5)In the process of delivery, the demand of the drugs

is described by fuzzy triangle variable. Only if the

nurse arrives the patient’s home, can she know the ex-

act quantity of the drugs.

Now, we give the mathematical notations as fol-

lows:

V : the set of the vehicles.

N: the set of the all the vertex in the graph, including

the depot, patients,and laboratory.

= (d

1,i

, d

2,i

, d

3,i

): the fuzzy demand of the patient i;

C: the set of the patients.

i, j = 0, 1, 2, . . . , n + 1 is the index of the all the nodes

in the graph. Especially 0 stands for the index of the

depot, n+1 stands for the index of lab, and others are

the patients.

k = 1, 2, . . . , K stands for the index of the vehicle.

q: the capacity of the vehicle.

p: the employee salary for every nurse in this task.

: it is an artiﬁcial variable which is used to construct

the sub-tour constraint.

: the additional distance caused by failure route.

let us describe the mathematical model with the

most common used 3-index method:

i jk

(

1 if the kth vehicle traveals from patient i to patient j;

0 otherwise.

The Home Health Care Routing Problem with

Fuzzy Demand (HHCRPFD) model can be mathe-

matically formulated as shown below:

min f =

∑

i∈V

∑

i∈C

∑

j∈N

i jk

∑

k∈V

∑

i∈N

∑

j∈N

i j

i jk

+ f

(3)

subject to,

∑

k∈V

∑

j∈N

i jk

= 1 ∀i ∈ C (4)

Cr(

∑

i∈C

∑

j∈N

i jk

− q ≥ 0) ≥ DPI ∀k ∈ V (5)

∑

j∈N

0 jk

= 1, ∀k ∈ V (6)

∑

i∈N

ihk

−

∑

j∈N

h jk

= 0, ∀h ∈ C, ∀k ∈ V (7)

∑

i∈N

i(n+1)k

= 1, ∀k ∈ V (8)

i jk

∈ {0, 1}, ∀i, j ∈ N, ∀k ∈ V (9)

− u

+ q ∗

∑

k=1

i jk

≤ q − d

1, j

, ∀i, j = 0, 1, . . . , N + 1, i 6= j;

(10)

The objective function (3) is to minimize the

total cost which includes the nurse employee cost,

planned transportation cost and the additional cost.

The constraints (4) donate that each customer is vis-

ited once and only once, and constraints(5) mean that

no vehicle is loaded with more than its capacity un-

der the fuzzy credibility theory. The constraints (6)

mean each vehicle starts from the depot. Constraints

(7)mean that each vehicle visits the patient and then

A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand

371

leaves the patient. Constraints (8) means that the ve-

hicle ends at the laboratory. Constraints (9) make the

decision-variable are binary. Constraints (10) are used

to eliminate sub-tours.

Our problem is neither like the classical VRP nor

like the Open Vehicle Routing problem (OVRP); all

the variations of VRP can be seen in the literature (Pil-

lac et al., 2013; Toth and Vigo, 2014). In the classical

VRP, each vehicle needs to return to the depot again,

while in the OVRP, each vehicle does not return to the

depot after servicing the last customer on a route, but

may end at a different location.

Remark 1: The mathematical is a typical Fuzzy

Chance Constraint Programming(FCCP), which is a

new branch in the uncertain programming. What’s

more, if we assume d

1,i

= d

2,i

= d

3,i

, the fuzzy

variable become a determinate variable, and the

HHCRPFD is reduced to a vehicle routing problem.

Considering that VRP has been proved to be the np-

hard problem, so there is no wonder that our model is

also a np-hard problem with fuzzy chance constraints.

Remark 2: If d

1,i

= d

2,i

= d

3,i

, this problem becomes

a deterministic model, and in this situation, the addi-

tional distance f

= 0.

3 HYBRID HEURISTIC

ALGORITHM

As mentioned above, the proposed model is a NP-

hard problem, which is difﬁcult to solve by the exact

method when the size of the problem becomes large.

Here, we propose a hybrid heuristic algorithm (HHA)

by integrating the stochastic simulation method and

hybrid genetic algorithm. Generally speaking, in the

ﬁrst stage, we apply the route construction method to

generate initial feasible solutions, then hybrid genetic

algorithm is employed to improve the initial solution.

To accelerate the convergence, elitism selection and

local search are employed, while to make the solution

escape from the local optima in advance, mutation op-

erators and crossover operators are designed. Consid-

ering that our problem is an instance of Fuzzy Chance

Constraint Programming (FCCP), Stochastic Simula-

tion method is designed to evaluate the each solution

candidate.

The detailed description of the HHA can are de-

scribed in the rest part of this section.

3.1 Individual Representation

In our research, each individual stands for feasible so-

lution, which is a vehicle routing arrangement. Here

we use a List to encode an individual which contains a

lot of routes. The process of the encoding and decod-

ing can be seen in Figure 1. The strength of this en-

code method is that each individual does not need to

be decoded and encoded alternately in the optimiza-

tion process.

Depot->2->3->7->5->11->lab

Depot->14->4->6->8->1->lab

Depot->9->10->13->12->lab

Solution

Individual

Figure 1: The representation of a individual.

3.2 Initial Population

The initial population are composed by two kinds,

most of them are ordinary individuals which are gen-

erate randomly, the rest are high-quality ones which

are obtained from the classical heuristic algorithm

called insertion.

Insertion heuristic are widely used to quickly con-

struct a feasible solution (Toth and Vigo, 2014). In

each iteration, a node is selected among all the unvis-

ited nodes, and then insert it in a right position which

makes the new route feasible and least cost, while a

insertion position couldn’t be found, a new route is

started. This insertion process repeated until all the

nodes are visited. Figure 2 illustrates the insertion of

the node k between i and j with the corresponding

insertion cost c

+ c

k j

− c

i j

depot

lab

depot

lab

Figure 2: The brief description of the insert method.

3.3 Fitness Evaluation

As mentioned above, the demand of drugs for each

patient is a triangular fuzzy number, so it cannot be

directly considered as a deterministic one.

Regarding the simulation of the phases of the

operation process of the HHC company and related

works, we derive an approximate estimate about ad-

ditional distances ( f

) due to route failures using a

stochastic simulation algorithm. We summarize the

stochastic simulation (Cao and Lai, 2010) as follows:

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

372

Step 1: For each patient, estimate the additional dis-

tance by simulating “actual” demand. The

“actual” demands were generated by the fol-

lowing process:

(1) randomly generate a real number of x in the

interval between the left and right bound-

aries of the triangular fuzzy number repre-

senting demand at the patient, and compute

its membership u.

(2) generate a random number a, a ∈ [0, 1];

(3) compare a with u, if a ≤ u, then “actual”

demand at the patient is adopted as being

equal to x; in the opposite case, if a < u, it

is not accepted that demand at the patient

equals x. In this case, random numbers x

and a are generated again and again until

random number x and a are found that sat-

isfy relation a ≤ u;

(4) check and repeat (1)–(3), and terminate the

process when each patient has a simulation

“actual” demand quantity.

Step 2: Move along the route designed by credibil-

ity theory and accumulate the amounts picked

up and calculate the additional distance due

to routes failure in terms of the “actual ” de-

mand..

Step 3: Repeat Step 1 and Step 2 for M times.

Step 4: Compute the average value of additional dis-

tance by M times simulation, and it is re-

garded as the additional distance f

In the fuzzy demand model, the objective is to

minimize the total cost, so the ﬁtness value here we

choose to use f =

AD+PD

. Specially, if d

1,i

= d

2,i

3,i

,this problem becomes a determinate model, and

in this situation, the additional distance f

= 0, the

ﬁtness can be described as f =

3.4 Selection

In the evolution process, sometimes the good indi-

viduals may lost due to the crossover and mutation,

which is not favorable to the convergence of ﬁtness

value. To overcome this drawback, we employ the

famous elitism strategy in the selection operator.

Here we pick out the top 2% individuals as elite

which are retained to the next generation directly

without taking part in the crossover, mutation and lo-

cal search operator.

3.5 Crossover

Crossover provides a chance to enhance the commu-

nication between different individuals, and aims to re-

produce new offspring. Ombuki proposed a effective

crossover operator named Best Cost Route Crossover

(BCRC) (Ombuki et al., 2006), the main idea is to

select nodes from one sub-route, and ﬁnd a best po-

sition to reinsert them into the other individual one

by one. We can call this operator as a global version

of BCRC. Although his operator can perform well, it

takes a long time to ﬁnd the best position of whole the

potential position. Inspired by his research, we pro-

posed a local version of BCRC. we insert each node

to the best position of a randomly selected route of the

individual. It is obviously that the local version saves

a lot of computing time.

The main steps of the crossover operator an be

found in Figure 3.

Figure 3: the crossover operator.

Speciﬁcally, for two individuals, the crossover is

undertaken as follows:

Step 1: For each individual, a route is selected ran-

domly. Before inserting the route into the

other individual, the repeating nodes would

be removed.

Step 2: The nodes selected from P1 (P2) are inserted

in to P2 (P1) one by one. Be attention that,

for the insertion of one node, one route is se-

lected randomly. The length of this route is

m, and the possible position for insertion is

m +1, and this node is inserted to the best po-

sition. This insertion is undertake until all the

selected nodes are inserted to the other indi-

vidual. while a insertion position could not be

found, a new route is started.

3.6 Mutation

Mutation is a divergence operation. It is intended to

occasionally break one or more members of a popu-

A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand

373

lation out of a local minimum space and potentially

discover a better minimum space. The mutation op-

erator is conducted by bring random, unrelated traits

into the present population and increase the variance

of the population. According to the characteristic of

individuals, two simple mutation operators are intro-

duced in our algorithm.

• inversion: two cut points are generated randomly,

and all the nodes between them will be inversed.

• single-point mutation: two nodes are randomly

generated, then they are swapped.

3.7 Local Search

In this paper, local search operator is employed to

improve the ﬁtness value of the individuals and ob-

tain better solutions. The most commonly-used 2-opt

method, or-opt method and their extension (Br

aysy

and Gendreau, 2005) are used to search the better so-

lution.

Note that the local search operators are quite not

the same with the mutation operator in two aspects:

(1) the aim of the local search operator is to make

an improvement of the solution, while the mutation

is just to make the population diversiﬁed which aims

to avoid the trapping into local optimal in advance.

(2) The mutation operator is executed just once in one

iteration, while the local search operators are executed

many times, until a solution deemed optimal is found

or a time bound is elapsed.

4 EXPERIMENTAL RESULTS

Because there was no one does the same work with

us, so it can not compare our work with the exist work

directly to validate the efﬁciency of the proposed Hy-

brid Heuristic Algorithm (HHA). Here, we ﬁrstly re-

duced our problem into the classical CVRP, experi-

mental results are compared with the optimal results.

After ensuring the algorithm is effective, we will use

it to solve the proposed fuzzy model. Here we need

to emphasize that the aim of our research is not to

design a new and efﬁcient algorithm to solve the clas-

sical CVRP, but just to design an efﬁcient algorithm

to solve the fuzzy model.

4.1 Experiments on Determinate Model

The fuzzy model is reduced to CVRP, if we assume

the following 3 points: (1) the fuzzy variables re-

duce to the deterministic ones, namely d

i,1

= di, 2 =

i,3

, i = 1, 2, . . . , N; (2) the laboratory is in the same

position with the depot. (3) the cost for each nurse is

0. In this situation, the fuzzy chance constraints be-

come the deterministic ones, additional distance be-

come 0.

Note that, when we apply the HHA to solve

the deterministic model, for there is no chance con-

straints, the additional cost is 0. And in the process of

ﬁtness evaluation, stochastic simulation doesn’t used.

Here we use one of the most famous bench-

mark instances called A series to test the pro-

posed HHA. The instances and the optimal re-

sult can not be downloaded from the web-

site http://neo.lcc.uma.es/vrp/vrp-instances/, and our

computing results and the comparison can be found

in Table 1.

Note that, in Table 1, NO means the ID of the

instance,and the name of instance is composed in 3

parts: for example, the instance named “A-n60-k9”,

“A” means the instance is from A-series, “n60” means

the size of the nodes is 60, and “k9” means that the

number of expected used vehicle is 9. As results

show, TD means the total distance (also called to-

tal cost in some literature), NV means the number

of the used vehicles, CT means the computing time,

and GAP means the percentage of the error between

our result and the optimal result (Juan et al., 2010;

MirHassani and Abolghasemi, 2011).

Table 1: Experimental results for the CVRP model.

HGA optimal result

GAP

No. NV TD CT(s) NV TD

A-n32-k5 5 787.20 10.54 5 784.00 0.41%

A-n33-k5 5 688.11 10.34 5 661.00 4.10%

A-n33-k6 6 745.80 9.56 6 742.00 0.51%

A-n34-k5 5 794.64 10.39 5 778.00 2.14%

A-n36-k5 5 819.93 11.34 5 799.00 2.62%

A-n37-k5 5 673.50 11.98 5 669.00 0.67%

A-n37-k6 6 961.68 19.77 6 949.00 1.34%

A-n38-k5 5 761.40 21.73 5 730.00 4.30%

A-n38-k5 5 845.00 19.84 5 822.00 2.80%

A-n45-k7 7 1216.56 20.47 7 1146.00 6.16%

A-n60-k9 9 1437.48 17.17 9 1408.00 2.09%

B-n31-k5 5 680.96 10.31 5 672.00 1.33%

B-n41-k6 6 875.31 11.02 6 829.00 5.59%

B-n50-k8 8 1373.56 12.09 8 1313.00 4.61%

B-n63-k10 10 1627.00 55.61 10 1537.00 5.86%

B-n78-k10 10 1305.00 66.50 10 1266.00 3.08%

We can conclude that: (1) the number of used ve-

hicle in our results are quite the same with the ex-

pected number; (2) our result is quite close to the op-

timal solution; (3) our result arrives convergence in a

reasonable time even for the big size instance, consid-

ering that CVRP is a NP-hard problem. So there is no

doubt that the proposed hybrid algorithm have a good

performance in solving the CVRP, and we will apply

our heuristic algorithm to the fuzzy chance constraint

programming in the next subsection.

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

374

4.2 Experiments on the Fuzzy Model

In this part, the experiments will be taken on the fuzzy

model. Because there is no corresponding benchmark

for this problem, we adopted the instance from the

exited instance named A-n32-k5, which is a small size

instance. The hybrid heuristic algorithm is encoded in

Matlab 2015b; in the process of computing the ﬁtness

value, the stochastic simulation for every individual is

500 times.

The Value for the Dispatcher’s Preference Index

(DPI) varied with the interval of 0.1 to 1 with the step

of 0.1. The computing results can be seen in Table 2 ,

Figure 4 and Figure 5.

Table 2: Experimental results for HHCRPFD model.

DPI NV TD PD AD

0.1 4 969.17 794.25 174.91

0.2 5 968.11 836.88 131.23

0.3 5 998.85 859.09 139.76

0.4 5 973.45 842.37 131.08

0.5 5 967.03 850.38 116.65

0.6 6 936.16 906.71 29.45

0.7 6 946.87 932.58 14.28

0.8 7 1063.27 1063.27 0.00

0.9 8 1163.66 1163.66 0.00

1 8 1203.94 1203.94 0.00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DPI value

200

400

600

800

1000

1200

1400

Cost

Total Distance

Planned Distance

Additional Distance

Figure 4: Cost changes for different DPI values.

We can ﬁnd that, as DPI rose, the planned dis-

tance is increasing. While the additional distance is

strictly decreasing as DPI value increases from 0 to

0.7. However, when DPI ∈ [0.7, 1], the additional dis-

tance becomes 0, that means there’s no failure route.

The total distance is increasing, but not in a very strict

tendency. We can also ﬁnd that, with the DPI rose,

the number of the needed nurses are increasing. It

concludes that more used nurses can help to decrease

the degree of the failure route.

As a consequence, lower values of parameter DPI

express our desire to use vehicle capacity the best we

can, so less nurses are needed. These values corre-

spond to routes with shorter planned distances. On

the other hand, lower values of parameter DPI in-

crease the number of cases in which vehicles arrive

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

the DPI value

4.5

5.5

6.5

7.5

the number of the used nurses

Figure 5: Number of the needed nurses changes for differ-

ent DPI values.

at a customer and are unable to service it, thereby in-

creasing the total additional distance they cover due

to the “failure”. Higher values of parameter DPI are

characterized by less utilization of vehicle capacity

along the planned routes and less additional distance

to cover due to failures, so more nurses are needed.

Therefore,when a HHC company makes decision on

this scenes (instance), the selected dispatcher prefer-

ence index should be 0.7 approximate.

5 CONCLUSIONS

Since transportation costs constitute one of the largest

forms of expenditure in the Home Health Care in-

dustry, it is of great signiﬁcance to research the op-

timization of the Home Health Care logistics. Based

on a survey of the Home Health Care companies,

the basic operational process illustrates that the de-

mand for the required drugs for each patient is non-

deterministic when the HHC company makes a de-

cision to arrange the vehicle routing. In this paper,

vehicle routing problem with fuzzy demand is con-

sidered, and a fuzzy chance constraint is constructed

based on the fuzzy credibility theory. Stochastic sim-

ulation method and hybrid genetic algorithm are inte-

grated to solve the proposed model.

In order to test the proposed model and algo-

rithm, the fuzzy chance constraints were reduced to

the deterministic ones. Hybrid heuristic algorithm

were applied to solve the benchmark instances, re-

sults show that the proposed hybrid heuristic algo-

rithm perform well. Then the algorithms are applied

to the HHCRPFD, the best Dispatcher’s Preference

Index (DPI) is obtained, and the inﬂuence of the pa-

rameters to the objective functions are analyzed.

In the future, we will consider some other uncer-

tain information in the routing optimization problem,

such as stochastic traveling time, or fuzzy traveling

time. Some other heuristic method like Simulated

annealing (SA), Tabu search (TS), will also be inte-

grated to solve the related problems.

A Fuzzy Chance-constraint Programming Model for a Home Health Care Routing Problem with Fuzzy Demand

375

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