Vehicle routing problem with time-windows for perishable food delivery

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Vehicle routing problem with time-windows

for perishable food delivery

Chaug-Ing Hsu

*

_{, Sheng-Feng Hung, Hui-Chieh Li}

Department of Transportation Technology and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30010, Taiwan, ROC Received 17 November 2004; received in revised form 9 May 2006; accepted 9 May 2006

Available online 7 August 2006

Abstract

This study has extended a vehicle routing problem, with time-windows (VRPTW), by considering the randomness of the perishable food delivery process, and constructing a SVRPTW model, to obtain optimal delivery routes, loads, fleet dispatching and departure times for delivering perishable food from a distribution center. Our objective was to minimize not only the fixed costs for dispatching vehicles, but also the transportation, inventory, energy and penalty costs for violating time-windows. We also discussed time-dependent travel and time-varying temperatures, during the day, modifying the objective functions as well as the constraints in the above mathematical pro-gramming models. Algorithms were developed to solve the proposed models; results indicated that inventory and energy costs can sig-nificantly influence total delivery costs. It was found that our proposed models yielded better results than the traditional VRPTW models.

Keywords: SVRPTW; Perishable food delivery; Soft time-window; Time-dependent travel

1. Introduction

Cold chain distribution is designed to keep temperature-sensitive food products in good condition from point of departure to final destination. Food products often deteri-orate, due to extended travel times and frequent stops to serve customers, during the delivery process. It is, there-fore, difficult to effectively manage cold chain distribution and ensure maximum freshness during hot or humid weather. Perishable food is delivered to retailers, using tem-perature-controlled vehicles; these vehicles have standard cold storage equipment and are usually more expensive, and consume more fuel, than regular vehicles. Due to changeable traffic conditions and the perishable nature of the food, travel time and food’s preservation have inher-ently been characterized as unpredictable. In addition, per-ishable food usually has a short shelf life; thus, timely delivery of perishable food not only significantly affects

the delivery operator’s costs, but also the revenues of retail-ers. Furthermore, the requirement to serving consumers with allowable delivery time-windows can increase the complexity of vehicle routing and scheduling problems for operators.

Perishable food deteriorates as a result of bacteria, light and air; the higher the temperature, the higher the rate of spoilage. In other words, the shelf life of perishable food depends on storage temperature; the lower the tempera-ture, the longer the shelf life. It is critical that perishable food with a short shelf life, such as milk or lunch box items, be delivered in as timely a manner as possible, in order to reduce spoilage. Perhaps the loss in retailers’ revenues ought to be transferred to distribution center operators as a penalty for delayed delivery. Outside temperatures can vary widely during the journey from the distribution center to the ﬁnal destination, with diﬀerent corresponding energy requirements for maintenance of proper tempera-tures. Thus, it is worth quantifying the changes in the qual-ity of perishable food with corresponding time-dependent temperatures.

*

Corresponding author. Tel.: +886 3 5731672; fax: +886 3 5720844. E-mail address:[email protected](C.-I. Hsu).

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Nowadays, many convenience store operators or larger retailers contract with distribution center operators to deli-ver perishable food within allowable delideli-very times, or time-windows. If vehicles arrive after a specified time-win-dow, a penalty cost may be incurred. Therefore, a well designed delivery route will not only ensure delivery of the freshest food, but also satisfy customers’ requirements in a cost-effective and timely manner. Fluctuating temper-atures, at different times of day, can complicate the situa-tion, however. Delivery time may be dependent on traffic conditions; travel time is longer during the rush hour in urban areas. Therefore, the consideration of time-depen-dent travel should be incorporated into determining opti-mal delivery routes, under time-window constraints. It also costs more to keep food from spoilage when a delivery vehicle is stuck in a traffic jam during hot spells. The impact of delays in delivering perishable food, is therefore, more critical than for general goods’ distribution.

Vehicle routing problems (VRP), related to goods

deliv-ery, have been extensively examined (e.g.,Belenguer,

Bena-vent, & Martinez, 2005; Chu, 2005; Daganzo, 1987a, 1987b; Prindezis, Kiranoudis, & Marions-Kouris, 2003;

Tarantilis, Ioannou, & Prastacos, 2005). Chu (2005)

addressed the problem of routing a ﬁxed number of trucks with limited capacity from a central warehouse to

custom-ers with known demand. Prindezis et al. (2003) presented

an application service provider (ASP) to coordinate and disseminate tasks and related information for solving the

VRP using appropriate metaheuristic techniques.

Belen-guer et al. (2005)developed the computer package RutaRep

as a decision support system to automatically generate

delivery routes in the meat industry. Moreover, Tarantilis

et al. (2005)surveyed the research eﬀorts on metaheuristics

solution methodologies for the most widely studied version of the VRP, i.e., the Capacitated VRP. Some researchers have considered vehicle routing problems with time-win-dow constraints, and have constructed penalty costs to

reﬂect violation of these time-windows (e.g., Koskosidis,

Powell, & Solomon, 1992; Sexton & Choi, 1986). These

studies have mainly focused on determining optimal routes by minimizing total routing costs, including total distance and time costs and the cost of waiting, due to a vehicle’s

early arrival (Solomon & Desrosiers, 1988). Taniguchi

and Shimamoto (2004)presented a dynamic vehicle routing

and scheduling model that incorporates real time informa-tion using variable travel times. The results showed that the total cost is decreased by the proposed model based on

var-iable travel times. Gendreau, Laporte, and Se´guin (1996)

reviewed the scientiﬁc literature on stochastic VRP. In the paper, the main problems are described within a broad classiﬁcation scheme and the most important contributions

are also summarized. Laporte, Louveaux, and Mercure

(1992) further examined vehicle routing problems, using

stochastic travel times, by formulating three stochastic

pro-gramming models.Ahn and Shin (1991)considered

tempo-ral issues in routing problems and discussed vehicle routing problems with time-windows, under time-varying

conges-tion. Considering time-varying and stochastic travel time,

Fu (2002)focused on the dial-a-ride paratransit scheduling

problems arising in paratransit service systems that are subject to tight service time constraints and time-varying, stochastic traﬃc congestion.

In another line of research, some studies formulated per-ishable food’s inventory models and discussed optimal

eco-nomic order quantity (EOQ) (e.g., Chakrabarty, Giri, &

Chaudhuri, 1998; Giri & Chaudhuri, 1998; Hariga, 1996).

Tarantilis and Kiranoudis (2001) developed a

threshold-accepting based algorithm to solve the heterogeneous ﬁxed

ﬂeet vehicle routing problem. Tarantilis and Kiranoudis

(2002)proposed an open multi-depot vehicle routing

lem (OMDVRP) to deal with a real life distribution prob-lem in Greece, in which the industry distributed fresh meat.

Burfoot, Reavell, Wilkinson, and Duke (2004) aimed at

estimating the energy savings that could be achieved using localised air delivery system based on experiments. Little has been done, however, to investigate inventory costs due to deterioration of perishable food and energy costs for cold storage vehicles, which are important issues in per-ishable food delivery.

In this study, perishable food were assumed to decrease

in value, throughout their lifetime (Raafat, 1991), and had

to be stored at chilled temperatures; the rate of deteriora-tion, at any moment, being dependent on the temperature. The customers of the distribution centers are retailers sell-ing perishable food to end-users. This study extended the

vehicle routing problem to include time-windows

(VRPTW), by considering the randomness of perishable food delivery process, and constructed a ‘stochastic vehicle routing problem with time-windows’ (SVRPTW) model, to obtain optimal delivery routes and vehicle loads, as well as fleet dispatching and departure times. Our objective was to minimize the fixed costs of dispatching vehicles, transporta-tion, inventory, energy and penalties for violating time-windows. Among these costs, transportation costs are dependent on distance traveled, while inventory costs are related to the deterioration of perishable food; both of these costs can be characterized as stochastic. Energy costs arise from the energy consumption of the vehicles’ cold storage equipment. To reflect the effects of time-dependent temperatures and travel, this study formulated a time-dependent deterioration function for perishable food and calculated the probability of deterioration and the losses involved. We further constructed a penalty cost for violat-ing time-windows, discussed time-dependent travel and time-varying temperatures during the day, and then modi-fied the objective functions, as well as the constraints in the above mathematical programming problem. Algorithms were developed to solve the proposed models and compare the results.

The remainder of this paper is organized as follows.

Section 2 formulates the vehicle routing problem with

time-windows for perishable food delivery. Section3

dem-onstrates development of the algorithms to solve the pro-posed models. A numerical example is provided in

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Section4, to illustrate the application of the models, and the eﬀects of changes in key parameters on the optimal

solutions. In Section5, we oﬀer concluding remarks.

2. Perishable food distribution model

This study focused on the delivery of perishable, temper-ature-sensitive food, from a distribution center, using vehi-cles with frozen storage equipment, to various retailers, in a variety of locations, with delivery time-windows. This study extends traditional VRPTW by giving further consid-eration to the characteristics of perishable food delivery, which include stochastic travel speed due to traffic conges-tion, the perishable features of food within the distribution process, energy consumed by storage equipment and soft time-window constraints. Furthermore, considering the influences of time-dependent temperature and travel, on total delivery costs, we revised the VRPTW according to time-sensitive spoilage rates of perishable food. Due to these perishable features, the amount of food carried by the vehicle decreases, as spoilage increases: these are defined as undeliverable food, due to spoilage. ‘‘Delivery failure’’ is defined as the condition where the customer did not receive the ordered food within the appointed time-window; this situation usually meant higher costs for the operator, who must pay a penalty for the loss incurred by the customer. In this study, an a priori strategy was provided, in response to the spoilage characteristics of perishable food; that is, extra food, added to prevent deliv-ery failure before departure of the vehicle from the distri-bution center, can be assessed and analyzed by the food spoilage rate, passage of time and temperature, during the delivery process.

Following the traditional VRP, this study deﬁned a completely symmetric graph G = (V, A) with node set V = {v0, v1, . . . , vn} and link set A = {(vi, vj) : vi, vj2 V,

i 5 j}. Let v0, vi, and di represent the distribution center,

the location and demand of customer i, i = 1, 2, . . ., n, respectively, where n is the number of customers. Further-more, let l denote the vehicle serving each route, l = 1, 2, . . ., m, where m represents the total number of vehi-cles required for serving all customer demands and is a decision variable in the model. The transportation cost on link (vi, vj) for vehicle l is deﬁned as clij and clij¼ clji.

2.1. Deterministic vehicle travel time

The total delivery costs for VRP, with hard time-win-dows, included ﬁxed costs for dispatching vehicles, trans-portation costs, inventory costs and energy costs. The ﬁxed costs for dispatching vehicles can be expressed as

Pm

l¼1fl, where f l

is the ﬁxed cost for dispatching vehicle l, l = 1, 2, . . . , m. Transportation costs related to distance

traveled were formulated as Pm_l¼1Pn_i¼0Pn_j¼0cl

ijg l

ij, where

gl

ij is an indicator variable; and g

l

ij¼ 1 for vehicle l

travel-ing via link (vi, vj), otherwise, glij¼ 0.

Inventory costs arise from the delivery process of perish-able food, with spoilage increasing as temperatures and time increase. We assumed that the vehicles’ refrigeration equipment was able to ensure optimal temperatures, after which, we concluded that the loss of food was attributable to the time accumulated during the delivery process; this was dependent on vehicle travel time, and the frequency of opening the cargo hold, while unloading perishable food

and serving customers.Fig. 1(a) and (b) show variations in

the remaining food, as time elapsed, in vehicles departing from the distribution center, and in situations both with and without spoilage features, respectively. In these ﬁgures,

yiand uirepresent the vehicle arrival times, and duration,

at customer i.

Considering the situation illustrated in Fig. 1(b), the

total food lost was the sum of the loss resulting from the

(a) Without perishing feature

Food remaining

Time

1

y y2 y3 y4 y5 Time

(b) With perishing feature

Food remaining 1 y y₂ y3 y4 y5 (1) (2) (3) (4)

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vehicle travel time between two adjacent customers, labeled (2), and the loss due to opening the cargo hold at customer stops, labeled (3). The total loss is labeled (1) and the amount of food delivered to customers is labeled (4), in

Fig. 1. Using the total load carried before departure as a

basis for comparison, the amount of loss for the situation considering the spoilage feature, is larger than that without

the spoilage feature, as shown inFig. 1(a) and (b).

Growth of metabolites, during the delivery process, is

characterized as a random variable (Chu, Cheng, & Lee,

1998), which inﬂuences the loss of food, as a function of

delivery time. Let ~bi represent the loss of inventory in the

vehicle from the time of departure from customer (i 1)

to the time customer i, had been served, which includes

vehicle travel time from customer (i 1) to i and serving

time at customer i. Since ~bi depends on the deterioration

function, it is also characterized as a random variable. The cost resulting from the above loss was deﬁned as inventory cost, which also represents the penalty paid for carrying the extra food. Thus, the total expected inventory cost is formulated as PX m l¼1 Xn i¼1 zl_ibi; ð1Þ

where bi denotes the expected loss from departure from

customer (i 1) to ﬁnishing serving customer i, P is the

cost per item of food and zl

i is an indicator; if vehicle l

serves customer i, then zl

i ¼ 1; otherwise, z

l i¼ 0.

When vehicles stop to serve customers, the spoilage rate increases, with the rising temperatures, due to the opening of the cargo hold, with corresponding heat transfer from cold to warm. This study has assumed the loss of food due to opening the cargo hold resulted mainly from the time it was left open, related to the amount of customer demand; that is, duration increased with an increase in

demand. Let G(di) represent the probability that the food

perished, due to an opened cargo hold, being a function

of customer demand di, i = 1, 2, . . . , n; and let d0be the total

amount loaded into the vehicle in the distribution center. Assume the spoilage of food had not yet begun at the

dis-tribution center, thus G(d0) = 0. Let f(y) denote the

proba-bility that the food had spoiled at time y and let F(Æ) be the cumulative probability density function of f(y).

Further-more, let yi, uiand Lldenote arrival time at customer i,

ser-vice time required to serve customer i and the load of vehicle l, respectively. Then, the expected loss for serving

customer i, bi can be formulated as

bi¼ Ll Z yiþui yl s fðtÞ dt þ GðdiÞ " # ; i¼ 1; 2; . . . ; n; ð2Þ where yl

sis the departure time from the distribution center

of vehicle l. As shown in Eq.(2), the ﬁrst term corresponds

to vehicle travel time from the distribution center to the location of customer i plus time spent serving customer i, while the second term is related to the time duration of an opened cargo hold, which further depends on the

de-mand volume of customer i. Let xl

0ibe the binary variable

which represents the relationship between vehicle l and

cus-tomer i; that is, for xl

0i¼ 1, vehicle l is assigned to serve

cus-tomer i, otherwise, xl

0i¼ 0. Since loss does not exist in the

distribution center, that is Fðyl

sÞ ¼ 0, then

Ryiþui

yl

s fðtÞ dt in

Eq.(2)can be expressed in the form of a cumulative

prob-ability density function F(Æ) as xl_0ibi¼ xl0iL l_{½F ðy} i y l sþ uiÞ þ GðdiÞ; i¼ 1; 2; . . . ; n; l ¼ 1; 2; . . . ; m: ð3Þ

Without loss of generality, the loss for serving customer i can also be modiﬁed in terms of the load of food after

serv-ing customer (i 1) and time spent from the point of

departure from customer (i 1) to ﬁnishing serving

cus-tomer i, which yields: xl_ði1Þibi¼ xlði1ÞiL l ði1Þ ½F ðyi y l sþ uiÞ F ðyði1Þ y l s þ uði1ÞÞ þ GðdiÞ; i¼ 2; 3; . . . ; n; l¼ 1; 2; . . . ; m; ð4Þ where Ll

ði1Þ is the load of vehicle l after serving customer

(i 1) and xl

ði1Þi represents the relationship between

vehi-cle l and customer i, i = 2, 3, ., n. The load of food in vehivehi-cle

l, after serving customer i can be calculated as

Ll i¼ L

l

ði1Þ bi di, which is constrained to be larger than

or equal to zero, or else customer i cannot be assigned to vehicle l.

Regarding energy costs, thermal load originates from the sun’s radiation heating the ground and warming the air; heat conduction results from the diﬀerence in tempera-ture between the inside and outside of the cargo hold

(MOEA-IDB, 2001). In practice, the thermal load,

result-ing from thermal convection by openresult-ing the cargo hold,

can be calculated as follows (MOEA-IDB, 2001):

Q_s¼ ð0:54Vlþ 3:22ÞðTO TIÞ b; ð5Þ

where Qsis the thermal load per hour (kcal/h), Vlis the

vol-ume of the cargo hold, TOand TIrepresent outer and inner

temperature and b is an indicator, reﬂecting the frequency of opening the cargo hold, respectively.

This study assumed homogenous vehicles, shipping the same kinds of food, so the volume and inner temperature of the cargo hold was the same for all vehicles. Symbol b represents the frequency of door openings and relates to the demand and spatial pattern of customers. This study also assumed that the operator had planned delivery routes in advance, so as to satisfy the time-constraints agreed to with customers. Therefore, the frequency of opening each vehicle’s cargo hold can be represented by its expected

value, b. Then, Eq.(5) can be written as follows:

Q_s¼ asbðTO TIÞ; ð6Þ

where asis a constant and equals (0.54Vl+ 3.22) in Eq.(5).

Furthermore, this study assumed that the outside tempera-ture was known; thus, the thermal load can be further sim-plified as a constant, i.e., the energy loss of opening the cargo hold, per hour, is fixed. For an operator with specific

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customers, if the demand pattern is ﬁxed, then the energy cost of each vehicle, due to opening the cargo hold, is merely a function of total travel time and time serving cus-tomers. In practice, thermal conduction, due to the diﬀer-ence between the cargo hold’s inside and outside temperatures can be estimated as

Q_T¼ U ffiffiffiffiffiffiffiffiffiffiAIAO

p

ðTO TIÞð1 þ qÞ; ð7Þ

where QTrepresents thermal load per hour (kcal/h), U

de-notes the conductivity of the cargo hold (kcal/h m2C),

while AI and AO represent the surface area of the inner

and outer cargo hold, respectively, and q denotes the de-gree of inferior quality of the cargo hold. We have ignored the impact of the cargo hold on conductivity, the surface area and degree of inferior quality on the thermal load. Therefore, thermal load is mainly dependent on the tem-perature diﬀerence between the inside and the outside of

the cargo hold. Thus, Eq.(7)is expressed as

Q_T¼ aTðTO TIÞ; ð8Þ

Similarly, the thermal load is a constant under the given outside temperature; total energy loss, due to thermal con-duction during the delivery process, is then dependent on total travel time. The energy cost of the vehicle depends on energy loss, energy cost per kcal and total travel time. Total energy cost for all vehicles can be expressed as

qX m l¼1 ½aðyl f y l sÞ; ð9Þ where yl

f denotes arrival time at the distribution center

after the delivery process for vehicle l, a is the thermal load,

a¼ ðasbþ aTÞðTO TIÞ and q represents the energy cost

per kcal. Let q denote the energy cost per hour, q¼ qa,

and then Eq.(9) can be simpliﬁed as qPm_l¼1ðyl

f y l sÞ.

From the discussions above, the VRPTW for perishable food delivery can be formulated as follows:

Min gl ij;yi;yls;ylf;b l_;zl i;m Xm l¼1 fl_þX m l¼1 Xn i¼0 Xn j¼0 cl ijq l ijþ P Xm l¼1 Xn j¼1 zl jbj þ qX m l¼1 ðyl f y l sÞ; ð10aÞ s:t: X m l¼1 zl_i¼ m i¼ 0; 1 i¼ 1; . . . ; n; ð10bÞ Xn i¼0 gl ij¼ z l j; j¼ 0; . . . ; n; l ¼ 1; . . . ; m; ð10cÞ y_ðiþ1ÞP yiþ uiþ tliðiþ1Þ ð1 xliðiþ1ÞÞM;

i¼ 1; . . . ; n; l¼ 1; . . . ; m; ð10dÞ y_iP yl sþ t l 0i ð1 x l 0iÞM; i¼ 1; . . . ; n; l¼ 1; . . . ; m; ð10eÞ

yl_f P y_ðiþ1Þþ uðiþ1Þþ tlðiþ1Þ0 ð1 x l ðiþ1Þ0ÞM; i¼ 1; . . . ; n; l¼ 1; . . . ; m; ð10fÞ ri6yi6si; i¼ 1; . . . ; n; ð10gÞ Ll_¼X n i¼1 zl idiþ bl6Kl; l¼ 1; . . . ; m; ð10hÞ xl_0ibi¼ xl0iL l_{½F ðy} i y l sþ uiÞ þ GðdiÞ; i¼ 1; . . . ; n; l¼ 1; . . . ; m: ð10iÞ

Eq.(10a)is an objective function that minimizes the sum of

ﬁxed costs for dispatching vehicles, transportation costs,

inventory costs and energy costs. Eqs. (10b)–(10h) are

constraints as described in the VRPTW formulations. Eq.

(10i) expresses the loss of food for serving customer i for

vehicle l. The decision variables are gl

ij; yi; yls; y l f; b l_{; z}l i and m.

That is, the operator can apply the model to optimally decide the vehicle delivery route, arrival time of each vehicle for serving each customer, departure time and return time of vehicles, the extra vehicle load, customers served by each vehicle, and the size of the delivery fleet. A trade-off rela-tionship exists between transportation and inventory costs. That is, a larger fleet incurs a higher fixed cost for dispatch-ing vehicles, but assigned customers and routdispatch-ing time are less for each vehicle within the fleet, thereby resulting in less inventory costs and less extra loads.

2.2. Stochastic vehicle travel time

The discussions in Section 2.1 dealt with deterministic

travel time; however, vehicle travel speed may be aﬀected by factors such as traﬃc volume, the weather and acci-dents, which are characterized as random in nature. Let ~tl

ij denote travel time on the link (vi, vj) for vehicle l. This

study adopted travel time, based on Lambert, Laporte,

and Louveaux (1993). Let A and A0_{denote the sets of links}

without traﬃc congestion and with the probability of traﬃc

congestion, respectively, A A0_{. For every link (v}

i, vj)2 A,

travel time on link (vi, vj) can be expressed as tlij¼ b0clij,

where b0 is a parameter. Assume some links of A0 have

the probability p of being congested. For every link ðvi; vjÞ 2 A0; tlijis equal to b1clijwith probability p and equal

to b2clij with probability (1 p), where b26b1. Then,

expected travel time on link (vi, vj) for vehicle l, tlij, is given

by tl ij¼ b₀cl ij if ðvi; vjÞ 2 A; ½pb1þ ð1 pÞb2clij if ðvi; vjÞ 2 A0; ( i¼ 1; . . . ; n; j¼ 1; . . . ; n; l¼ 1; . . . ; m; ð11Þ where cl

ij represents travel costs on link (vi, vj).

Because of the randomness of travel time on links, arri-val time at each customer location is also characterized as a random variable. Since the real-time traﬃc conditions for every link is unknown before departure from the distribu-tion center, arrival time at each customer is diﬃcult to pre-dict. Rather than trying to identify this uncertain travel

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time, we have employed the expected value of arrival time

and revised Eqs.(10d)–(10f), which yield:

y_ðiþ1ÞP yiþ uiþ tliðiþ1Þ ð1 xliðiþ1ÞÞM;

i¼ 1; . . . ; n; l¼ 1; . . . ; m; ð12Þ y_iP yl sþ t l 0i ð1 x l 0iÞM; i¼ 1; . . . ; n; l ¼ 1; . . . ; m; ð13Þ yl_f P y_ðiþ1Þþ uðiþ1Þþ tlðiþ1Þ0 ð1 x

l ðiþ1Þ0ÞM;

i¼ 1; . . . ; n; l¼ 1; . . . ; m: ð14Þ

Due to stochastic travel time, the time-window constraint

of Eq. (10g) can be revised according to Lambert et al.

(1993)as ðri ylsÞ p b1 b2 þ ð1 pÞ þ yl s6yi 6_ðs_i_yl sÞ p þ ð1 pÞ b₂ b1 þ yl s; i¼ 1; . . . ; n: ð15Þ Sinceb1

b2>1, the left hand side of Eq.(15)is larger than the

lower bound of time-window, ri; the right hand side of Eq.

(15)is also smaller than the upper bound of time window,

si. Therefore, the duration of the time-window, considering

stochastic travel time, is narrower than when no consider-ation was given. Moreover, the equconsider-ations involved with ar-rival time and travel time have been revised accordingly;

i.e., the loss of food during delivery process as Eq. (10i);

and the inventory and energy cost in the objective function

as Eq. (10a).

2.3. Relaxation from hard windows to soft time-windows

The time-window constraints, discussed in Section 2.1,

were ‘‘hard’’ constraints, which cannot be violated. These hard time-window constraints increase the complexity of determining optimal delivery routing, however. In contrast, in the case of soft time-windows, constraints can be vio-lated, but with a penalty cost. When a vehicle arrives early, or with an acceptable delay, the food can still be delivered,

with a penalty cost. The relationship between penalty cost

and arrival time can be seen inFig. 2(a).

Let R and S denote the earliest acceptable time for early arrival and the latest acceptable time for late arrival, R 6 r

and S P s, respectively. As shown inFig. 2(a), the

accept-able periods for early arrival and delay are [R, r) and (s, S], respectively; within each range, there are diﬀerent penalties. When arrival time is beyond [R, S], customers may refuse to receive the food, and a large M, representing a huge pen-alty, has been introduced to avoid this occurrence. When early arrival lies within [R, r), the operator must decide whether to immediately serve the customer or wait until time r. In practice, the increased cost resulting from waiting until the beginning of the time-window is very low; this is because the diﬀerence between the earliest acceptable time, R, and the beginning of the time-window, r, is usually rel-atively small. Therefore, this study assumed the operator would rather wait and serve on time, since the increased cost is negligible; consequently, R is approximated to r, similar to hard time-window constraints.

To avoid double counting, the penalty cost for violating the time-window was considered as the revenue lost due to late delivery. The probability, that perishable food can be sold, depends on the time between purchasing and expira-tion date; this probability decreases, at an increasing rate, as the time of purchase nears the expiration date. Thus, customer revenue may be reduced due to late delivery. The penalty cost, due to violating the upper bounds of

the time-window, speciﬁed by customer i, si, can be

formu-lated as g(yi si)t· P · di, where t and g represent

param-eters, and t > 1. Substituting k for g· P, the penalty cost

can be simpliﬁed as kdi(yi si) m , and is given by Q_iðyiÞ ¼ M ; y_i< ri 0; ri6yi6si kdiðyi siÞ v ; si< yi6Si M ; y_i> Si 8 > > > < > > > : 9 > > > = > > > ; ; i¼ 1; . . . ; n; ð16Þ

where Qi(yi) represents the penalty cost of customer i, and

is a function of vehicle arrival time at customer i, yi. The

R r s S

M

0 _Time

Penalty cost

Fig. 2(a). The relationship between arrival time, time-windows and penalty cost. M Time R r s S Penalty cost 0

Fig. 2(b). The revised relationship between arrival time, time-windows and penalty cost.

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revised relationship between arrival time, time-window and

the penalty cost is illustrated inFig. 2(b).

The vehicle routing problem with soft time-window con-straints (VRPSTW) discussed above can be formulated as follows: Min gl ij;yi;yls;ylf;b l_;zl i;m Xm l¼1 fl_þX m l¼1 Xn i¼0 Xn j¼0 cl ijq l ijþ P Xm l¼1 Xn j¼1 zl jbj þ qX m l¼1 ½aðyl f y l sÞ þ k Xn i¼1 di½ðyi siÞþ m ; ð17Þ s:t: ð10bÞ–ð10fÞ; ð10hÞ and ð10iÞ; ri6yi6Si; i¼ 1; . . . ; n; ð18Þ where (yi si)+= max{0, (yi si)}.

2.4. Time-dependent temperatures and vehicle travel time This study further relaxes the assumption of a constant

temperature, discussed in Section2.1. Let H(y), DH(y) and

H0denote the temperature at time y, the diﬀerence in

tem-perature between the outer and inner cargo hold at time y and the optimal inner temperature, required to keep the

food fresh, respectively, where DH(y) = H(y) H0. The

spoilage rate, due to an opened cargo hold to serve

cus-tomer i can, thus, be revised as G0_(d

i) = g(di)DH(y), where

g(di) is the average rate of spoilage per unit temperature

diﬀerence and is a function of the demand of customer i, di.

As stated, the total energy cost arises from the thermal load, due to an opened cargo hold and travel time; both of these inﬂuences depend on the diﬀerence between

out-side and inout-side temperatures, as shown in Eqs. (6) and

(8), respectively. Then, the energy cost of vehicle l, during

one routing period, from distribution center departure to

return, can be formulated as qRy

l f

yl s a

0_DH_{ðyÞ dy, where a}0

denotes the energy loss per hour per unit temperature dif-ference. The energy cost per hour, under one unit temper-ature diﬀerence, can be further expressed by energy loss per

hour under one unit temperature diﬀerence, a0_{, and energy}

cost per kcal, q, that is, q0_{¼ qa}0_{. Moreover, the total energy}

cost of m vehicles can be expressed as

q0Pm_l¼1 Ry l f yl s DHðyÞ dy

. To show how time-dependent traﬃc can aﬀect travel time on a link, this study considered travel

time on link (vi, vj) as a function of entering time on link

(vi, vj), y0i, that is, t l

ijðy0iÞ. If traﬃc is heavy on link (vi, vj) at

time y0

i, more time is spent navigating that link.

3. Algorithm

The VRP inherently belongs to the NP-hard problem (Golden & Assad, 1988) and a VRP with a hard

time-win-dow is more complex than the simple VRP (Solomon,

1987). This study adopted a heuristic method, which

extended the ‘‘Time-Oriented Nearest-Neighbor Heuristic’’

bySolomon (1983). The heuristics for the VRP with a hard

time-window consist of the following steps:

Step 1. Input basic data, such as demand, supply parame-ters and network, G = (V, A).

Step 2. Denote the distribution center as the beginning of a route.

Step 3. Determine the customer closest to the last cus-tomer added to the route.

Step 4. Repeat Step 3 until the vehicle is ﬁlled to capacity. Step 5. Assign another vehicle and repeat Step 2 until all

customers have been served.

The details for Step 3 are described as follows:

Assume customer i is the first customer added to the route and customer (i + 1) represents the next customer added to the route. Two constraints must be satisfied before the closest customer can be determined: (1) the time-window constraint, specified by customer (i + 1); (2) the food remaining in excess of the demand of customer (i + 1). In this study, the factors determining whether the closest customer can be added to the route included the

demand of customer (i + 1), d(i+1), travel costs from

cus-tomer i to cuscus-tomer (i + 1), hi(i+1), time duration between

ﬁnishing serving customer i to arrival at customer (i + 1), Dyi(i+1), and duration from the end of the time-window

of customer (i + 1) to the earliest service time for customer

(i + 1), ai(i+1). Among these factors, hi(i+1)was classiﬁed as

the spatial distance factor, while Dyi(i+1)and ai(i+1)are time

distance factors. The deﬁnitions of Dyi(i+1) and ai(i+1) are

Dyi(i+1)= y(i+1) (yi+ ui) and ai(i+1)= s(i+1) (yi+ ui+

ti(i+1)), respectively, where y(i+1)represents arrival time at

customer (i + 1). Then, the cost function determining the closest customer can be formulated as

C_iðiþ1Þ¼ d1hiðiþ1Þþ d2Dyiðiþ1Þþ d3aiðiþ1Þþ d4dðiþ1Þ; ð19Þ

where Ci(i+1)is the cost function for customer (i + 1), while

d1, d2, d3and d4express the weights of these inﬂuences, and

represent the marginal cost of the objective function in the constructed model with respect to the addition of one unit of hi(i+1), Dyi(i+1), ai(i+1) and d(i+1), respectively. The

cus-tomer with the smallest Ci(i+1) is closest. Note that

d1+ d2+ d3+ d4= 1 and d1P0, d2P0, d3P0, d460.

The weights of d1, d2, d3and d4can be further evaluated as:

(1) Travel cost from customer i to (i + 1), hi(i+1). An

increase of one unit of hi(i+1) means customers i and

(i + 1) are one more distance apart, and the addition of customer (i + 1) will further increase the total delivery cost

by d1units.

(2) Time duration from ﬁnishing serving customer i to

arrival at customer (i + 1), Dyi(i+1). The costs incurred by

one more unit of Dyi(i+1) can be divided into energy and

inventory costs. Suppose there is an increase of one unit of Dyi(i+1), meaning that customer (i + 1) and i are one more

time-distance apart; the energy costs are increased by d units. Moreover, the increased inventory cost is brought about by the loss of food due to one additional unit of travel time. Let

(8)

lrepresent the average life of the perishable food: the

aver-age spoilaver-age rate is 1/l. And, let Ll_i denote the food

remain-ing after servremain-ing customer i usremain-ing vehicle l. The increased inventory costs, due to an increase of one unit in travel time

can be formulated as p Ll

i=l. One procedure in the

algo-rithm is the addition of customers into the routes, one after the other, implying that the remainder of food, after serving customer i is known before construction of the route has

been completed. Let /ibe the total amount of food delivered

after serving customer i, then the remainder of food becomes

Kl /i. In sum, the total delivery costs are increased by

a+ p(Kl /i)/l due to one additional unit of Dyij.

(3) The time duration from the end of the time-window of customer (i + 1) to the earliest service time for customer

(i + 1), ai(i+1). Factor ai(i+1)represents the inﬂuence

result-ing from the order of the customers’ time-windows on the delivery routes. Because of the order of the time-windows, adding customer (i + 1) may mean that other customers

cannot join the route. Only if ai(i+1)P0, will the

time-win-dow of customer (i + 1) be satisﬁed and customer (i + 1) will be considered to be added into the route.

(4) Demand of customer (i + 1), d(i+1). Once customer

(i + 1) has been added into the route, the load carried by

the vehicle is increased by one additional unit of d(i+1)

and the inventory cost, per unit, reduced accordingly. The decrease in total delivery cost due to the increase of

one unit, d(i+1), can be estimated by the purchasing cost

per unit of food, and the average loss of food per hour, due to serving customer (i + 1).

The modifications in Step 2 of the proposed heuristics were necessary to solve the VRP models with soft time-win-dows, time-dependent temperatures and vehicle travel times. To reflect the impact of soft time-window customer constraints in determining the delivery routes, the time duration from the end of the time-window of customer (i + 1) to the earliest service time for customer (i + 1), a0_iðiþ1Þ, was modified as a0

iðiþ1Þ¼ Sðiþ1Þ ðyiþ uiþ tl_iðiþ1ÞÞ P

0, where S(i+1)denoted the end of soft time-window of

cus-tomer (i + 1). The condition a0

iðiþ1Þ<0 holds for models

with soft time-windows, and the penalty cost, due to adding customer (i + 1) into the route, with a violation of the

time-window, is kdðiþ1Þa0m_iðiþ1Þ, as shown in Eq.(16). The marginal

cost of the objective function in the constructed model, with

respect to the additional unit of a0

iðiþ1Þ, can then be

formu-lated as kdðiþ1Þa0m1iðiþ1Þ. In practice, the operator will try to

avoid an enormous penalty cost; kdðiþ1Þa0miðiþ1Þwas assumed

to be ten percent, or less, of the ﬁxed cost fl. In response

to time-dependent temperatures and travel time, related parameters and variables of the cost function in Step 2 must be revised accordingly, i.e., using time-dependent penalty

cost, a0DH(y), and time-dependent travel speed, s tlij

tl ijðy 0 iÞ . 4. Numerical example

This section presents an application of the proposed models, using a numerical example. A rectangular grid

net-work was used in this study, with one depot at coordinate (0, 0), representing the distribution center, which dispatches vehicles to deliver lunch box items to local retail customers. The study covered an area of ﬁfty square kilometers and comprised a random extraction of the characteristics of ﬁfty customers, which included locations, time-window constraints and demand; customers’ time-windows were randomly generated between 6:20–11:00 a.m. and customer demand ranged between ten items to items making up one-quarter of vehicle capacity. Identical customer time-win-dow duration, i.e., one half hour, was assumed, in order to simplify the problem, while service times for customers

were demand-dependent, i.e., ui¼₂₀di (min). The life cycles

of the lunch box items, and the required temperature to

preserve these items, were 24 h and 18 C, respectively.

For simpliﬁcation, the perishable rate of the lunch box items was estimated by the reciprocal of the life cycle, which was a constant, under the appropriate temperature; in addition, the cumulative probability function, F(Æ), can

be simpliﬁed as Dy

1440, where the denominator represents

the life cycles of the lunch box items (min). We further assumed the spoilage of food, due to the opened cargo hold, was double that due to vehicle travel time: i.e., GðdiÞ ¼₁₄₄₀ui ¼_28;800di . Base values for the parameters in the

total delivery cost function, and time-window constraints, were estimated by interviewing the distribution center

oper-ator, as listed inTable 1.

For comparisons, the traditional VRPTW, giving no consideration to either energy or inventory costs, was applied here, in order to determine optimal delivery routes,

where three sets of weights, i.e., (d1, d2, d3) = (0.5, 0.5, 0),

(0.4, 0.4, 0.2), (0.3, 0.3, 0.4) were employed to explore changes to these optimal solutions, due to variations in

key parameters.Table 2shows the results for a basic model

with deterministic and stochastic travel time, respectively, together with models, both with and without consideration being given to energy and inventory costs, using various

values of d3. The total inventory cost was divided into:

(1) inventory cost due to vehicle travel time; (2) inventory

cost due to opening of the cargo hold as shown inTable 2.

Table 1

Initial values of parameters

Symbol Deﬁnition Initial value

s Average travel speed (km/h) 30 fl _{Fixed cost for dispatching vehicles}

(NT $/vehicle)

750 P Purchasing cost per item (NT $/item) 50

q Energy cost per hour (NT $/h) 30 Kl Vehicle capacity (items) 300 F(Æ) Cumulative probability density function of f(y) ₁₄₄₀Dy G(di) The probability that the food is

perished due to an opened carriage

di 28;800

p The probability that the link of A0_{is congested} _0.5

b0 1/100

b1 1/100

b2 1/120

(9)

Table 2, compares diﬀerent costs, using percentage of total delivery cost. Total inventory cost accounts for the highest percentage, i.e., 33%, with inventory cost due to routing time with time-dependent vehicle travel time and inventory cost due to opening the cargo hold, accounting for approximately 29% and 4%, respectively. The percent-age total, of both inventory and energy costs, was 37%.

Parameter d3represents the weight that the operator placed

on customers’ time-window constraints during the delivery route design, regarding sequence of service. With a larger

value of d3, there were less vehicles required, as well as a

lower ﬁxed cost for dispatching vehicles. On the other hand, without considering the order of time-windows,

i.e., d3= 0, more vehicles would have to be dispatched;

however, inventory costs, due to vehicle routing time,

was higher for models with a higher value of d3 than for

those with a smaller value of d3, which shows that a

trade-oﬀ relationship exists between inventory costs and total costs for dispatching vehicles. An appropriate value

of d3may not only reﬂect the impact of time-window

con-straints on the service sequence of customers, but may also

result in the lowest total delivery costs. As shown inTable

2, d3= 0.15 and (d1, d2, d3) = (0.4, 0.4, 0.2) yielded the

low-est total delivery costs for the deterministic models, with and without consideration being given to energy and inven-tory costs, respectively.

As for the traditional VRPTW with no consideration for energy and inventory costs, transportation and ﬁxed costs, for dispatching vehicles, were the most inﬂuential factors in total delivery costs. However, the revised results, after applying the proposed model in this study, i.e., Eqs.

(1) and (9), show that total energy and inventory costs

account for a signiﬁcant percentage of the total costs, i.e., 37%, which implies that the delivery route, using the traditional VRPTW, is not optimal, since neither energy nor inventory costs, critical to delivery of perishable food, were considered. Furthermore, the average total delivery costs, using models with no consideration given to energy

and inventory costs was NT $15,440, which is greater than for those models which took these costs into consideration, i.e., NT $15,116. Further comparison of models with deterministic and stochastic travel time, where energy and inventory costs were both considered, showed that total delivery costs were higher for the model with stochastic vehicle travel time, than for the model with deterministic vehicle travel time. This ﬁnding implies that time-window constraints are more rigid, with respect to stochastic travel time, with more vehicles required to sat-isfy customers’ demand. In addition, the inventory, energy and transportation costs, for models using stochastic travel time, were higher than for those using deterministic travel time.

Table 3 shows the results of revised models with soft

time-windows and time-dependent temperature and travel.

As shown inTable 3, penalty costs arose, due to violations

of time-window constraints, which were relaxed here, as soft time-windows. Fixed costs for dispatching vehicles, using the model with soft time-windows were less than for the model which considered hard time-windows for

d3= 0.15, shown inTable 2, respectively; this indicates that

the release of hard time-window constraints resulted in a smaller number of vehicles being required to serve custom-ers. However, a delay in delivery may result in higher inventory costs than when delivery is achieved within time-window constraints. The models, which considered time-dependent travel and temperature, captured the impact of variations in these functions, at diﬀerent times, on optimal decisions; therefore, total delivery costs as well as inventory, energy and transportation costs were lower in these models, than in those which did not consider

time-dependent travel and temperatures. Table 3 also shows

the service sequence of vehicle routing and cost compo-nents related to each vehicle for models with soft time-win-dows and time-dependent temperature and travel time, where the parentheses denote customer demand, and aster-isks show that customer time-windows have been violated,

Table 2

Results from basic models with hard time-windows (unit: NT $)

Parameter Inventory cost Energy cost Transportation cost

Fixed costs for dispatching vehicles Total delivery cost (1) (2) Deterministic travel time

With considering the loss of food and energy cost

d3= 0 3600 500 549 1530 9000 15,180

d3= 0.15 4100 550 556 1557 8250 14,963

d3= 0.3 5050 450 568 1639 7500 15,206

Without considering the loss of food and energy cost

(d1, d2, d3) = (0.5, 0.5, 0) 3550 550 585 1597 9000 15,281 (d1, d2, d3) = (0.4, 0.4, 0.2) 4350 650 624 1397 8250 15,270 (d1, d2, d3) = (0.3, 0.3, 0.4) 5550 650 512 1557 7500 15,769 Stochastic travel time d3= 0.15 5350 500 708 1750 9000 17,308 Percentage of the total delivery cost

(10)

respectively. Ten customers were identified as having had their time-window constraints violated, with the operator having to pay a small penalty, i.e., NT $178; this shows that most customers can put up with time-window viola-tions. The VPRSTW is capable of finding solutions in cases where a hard time-window formulation would fail. Prob-lems resulting from tight time-windows and a small fleet may not be able to satisfy all customers, while a small fleet may not be able to satisfy all customers on time. In this case, the VRPSTW would yield a solution where some of the customers would not be serviced on time. Naturally, this solution is not feasible for hard time-window models, but the operator would, at least, have a solution at hand. This solution can be either accepted as is, or can be improved by adjusting the appropriate time-windows, to produce routes to service more, or all of the customers, on time. The VRSPTW solution can provide ample infor-mation on the customers to whom the schedule is infeasi-ble; the ‘‘trouble maker’’ can be easily identified from the

routes at hand (Koskosidis et al., 1992).

5. Conclusions

This study has focused on determining the optimal delivery routing, loads and departure times of vehicles, as well as the required number of vehicles for delivering per-ishable food to many customers, from a distribution center. Features related to delivery of perishable food were consid-ered, such as the time-window constraints of customers and the stochastic characteristics of travel time and food’s pres-ervation. Models, using stochastic vehicle routing problems with time-windows, for perishable food, were constructed using mathematical programming methods. Time-depen-dent temperatures and travel time, and soft time-windows with penalty costs, were further discussed and the objective functions, as well as the constraints, in the mathematical programming models were modiﬁed, accordingly.

The results showed that the sum of inventory and energy costs constitutes a significant percentage of total costs, which cannot be ignored. We also discovered a trade-off relationship between the fixed costs of dispatching vehicles

Table 3

Results from revised models with soft time-windows and time-dependent temperature and travel (unit: NT $) Inventory cost Energy cost Penalty cost

Transportation cost Fixed costs for dispatching vehicles Total delivery cost (1) (2) Revised modela ₅₀₅₀ ₄₅₀ ₆₁₉ ₁₀₃ ₁₈₃₄ ₇₅₀₀ _15,453 Revised modelb ₄₅₀₀ ₅₀₀ ₅₃₈ ₁₇₈ ₁₄₂₆ ₆₇₅₀ _13,713 Route sequences 1 0! 35 ! 40 ! 1 ! 30*_{! 24 ! 50} ₆₀₀ ₅₀ ₆₇ ₁₆ ₁₈₄ ₇₅₀ ₁₆₅₁ (16)(26) (56) (32) (14) (42) !10 ! 49 ! 46 ! 23 ! 0 (21) (24) (13) (35) 2 0! 32 ! 5 ! 42 ! 28 ! 25! 750 50 63 0 168 750 1781 (54) (43) (27) (49) (27) 16! 0 (74) 3 0! 7 ! 0 50 0 14 0 82 750 896 (31) 4 0! 15 ! 11*_{! 6}*_{! 12}*_{! 13!} ₃₀₀ ₁₀₀ ₄₉ ₃₃ ₁₃₈ ₇₅₀ ₁₃₃₇ (58) (63) (74) (46) (22) 36! 0 (18) 5 0! 43 ! 20 ! 14*_{! 27 ! 37 !} ₆₀₀ ₁₀₀ ₆₄ _0.4 ₁₁₅ ₇₅₀ ₁₆₂₉ (65) (60) (41) (36) (58) 26! 0 (23) 6 0! 33 ! 31 ! 38 ! 3 ! 47 ! 2 ! 550 50 69 52 196 750 1615 (16) (17) (61) (31) (66) (32) 39! 29*_{! 0} (22) (42) 7 0! 48 ! 21 ! 4*_{! 41}*_{! 44!} ₉₅₀ ₅₀ ₉₇ ₇₆ ₂₅₃ ₇₅₀ ₂₁₀₀ (36) (50) (25) (39) (21) 17*_{! 19}*_{! 0} (24) (35) 8 0! 18 ! 34 ! 0 200 0 48 0 105 750 1103 (54) (51) 9 0! 9 ! 45 ! 22 ! 8 ! 0 500 100 66 0 186 750 1602 (67) (72) (61) (71)

Note: Parentheses denote customer demand and customers with asterisks show that the time-window speciﬁed by that customer is violated.

a

Model with soft time-windows.

b

(11)

and inventory costs, showing that delivery, using a smaller number of vehicles, may result in lower fixed costs, but higher inventory costs. The models we have proposed, which take the energy and inventory costs, related to deliv-ery of perishable food, into consideration, yielded better results for deciding optimal delivery routes than the tradi-tional VRPTW. These results also showed that, when no consideration was given to the order effects of time-win-dows on total delivery costs, the operator had to dispatch more vehicles to satisfy customer time-windows; an appro-priate setting of parameters may reflect not only the impact of time-window constraints on customer service sequenc-ing, but may also result in the lowest delivery costs. The results from the models using stochastic travel times implied that time-window constraints were more difficult to satisfy than models using deterministic travel times, requiring more vehicles to be dispatched, in order to satisfy customers’ needs. The models with soft time-windows also yielded a smaller vehicle requirement than those incorpo-rating hard time-windows; a delay in delivery, however, may result in higher inventory and penalty costs. Models considering time-dependent travel and temperatures were shown to result in lower total delivery costs, as well as lower inventory, energy and transportation costs, than those which did not consider these factors.

In summary, this study has shown how crucial charac-teristics, related to the delivery of perishable food may be considered in formulating vehicle routing solutions, with time-window constraints. The proposed models provide eﬀective tools, which may enable operators to make eﬀec-tive delivery decisions, under time-varying temperatures and time-dependent travel, by assessing the impact of ran-dom delivery times, food’s spoilage and time-windows on vehicle routing and the resultant costs.

Acknowledgements

The authors would like to thank the National Science Council of the Republic of China for ﬁnancially supporting this research under Contract No. NSC 92-2416-H-009-014. References

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