Optimal delivery cycles for joint distribution of multi-temperature food

(1)

Optimal delivery cycles for joint distribution of multi-temperature

food

Chaug-Ing Hsu

*

_{, Wei-Ting Chen}

1

_{, Wei-Jen Wu}

1

Department of Transportation Technology and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC

a r t i c l e i n f o

Article history:

Received 20 June 2011 Received in revised form 27 March 2013 Accepted 2 April 2013 Keywords:

Multi-temperature joint distribution Perishable food delivery

Time-dependent consumer demand

a b s t r a c t

The need for fresh, refrigerated, and frozen food has been continuously growing due to high demand for healthy and convenient diets in urban fast-paced daily living. Correspondingly, the market for low temperature logistics is expanding due to demand for low-temperature food, and the process of deliv-ering food requiring storage and shipping in containers with different temperature ranges has become an important issue for carriers. This study analyzes optimal delivery cycles for jointly delivering multi-temperature food using Traditional Multi-Vehicle Distribution and Multi-Temperature Joint Distribu-tion systems. We formulate mathematical models for the systems considering a variety of time-dependent demands and time-windows for delivering different temperature range foods to various customers. The models provide effective tools that determine delivery cycles and dispatching lists for carriers. The results show both carriers and shippers beneﬁt from jointly delivering different tempera-ture range foods using a single vehicle to ensure the freshness of the food.

1. Introduction

The need for fresh, refrigerated, and frozen food has continuously grown in recent years due to high demand for healthy and

conve-nient diets in urban fast-paced daily living. According toGlobal Cold

Chain Alliance (2008), demand for temperature-controlled food is increasing in many markets across the globe; thus, the market for

low-temperature logistics is expanding.Hsu and Liu (2011)deﬁned

multi-temperature logistics as encompassing all processes

involving the movement and storage of cargos in an efﬁcient and

cost-saving manner, where optimal temperature control is

neces-sary to maintain the cargos’ original value and quality. However,

time-dependent demand patterns may vary widely for different temperature range foods. For example, demand for deep frozen food, like tuna, usually occurs in the early morning, but some fresh food served during lunchtime is needed just before noon; therefore, the optimal delivery cycle for each temperature range may be different. This study aims to analyze optimal delivery cycles for jointly delivering multi-temperature food using both the Traditional Multi-Vehicle Distribution (TMVD) and Multi-Temperature Joint Distribution (MTJD) systems. We formulate a mathematical model for each system considering a variety of time-dependent demands

and time-windows for delivering different temperature range foods to various customers.

TMVD uses one type of refrigerated vehicle to distribute cargos in a single temperature range around a set-point, such as vehicles

with temperatures set at20_{C, 0}_{C or}_þ12_{C. Refrigerated}

ve-hicles maintain the required temperature using a mechanical compression refrigeration unit driven by an engine. This tempera-ture control system is usually affected by the frequency and dura-tion of vehicle door opening, thus it cannot be tuned precisely. However, in multi-compartment vehicles, the refrigerated space is subdivided into a number of compartments with individual

tem-perature set-points to provide ﬂexibility for business operations

(Tassou, De-Lille, & Ge, 2009). Compared with TMVD, the MTJD technique can simultaneously transport goods at two or more

temperature ranges in a single vehicle. Kuo and Chen (2010)

pointed out the logistical costs of handling frequent deliveries in small lots using less than truckload (LTL) transportation can be

signiﬁcantly reduced using the MTJD model, while maintaining

customer satisfaction. In this study, the MTJD system uses replaceable cold accumulators inside insulated boxes and cabinets to maintain precise temperatures and then uses these cold boxes and cabinets to carry different temperature range foods in regular vehicles for shipping. Therefore, in a regular vehicle, the proportion of space used by each temperature range food can be allocated based on dynamic demand during each period. Since temperature control relies on cold boxes and cabinets, the technique can avoid food deterioration resulting from bacteria due to repeated opening

* Corresponding author. Tel.: þ886 3 5731672; fax: þ886 3 5720844. E-mail address:[email protected](C.-I. Hsu).

1 _{Present address.}

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of the vehicle’s doors. Moreover, with MTJD, a carrier can unload

foods of all temperature ranges at a single customer’s location at

the same time; thus, it not only saves carrier costs but also reduces handling time and enhances the level of customer service.

For research regarding transportation networks for perishable

goods,Zhang, Habenicht, and Spieß (2003)presented a tabu search

algorithm to optimize the structure of cold chains.Hsu, Hung, and

Li (2007)studied a vehicle routing problem with time-windows for delivering perishable food. In addition to distribution networks, many studies have focused on the phenomenon of quality and shelf

life decay over time (e.g.Bogataj, Bogataj, & Vodopivec, 2005;Likar

& Jevsnik, 2006). There are numerous related studies in the

chemical and process engineeringﬁeld (e.g.Borghi, Guirardello, &

Filho, 2009). Food distribution strategies are currently tending to-ward the use of shipments containing a variety of food types in small amounts and at varying temperatures, but there are few studies that provide a rationale for this shift. In response, this study constructs a mathematical programming model to optimize de-livery cycles for multi-temperature food considering time-dependent demands and joint distribution. Through the proposed model, carriers can determine what temperature range foods should be loaded on the vehicles during different periods to minimize costs.

The reminder of this paper is organized as follows. Sections2

and 3describe the formulation of the MTJD and TMVD systems, respectively. An algorithm to solve the models in this study is

developed in Section4, and Section5presents a numerical example

to illustrate the feasibility and results of the models. Finally,

con-clusions are summarized in Section6.

2. Model formulation for the MTJD system

This section examines how time-dependent demand for

different temperature-range foods inﬂuences operation costs for

carriers. The delivery cycle (or frequency) for each temperature range food is an importation issue for a carrier because these

de-cisions directly inﬂuence operating costs of the carrier and the

quality of service provided to shippers. Carriers generally enhance their transportation offerings by providing high frequency (low cycle) service to reduce transportation time. On the other hand, inventory costs related to cargos in a distribution center waiting to be shipped also depend on the delivery cycle, and are borne by both carriers and shippers. Therefore, in this study, transportation and inventory costs are regarded as two of the major factors affecting delivery service decisions. Furthermore, this study extends the cost formulation to include energy costs due to using cold boxes and cabinets for storing perishable foods, and penalty cost for violating delivery time windows. Therefore, the optimal delivery cycle for each temperature range food and shipping list for each period are generated by minimizing the total cost, which includes trans-portation, inventory, penalty, and energy costs. In this study,

car-riers are assumed to provideﬁxed delivery cycles and have their

own distribution centers, vehicle_{ﬂeets, and temperature control}

equipment for delivering food to their customers. Customers in this

study are deﬁned as general retailers and are assumed to know the

carrier’s service level (i.e., delivery cycle for each temperature

range) when choosing a carrier. This study explores how carriers decide on a delivery list for each period.

In this study, ordered food will be shipped in the delivery cycle that is closest to the demand time for the corresponding temper-ature range. Thus, shippers can forecast before deciding whether

theﬁxed cycles will ensure the food will be delivered in time. In

such situations, the costs associated with routing distance have no

direct inﬂuence on determining delivery cycles. Therefore, the

is-sues related to the vehicle routing problem, which are usually

solved when the delivery list is known, are not taken into account

in this study. Let Drbe the delivery cycle of temperature range r

food. Thus, if food i needs to be stored in temperature range r, the time food i, needed by retailer j at period t, leaves the distribution

center, ysijt, can be expressed as nDr, where n is a natural number

because the time when food i leaves the distribution center must be

a multiple of Dr. The value of n for each order is discussed in Section

2.4. Moreover, this study formulates a mathematical programming

model for determining the optimal delivery cycle for each tem-perature range considering dynamic demand and different com-ponents of total operation cost; we assume the carriers are seeking to minimize total cost.

2.1. Transportation and energy costs

In this study, transportation costs involve both aﬁxed cost for

dispatching vehicles and a cost for loading/unloading cold boxes

and cabinets. This study assumes that carriers have sufﬁcient

ve-hicles to carry all ordered food at each period. Theﬁxed cost for

dispatching regular vehicles can be expressed asP

t

ktf , where ktis

the number of vehicles dispatched at period t, and f is theﬁxed cost

for dispatching a regular vehicle.

The loading/unloading cost depends on the quantity

trans-ported per shipment. We denote Nrtsand NrtGas the numbers of

temperature range r cold boxes and cabinets used at period t, and

d

_s

and

d

Gas the loading/unloading costs per unit cold box and cabinet,

respectively. Furthermore, the total loading/unloading cost, CL, can

be formulated as CL ¼ X r X t

d

sNrtsþ

d

GNrtG (1)

This study focuses on the tradeoff between using cold boxes and cabinets, the capacities of which are 90 L and 936 L, respectively.

However, the numbers of cold boxes and cabinets inﬂuence not

only the loading/unloading cost but also the energy cost of the MTJD system. The energy cost arises from the energy consumption of the cold accumulators inside the cold boxes and cabinets. Therefore, the energy cost depends on the numbers of cold boxes

and cabinets used. Let

f

rand

F

rdenote the energy cost per unit box

and cabinet, respectively, for temperature range r. Then the total

energy cost, CE, can be expressed as

CE ¼ X r X t ð

f

_rNrtsþ

F

rNrtGÞ (2) For this study, a survey was conducted to identify factors affecting costs for using cold boxes and cabinets. The data provided evidence that the loading/unloading cost and energy cost per unit capacity for one cold cabinet are less than those for one box. Therefore, we assume all food is packed into cold cabinets, but if the carrier uses a cold cabinet that is not full, then consideration should be given as to whether the food should be moved to cold boxes. That is, the study derives the critical volume that determines whether to use several boxes to replace a cold cabinet to yield the lowest loading/unloading and energy costs. Since the cost for using

a box and a cabinet can be formulated as (

d

_sþ

f

r) and (

d

Gþ

F

r),

respectively, the loading/unloading and energy costs of [(

d

_G_þ

F

r)/

(

d

_sþ

f

r)] units of cold boxes for temperature range r is equal to that

of one cold cabinet for the same temperature range. Let V_sdenote

the capacity of one unit cold box. Therefore, the critical volume for

using cold boxes can be derived as V_s[(

d

Gþ

F

r)/(

d

sþ

f

r)], which is

the total capacity of [(

d

G þ

F

r)/(

d

s þ

f

r)] units of boxes. If the

(3)

volume, the food should be moved to cold boxes; otherwise, that food should be kept in the cold cabinet. We assume approximately full capacity utilization for all boxes and cabinets. Furthermore, the number of boxes and cabinets used at period t for temperature

range r, Nrtsand NrtG, can be expressed as

and

respectively, where Qijtis the amount of food i needed by retailer j

at period t;

g

idenotes the temperature range in which food i needs

to be stored. Symbol VGdenotes the capacity of one unit cold

cab-inet. Consequently, the transportation cost of the MTJD system

during the entire study period, CT, can be expressed as

C_T ¼ X t ktfþ X r X t ð

d

_sNrtsþ

d

GNrtGÞ (5) 2.2. Inventory cost

In this study, inventory cost is determined by the difference between the time food arrives at the distribution center and leaves the distribution center. Furthermore, the total inventory cost of the

MTJD system, CI, can be presented as

CI ¼ X i X j X t Qijt

b

i ysijt yfijt (6)

where y_ﬁjtand ysijtare the times when food i, needed by retailer j at

period t, arrives at and leaves the distribution center, respectively.

Symbol

b

idenotes the inventory cost per unit of time per item of

food i stored in the distribution center, which involves the cost for storage and temperature control.

2.3. Penalty cost

This study assumes retailers accept soft delivery time-windows. When a vehicle arrives early, or within an acceptable period of delay, the food can be still delivered with a penalty cost. Let

Puijt; sijtR be the soft time-window for food i, needed by retailer j at

period t. Symbols Uijtand Sijtdenote the earliest acceptable time for

early arrival and the latest acceptable time for late arrival,

respec-tively, of food i, needed by retailer j at period t, and Uijt uijt,

Sijt sijt. The relationship between penalty cost and arrival time can

be seen in Fig. 1, which shows the acceptable periods for early

arrival and delay for food i needed by retailer j at period t arePUijt,

uijt) and (sijt, SijtR, respectively. There are different penalties for each

range. When arrival time is beyondPUijt; SijtR, the customer may

refuse to receive the food and the carrier should pay the customer a

penalty, ci, for each item of food i. Hence, the penalty cost due to

violating the earliest and latest acceptable times in the time-window for food i ordered by retailer j at period t can be

expressed as Qijtci.

According toHsu et al. (2007), the penalty cost, due to violating

the upper bounds of the soft time-window,Puijt; sijtR, speciﬁed by

retailer j who needs food i at period t, can be formulated as

QijtHidi(Drnþ

r

j sijt)vi, where

r

jis the expected travel time from the

distribution center to retailer j’s location, and Hiis the value of unit

item of food i. Symbols dijand viare parameters; di 1 and vi 1.

Furthermore, the penalty cost for food i needed by retailer j at

period t, CP1(Qijt), can be formulated as

Nrts ¼ 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 0; if P gi¼ r P j P ysijt¼ t QijtVi ! modVs>

d

_Gþ

F

r

d

rþ

f

r 2 6 6 6 6 6 4 P gi¼ r P j P ysijt¼ t Q_ijtV_i ! modV_s Vs 3 7 7 7 7 7 5; if P gi¼ r P j P ysijt¼ t Q_ijtV_i ! modV_s

d

Gþ

F

r

d

rþ

f

r (3) N_rt_G ¼ 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : 2 6 6 6 6 6 4 P gi¼ r P j P ysijt¼ t Q_ijtV_i Vs 3 7 7 7 7 5; if P gi¼ r P j P ysijt¼ t QijtVi ! modVs>

d

_Gþ

F

r

d

rþ

f

r 2 6 6 4 P gi¼ r P j P ysijt¼ t QijtVi V_s ; 3 7 7 5 if P 0 gi¼ r P j P ysijt¼ t QijtVi ! modVs

d

Gþ

F

r

d

rþ

f

r (4)

Time

P

en

alty

co

st

ijt

U

u

ijt

s

ijt

S

ijt

(4)

CP1 Qijt ¼ 8 > > > > < > > > > :

Qijtci; if nDrþ

r

j<Uijt

0; 0; QijtHidi nDrþtjsijt yi ; Qijtci;

if UijtnDrþ

r

j<uijt

if uijtnDrþ

r

jsijt

if s_ijtnDrþ

r

jSijt

if nDrþ

r

jSijt

(7)

and the total penalty cost for the MTJD system, CP, can be calculated

asP i P j P t CP1ðQijtÞ.

2.4. Formulation of the optimal problem

As regards a shipment dispatched at each period, this study assumes the carrier dispatches orders whose demand time falls in

[(nDr Dr/2),(nDrþ Dr/2)) at nDr. Let Mr,ndenote the time interval

during which the carrier accumulates temperature range r food to

dispatch at nDr. This study divides operation duration into m

pe-riods. Furthermore, Mr,nfor even and odd n can be formulated as

follows.

If cycle Dris an even number i.e., Dr¼ 2N, N˛[1,2,.m/2],

Mr;n¼ 8 > > > > > > > > < > > > > > > > > : Drn Dr 21 ; DrnþD₂r if n¼ 2;3;:::::;m Dr1 1; DrnþDr 2 Drn Dr 21 ; 24 if n¼ 1 if n¼_Dm r 9 > > > > > > > > = > > > > > > > > ; (8)

If cycle Dris an odd number, i.e., Dr¼ 2N 1, N˛[1,2,.m/2],

Mr;n¼ 8 > > > > > > > > < > > > > > > > > : Drn Dr1 2 ; Drnþ Dr1 2 if n¼ 2;3;:::;m Dr1 1; Drnþ Dr1 2 Drn Dr1 2 ; 24 if n¼ 1 if n¼_Dm r 9 > > > > > > > > = > > > > > > > > ; (9)

Furthermore, the time that food i needed by retailer j at period t

leaves the distribution center, ysijt, can be determined as

y_sijt ¼ nDr (10)

where n is a positive integer such that (nDr Dr/2) t < (nDrþ Dr/2).

To avoid increasing penalty and other costs for orders that cannot be delivered between the earliest and latest acceptable times, this study assumes the carriers would not dispatch food that

retailers would refuse to receive. Let 3 ijtbe a binary variable, if the

penalty for food i, needed by retailer j at period t, is Qijtci, 3 ijt¼ 0;

otherwise, 3 ijt¼ 1.

Furthermore, a nonlinear programming problem can be formulated here for determining the optimal delivery cycle for each

temperature range, Dr,cr, for the MTJD system, by minimizing the

total operation cost subject to the delivery time-window for each order. From the above discussion, the nonlinear programming problem for minimizing cost throughout the study period is as follows. Min Dr;cr CTþ CEþ CIþ CP (11-a s.t. CT ¼ X t ktfþ X r X t ð

d

sNrtsþ

d

GNrtGÞ (11-b) CE ¼ X r X t ð

f

_rNrtsþ

F

rNrtGÞ (11-c) CI ¼ X i X j X t Qijt

b

i ysijt yfijt (11-d) CP ¼ X i X j X t CP1 Qijt (11-e) CP1 Qijt ¼ 8 > > > > < > > > > :

Q_ijtc_i; nDrþ

r

j<Uijt

0; U_ijtnDrþ

r

j<uijt

0; u_ijtnDrþ

r

jsijt QijtPidi nDrþ

r

jsijt yi ; sijtnDrþ

r

jSijt Q_ijtc_i; nDrþ

r

jSijt (11-f) ysijt ¼ nDr ∍ðnDr Dr=2Þ t < ðnDrþ Dr=2Þ ci; j; t n˛Nþ (11-g) Nrts ¼ 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 0; if P gi¼ r P j P ysijt¼ t Q_ijtV_i ! modV_s>

d

_Gþ

F

r

d

rþ

f

r 2 6 6 6 6 6 4 P gi¼ r P j P ysijt¼ t Q_ijtV_i ! modV_G V_s 3 7 7 7 7 7 5; if P gi¼ r P j P ysijt¼ t Q_ijtV_i ! modV_s

d

Gþ

F

r

d

rþ

f

r (11-h)

(5)

Eq.(11-a)represents the objective function that minimizes costs

throughout the study period. Eqs. (11-b)eEq. (11-d) deﬁne the

transportation, energy, and inventory costs as Eq.(5), Eq.(2), and

Eq.(6), respectively. Eq.(11-e)deﬁnes the penalty cost. Eq.(11-f)

represents the relationship between penalty cost and delivery time. Eq.(11-g)presents the relationship between optimal delivery cy-cles and the time food leaves the distribution center for each order.

Finally, Eq. (11-h) and Eq. (11-i)express the formulation of the

numbers of cold boxes and cabinets used, respectively, during each period.

3. Model formulation for the TMVD system

In order to understand the advantages of the MTJD system, this study constructs a mathematical programming model for deter-mining the optimal delivery cycle for the TMVD system. Thus, the cost structures and service levels of the two different systems can

be compared. As discussed in Section1, the differences between the

MTJD and TMVD systems involve facility ﬂexibility, cost for

pur-chasing and using vehicles, and energy resources for temperature control. In practice, the temperature ranges for refrigerated vehicles can be modulated by vehicle freezer systems. However, the cost for changing the temperature range by replacing freezer systems is very large; therefore, the temperature range for refrigerated

vehi-cles is usuallyﬁxed. To accurately compare the operation costs of

the MTJD and TMVD systems, this study assumes the temperature range divisions for the two systems are the same. Therefore, the mathematical programming model for determining optimal de-livery cycles for the TMVD system is similar to the MTJD system, except for some equations. This section illustrates the differences in model formulation between the MTJD and TMVD systems.

Let Nr

tbe the number of normal containers used for temperature

range r food at period t without the function of temperature con-trol. In practice, the normal containers are made from thick paper or plastics, but this study assumes they are made of plastic and are of identical size. Similar to the MTJD system, the number of

con-tainers used at period t, Nr

t, can be calculated as N_tr ¼ 2 6 6 4 P gi¼ ¼ r P j P ysijt¼ t QijtVi V_N 3 7 7 5 (12)

where VN is capacity of one normal container;

d

denotes the

loading/unloading cost per one unit normal container.

Further-more, the transportation cost of the TMVD system, C_T0, can be

expressed as C0_T ¼ X t X r kr_tfrþX t X r

d

N_tr (13) where kr

tis the number of temperature range r vehicles dispatched

at period t, and fris theﬁxed cost for dispatching a temperature

range r vehicle.

Regarding energy cost, let Frbe the energy cost for using a

temperature range r vehicle; furthermore, the energy cost of the

TMVD system can be expressed asP

r

P

t

kr

tFr. However, in practice,

there exists a loss of energy due to opening the cargo hold because the temperatures inside and outside refrigerated vehicles are different. Such loss of energy depends on the amount of time the cargo hold is open, which is related to unloading time at a

cus-tomer’s location. Let tN denote the time duration to unload a

container from the vehicle, and

a

rbe the cost for lost energy per

unit of time for temperature range r vehicle. Thus, the cost of en-ergy loss due to opening the cargo hold in the TMVD system can be

calculated asP

r

P

t

a

rNtrtN. Furthermore, the total energy cost of the

TMVD system, C_E0, can be formulated as

C_E0 ¼ X r X t kr_tFrþX r X t N_rttN

a

r (14)

Hsu et al. (2007)formulated a loss of inventory cost using a

probability density function. Let G(Qijt) be the probability that Qijt

items of food i, ordered by retailer j at period t, perished due to opening the cargo hold per unit of time. The greater the difference in temperature inside and outside the vehicle, the higher the probability that food perished; that is, the probability that food perished at noon is higher than at other times during the day. The loss of inventory for the TMVD system can be expressed as P i P j P t

HiQijtGðQijtÞNtrtN. Furthermore, the total inventory cost of

the TMVD system, C_I0, can be expressed as

C_I0 ¼X i X j X t Qijt

b

i nDr yfijt þX i X j X t HiQijtG Qijt Nt_rtN (15)

The programming model for determining the optimal delivery cycle for different temperature ranges by minimizing cost through the study period for the TMVD system can be formu-lated in the same manner as for the MTJD system. However, the transportation, energy, and inventory cost functions would be

replaced by Eqs.(13)e(15), respectively. In addition, the

formu-lation related to cold boxes and cabinets would be replaced by

Eq. (12). This study further compares the objective value

be-tween the MTJD and TMVD systems by a numerical example in

Section5.

4. Algorithm

We assume delivery cycles for all temperature ranges are natural numbers, in terms of the unit of time being studied; therefore, a general integer programming model is formulated since all decision variables are positive integers. The solutions for the proposed models include optimal delivery cycles for each temperature range.

NrtG ¼ 8 > > > > > > > > > > > < > > > > > > > > > > > : 2 6 4 P gi¼ r P j P ysijt¼ t Q_ijtV_i VG ; 3 7 5 if P gi¼ r P j P ysijt¼ t QijtVi ! modVs>

d

_Gþ

F

r

d

rþ

f

r 2 6 4 P gi¼ r P j P ysijt¼ t QijtVi V_G ; 3 7 5 if P0 gi¼ r P j P ysijt¼ t QijtVi ! modVs

d

Gþ

F

r

d

rþ

f

r (11-i)

(6)

This study divides operation duration into m periods. For a carrier

transporting[ different ranges food, there are [ integer decision

variables and m[feasible solution combinations. Time for solving the

proposed models exponentially increases with the number of vari-ables. Furthermore, if we assume the delivery cycle must be a factor number of m (i.e., the domain of decision variables is a combination

of factor numbers of m), for[ temperature ranges, there are [ explicit

constraints, and the number of feasible solutions decreases. On the other hand, many variables in the cost functions depend on the decision variables. For example, the numbers of cold boxes and cabinets at each period depend on delivery cycle combinations.

Therefore, for a problem with m periods and[ ranges, there are m[

element constraints for cold boxes and cabinets, respectively. In addition, the penalty cost of each shipment also depends on delivery

cycles, as shown in Eq.(7). For each shipment, there is an element

constraint for penalty calculation. According to Hillier and

Lieberman (2009: chap. 12), the process of applying constraint

programming to integer programming problems involves efﬁciently

ﬁnding feasible solutions that satisfy all constraints and searching for the optimal solution among these solutions. The methods include enumerating solutions and adding a constraint that tightly bounds the objective function to values that are very near to what is anticipated for the optimal solution. In sum, due to the large

numbers of constraints and feasible solutions, it is dif_{ﬁcult and}

time-consuming toﬁnd an optimal solution; thus approximate methods

are required. The most commonly used approaches are a genetic algorithm (GA) and simulated annealing (SA). However, adapting GA

tends to be computationally expensive (Mishra, Dutta, & Ghosh,

2003), and the crossover rule is not suitable for the proposed

models because they are not sequence problems, and the delivery

cycles for each temperature do not inﬂuence each other due to the

assumption of sufﬁcient vehicles and shipping equipment. As for SA,

it has been extensively used in solving many difﬁcult optimization

problems. The major advantage of the SA algorithm is the ability to avoid becoming trapped in the local optimal. Therefore, this study adopted the SA algorithm to solve the optimal solution for each

system. In this section, weﬁrst develop an approach to generate an

initial solution, and then use the SA algorithm to develop a heuristic to improve the initial solution. The heuristic for improving the so-lution is described as follows.

4.1. Initial solution (INIT)

Since local improvement methods must start with a feasible solution, this study develops a heuristic to generate initial solu-tions. Based on the characteristics of transportation with econo-mies of scale, the average delivery cost per unit item can be reduced if more food is assigned to a vehicle with a larger capacity. However,

for perishable food, service level (i.e., delivery time) inﬂuences the

shipment and revenue of a carrier more than other categories of

cargos because such food usually decays with time. This study considers the time-dependence of food demand and vehicle ca-pacity to design a procedure to generate initial solutions that minimize late delivery and ensure the greatest capacity of cold boxes or cabinets that can be used. The procedure is described as follows.

Step 1. Calculate the average duration between two shipping

demands, Xrfor each temperature range (i.e., r¼ 1w[).

Step 2. Calculate the average shipping demand during Xr, 6Xr,

that is, 6X r ¼ P i P j P t

QijtVi=ðm=XrÞ; r ¼ 1w[, where m is the

number of periods. If 6X

r > Vs, the initial delivery cycle of

temper-ature range r, is the factor number of m, which is both closer and

larger than Xr. If 6Xr Vs,ﬁnd the smallest natural number n’such

that n0Xr> Vs, and let the initial delivery cycle of temperature range

r be the factor number of m, which is both closer and larger than n0Xr.

4.2. Simulated annealing (SA)

The values of the SA algorithm parameters include (1) the initial

temperature Z0¼ 99; (2) the decreasing ratio of temperature is

0.95, and the stop temperature is 0.1; and (3) the number of moves at each temperature is 50.

Referring toHeragu and Alfa (1992)andYan and Luo (1999), the

SA algorithm can be described as follows.

Step 0 Employ INIT toﬁnd an initial feasible solution, A, and

calculate its objective function, z(A).

Step 1. At temperature Zx, implement the Metropolis algorithm

(Metropolis, Rosenbluth, Rosenbluth, & Teller, 1953):

1.1. Randomly choose a temperature range r and randomly

generate a variable

p

wU(0,1); if y 0.5, Tr¼ Trþ 1;

other-wise, Dr¼ Dr 1. Let the altered solution be adjacent

solu-tion, A0. Calculate the objective value z(A0) for adjacent

solution S0.

1.2 Determine whether the new solution is accepted. 1.2.1 Calculate the difference between the objective

function of A and A0,

D

¼ z(A0) z(A).

1.2.2 If

D

_{< 0, then A ¼ A}0; else randomly generate a

var-iable

p

1wU(0,1). If expð

D

=ZxÞ

p

1, then A¼ A

0

; else go to Step 1.

1.2.3 If the stop criteria of the Metropolis algorithm are

satisﬁed, then go to Step 2, else go to Step 1.

Step 2. If the stop criteria of the SA algorithm are satisﬁed, then

go to Step 3; else let x¼ x þ 1 and Zxþ1¼ 0.95Zx, and go to Step 1.

Step 3. Output the optimal delivery cycle for each temperature

range food, A*¼ (D1,D2,D3,D4,D5).

Table 1

Initial values of food demand.

Temperature range r Food code i Food Pi(NT$) Vi(L) bi ci(NT$) mi(h) vi

Range 1 :30_C ₁ _Tuna ₂₀₀₀ ₂₅ ₁ ₃₀₀₀ ₂₀ ₂

Range 2 :30_{C to}₁₈_C ₂ _{Ice cream} ₂₀₀ _0.08 _0.8 ₃₀₀ ₂₄ _1.2

3 Ice cube 150 2.5 2 225 12 1.1 4 Frozen dumpling 100 2 1.5 150 24 1.3 Range 3 : 0_{C to 7}_C ₅ _Milk ₁₂₀ _0.75 ₁ ₁₈₀ ₁₂ _1.2 6 Juice 20 0.4 0.8 30 24 1.2 Range 4 : 18_C ₇ _Cookie ₃₀ ₂ _0.1 ₄₅ ₂₄ _1.5 8 Medicine 80 0.05 0.2 120 12 1.05 Range 5 : 40_C ₉ _Lunchbox ₆₀ _0.75 _0.4 ₉₀ ₁₂ _1.5 10 Hot meal 300 10 0.5 450 6 1.5

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5. Numerical example

This section presents an application of the proposed models, using a numerical example. This study generates a random extraction of the characteristics of 10 customers, which include locations, demand times, time-window constraints and items, and amount of food demand in different temperature ranges.

Cus-tomers’ time-windows are generated between 0:00 and 24:00. For

simplicity, this study assumes one operating day, namely 24 h, as the study period, with the unit of time for the study being

1-h intervals. T1-his study usesﬁve temperature ranges, with the food

list for each range shown inTable 1, whileTable 2shows the

ex-pected travel time to each customer.Table 3lists the parameters

related to operation of the distribution center. Fig. 2 illustrates

time-dependent demand for each temperature range food during

the entire study period. The demand time for each order is deﬁned

as the midterm of its time-window. Using the algorithm presented

in Section 4, the proposed models in Sections 2 and 3 can be

implemented and the optimal delivery cycle for each temperature range can be determined for the MTJD and TMVD systems, respectively.

Table 4lists the results and optimal objective function values for MTJD and TMVD system, respectively. The optimal delivery cycles

for the MTJD system for_{ﬁve temperature ranges (TRs) are 4, 2, 1, 2,}

1 h(s), respectively. For the MTJD system, the greater the shipping demand, the shorter the delivery cycles for the temperature range. However, comparing the two systems, the delivery cycles for TRs 2, 3, 4 under TMVD are much longer than the MTJD system because the carrier should accumulate a greater shipping volume to realize economies of scale with refrigerated vehicles used for only one temperature range. For TR 1, the lowest range, food in this range is most perishable, and its penalty cost per item would be much higher than other temperature range foods when the delivery time-window is violated. Therefore, the optimal delivery cycle is the same in the two systems. For TR 5, the delivery cycle for both systems is 1 h because shipping demand for this temperature range food appears high and frequent during the entire study period.

Table 4also compares different costs for the MTJD and TMVD systems with percentage of total cost. For both systems, inventory costs account for the highest percentage of the total cost. However, the inventory cost for MTJD is NT$1,351,631, which is much lower than that for TMVD, NT$3,104,008. This indicates that using the MTJD system can reduce the time difference between when the shipper orders the food and when the carrier dispatches the food. Hence, the carrier cannot only lower the inventory cost but also enhance the service level to increase competitiveness. As for transportation cost, there exists a tradeoff between transportation and inventory costs, which are linked by vehicle dispatching fre-quency (or delivery cycle). However, the transportation cost in this study depends only on the total vehicle dispatching frequency; transportation cost varying with routing distance is not taken into

account, as discussed in Section2. Therefore, the tradeoff between

transportation and inventory costs is less obvious. Furthermore, the transportation cost for TMVD, NT$54,750, is lower than for MTJD, NT$65,000, due to the lower dispatching frequencies for TRs 2, 3

and 4. As regards penalty costs, as shown inTable 4, the TMVD

system results in many more penalties, NT$2,711,689, than MTJD, NT$448,078, due to increased violations of time-windows. The huge penalty cost in the TMVD system is also related to the lower dispatching frequencies (i.e., longer delivery cycles for TRs 2, 3 and 4). Finally, for energy costs, even though there are more items included in the energy cost in TMVD, the energy cost yielded by

that system, NT$42,315, is not signiﬁcantly greater than MTJD,

NT$50,735, due to economies of scale resulting from accumulating shipments for TRs 2, 3 and 4.

In sum, total operation cost as well as inventory and penalty costs are much lower in MTJD than in TMVD. As for transportation and energy costs, the differences between two systems are negli-gible. Overall, the MTJD system can reduce total cost by NT$3,998,217 over the TMVD system. This indicates that carriers can effectively lower operation costs by using the MTJD system.

Fig. 2. Time-dependent demand for different temperature range foods.

Table 4

Overall results from MTJD and TMVD systems.

MTJD system TMVD system Optimal delivery cycles of

ﬁve temperature ranges (h)

(4,2,1,2,1) (4,24,24,24,1) Total cost (NT$) 1,915,445 5,913,662 Transportation cost (NT$) 65,000 (3.39%) 54,750 (0.93%) Inventory cost (NT$) 1,351,631 (70.56%) 3,104,008 (52.49%) Penalty cost (NT$) 448,078 (23.39%) 2,711,689 (45.85%) Energy cost (NT$) 50,735 (2.65%) 43,215 (0.73%) Table 3

Values of parameters related to carrier.

Symbol Equipment Value

f Fixed cost for dispatching a regular vehicle (NT$)

750 fr _{Fixed cost for dispatching a refrigerated}

vehicle (NT$)

900 ds Loading/unloading cost per box (NT$) 50 dG Loading/unloading cost per cabinet (NT$) 100

Vs Cold box capacity (L) 90

VG Cold cabinet capacity (L) 936 fr Energy cost per cold box (NT$)

(TRs 1, 2, 3, 4, 5)

(95, 91, 90, 86, 83) Fr Energy cost per cold cabinet (NT$)

(TRs 1, 2, 3, 4, 5)

(950, 900, 850, 800, 750) Fr _{Energy cost per refrigerated vehicle (NT$)}

(TRs 1, 2, 3, 4, 5)

(998, 956, 945, 903, 871) Table 2

Expected travel time from distribution center to retailers. Retailers Expected travel

time (min)

Retailers Expected travel time (min) 1 5 6 30 2 10 7 35 3 15 8 40 4 20 9 45 5 25 10 50

(8)

Table 5shows the temperature ranges of delivered food and the numbers of stops, vehicles used, and cold boxes/cabinets used for

each period for both the MTJD and TMVD systems.Table 6shows

detailed service lists for each period for the MTJD system. The detailed service lists include the retailers served, food items dis-patched, and shipping amount for each period. The results in

Table 5show the MTJD system delivered more different tempera-ture range foods and served more retailers than the TMVD system

during most periods. Thisﬁnding indicates that a carrier can unload

different temperature range foods simultaneously at a retailer’s

location using a single regular vehicle, which indicates both unloading time and routing time can be reduced substantially for the retailer and/or carrier.

AsTable 5andFig. 2show, for TR 1, vehicles are dispatched before periods with high demand. For example, demand for TR 1 peaks at periods 6, 11, 15, 18, and 22 and vehicles are dispatched for TR 1 at periods 4, 8, 12, 16, 20, and 24. The difference between the peaks of demand and dispatching time is due to expected vehicle travel time during the transportation process, which is listed in

Table 2. For TRs 2 and 4, shipping demand appears frequent but is nil during some periods. Such demand patterns result in a vehicle being dispatched every two periods for the MTJD system. However, demand per unit period for these two temperature ranges is much lower than the capacity of a refrigerated vehicle, so carriers using the TMVD system should dispatch these two temperature range foods only once in 24 h (i.e., the entire study period). In that way, refrigerated vehicles of greater capacity can be used and the de-livery cost per unit item of food can be reduced due to economies of scale.

For TR 3, shipping demand appears frequently before 12:00 and is nil at some late period. The optimal delivery cycle for TR 3 for the

MTJD system is 1 h, which is the shortest cycle among the ﬁve

temperature ranges, thereby satisfying the frequent demand before

12:00. However, as shown inTable 6, for periods without demand

for TR 3, the carrier does not dispatch this food range before such periods. On the other hand, the optimal delivery cycle for TR 3 in

Table 6

Results of stop locations for each period for the MTJD system. Period Delivered

ranges

Stop codes (food category code, number of units of food) 1 3 10(5,20;6,40) 5 9(9,200) 2 2 1(4,20); 2(2,100;4,400); 4(2,50;4,200); 10(2,100;3,20;4,100); 9(4,40) 3 2(5,300;6,300); 9(5,30;6,50) 4 5(8,500); 9(7,50;8,25); 10(7,200) 3 5 10(10,20) 4 1 1(1,5); 9(1,10); 10(1,100) 2 9(2,10;3,50); 3(2,50;4,200); 10(2,100) 4 9(7,10); 10(7,30;8,100;9,20) 5 3 3(5,50;6,200); 9(5,200;10,50); 10(5,20;6,40) 6 2 3(3,150); 10(4,30;3,100); 2(4,200) 4 9(8,150;7,50) 5 3(10,100); 9(9,200) 7 3 4(5,100;6,300) 5 4(9,400); 5(10,100); 10(10,50) 8 2 9(3,50); 10(2,100;4,100) 3 10(5,20;6,40) 4 10(7,200); 5(8,400) 5 9(10,50) 9 3 6(5,300;6,300); 9(5,250;6,200) 5 9(9,50) 10 2 7(2,20); 9(2,150); 10(2,100;3,50); 1(3,500); 6(2,200;4,100) 3 4(5,150); 10(5,20) 4 7(8,40); 8(7,30); 9(7,150;8,200); 10(7,40) 5 2(10,200); 6(10,150) 11 3 1(6,150) 5 9(10,50); 10(10,200) 12 1 10(1,30) 2 1(4,15); 2(2,100); 10(2,100;4,150) 3 9(5,150); 10(5,20) 4 3(8,50); 4(7,60); 5(7,100); 6(7,90); 10(8,100); 2(7,100); 8(8,50); 9(7,80) 5 2(9,50); 9(9,200); 10(9,200) 13 3 10(6,50) 5 7(9,40;10,20); 9(10,250); 10(10,120) 14 2 9(4,50;3,150); 10(3,100;2,50) 3 9(6,200); 10(5,120) 4 3(7,150); 5(8,100); 4(8,100); 10(7,50) 15 3 10(6,40) 5 9(9,100;10,50) 16 1 10(1,80;1,80); 7(1,10); 1(1,5) 2 10(4,100;2,100); 1(3,500); 2(2,100;3,50;4,400) 4 6(8,20); 9(8,50;7,250); 10(7,200) 5 1(9,50) 17 3 1(6,150); 10(5,20;6,40) 5 9(10,150) 18 2 1(4,20); 6(4,200); 7(3,120); 9(2,200); 10(3,150) 3 6(5,400) 4 3(7,120); 4(7,70); 9(8,150) 5 2(9,400;10,90); 5(10,120); 9(10,50); 10(9,70) 19 3 9(5,150); 10(6,40) 5 4(9,80); 10(10,100) 20 1 9(1,50) 2 2(2,50); 4(2,50); 10(3,100;2,100); 8(4,100) 3 2(6,80); 4(5,100); 10(5,20) 4 6(7,100); 9(7,50;8,200); 10(7,40;8,20); 7(7,20) 5 6(10,150); 8(10,100); 9(9,200;10,50) 21 3 9(6,50); 10(10,40) 22 2 2(4,200); 9(3,200); 10(3,100;4,100;2,100); 1(2,80;4,40); 6(3,50;4,150) 3 10(6,40;7,50) 5 1(10,30) 23 3 9(5,250;6,200); 10(5,120) 5 9(10,50) 24 1 10(1,10) 2 10(3,20) 3 10(6,40) 4 5(8,200); 9(7,200); 10(7,30) 5 9(9,200) Table 5

Results from each period for the MTJD and TMVD systems. Time period MTJD system TMVD system Delivered ranges Optimal number of stops, vehicles, cold cabinets and cold boxes

Delivered ranges Optimal number of stops, vehicles and boxes 1 3,5 (2,2,1,1) 5 (1,1,7) 2 2,3,4,5 (6,3,2,11) 5 (0,0,0) 3 3,5 (1,1,0,3) 5 (1,1,2) 4 1,2,3,4,5 (3,4,3,8) 1,5 (3,4,29) 5 3,5 (3,1,0,10) 5 (1,1,5) 6 2,3,4,5 (4,2,1,6) 5 (2,1,2) 7 3,5 (3,3,2,3) 5 (3,2,18) 8 1,2,3,4,5 (3,2,0,16) 1,5 (1,1,5) 9 3,5 (2,2,1,1) 5 (1,1,1) 10 2,3,4,5 (8,7,6,8) 5 (2,4,35) 11 3,5 (3,3,3,1) 5 (2,3,25) 12 1,2,3,4,5 (9,3,2,9) 1,5 (3,2,12) 13 3,5 (3,5,4,4) 5 (3,4,40) 14 2,3,4,5 (5,2,1,7) 5 (0,0,0) 15 3,5 (2,2,1,1) 5 (1,1,6) 16 1,2,3,4 (6,9,8,7) 1,5 (3,6,45) 17 3,5 (3,3,2,2) 5 (1,2,15) 18 2,3,4,5 (8,3,1,16) 5 (4,1,4) 19 3,5 (3,2,1,4) 5 (2,2,11) 20 1,2,3,4,5 (7,5,3,17) 1,5 (3,4,32) 21 3,5 (2,1,0,6) 5 (1,1,4) 22 2,3,4,5 (5,3,2,7) 5 (1,1,3) 23 3,5 (2,1,0,10) 5 (1,1,5) 24 1,2,3,4,5 (2,1,0,4) 1,2,3,4,5 (10,4,24)

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the TMVD system, 24 h, is much longer than for the MTJD system due to its considerably smaller shipments during most periods. Finally, for TR 5, its high and frequent shipping demand leads to more frequent dispatching for both systems.

Table 5also lists the numbers of stops, vehicles, and containers. This information can highlight the content of shipments in each period. Except for period 24, the numbers of stops for MTJD are greater than for TMVD because food for TRs 2, 3, 4 is not dispatched

during periods 1e23 in the TMVD system. For TMVD, food of TRs 2,

3, and 4 is only dispatched at period 24. Hence, the carrier has to stop at many more locations to deliver food in these ranges. Therefore, the number of stops for TMVD is greater than for MTJD at

period 24. The results noted inTable 6enable a carrier to effectively

prepare the_{ﬂeet, cold cabinets, and boxes for each period.}

6. Conclusion

This study develops a mathematical programming model that can determine the optimal delivery cycle for each temperature range food by taking into account time-dependent demand for different temperatures range foods. Though the proposed model, time-varying demand and equipment usage can be analyzed (i.e., demand uncertainty and equipment are dealt with using a multi-periods approach, and costs for different carrier sizes can be determined). The proposed models provide effective tools to determine delivery cycles and dispatching lists for carriers with time-dependent demand by assessing the impact of transportation and inventory costs, as well as energy consumption for temperature control and delivery time-windows, and those resultant costs. References

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Nomenclature

Dr: delivery cycle of temperature range r food

ysijt: time when food i, needed by retailer j at period t, leaves distribution center

n: natural number

kt: number of vehicles dispatched at period t

f:ﬁxed cost for dispatching a regular vehicle

Nrts: number of temperature range r cold boxes used at period t

NrtG: number of temperature range r cold cabinets used at period t ds: loading/unloading cost per cold box

dG: loading/unloading cost per cold cabinet CL: loading/unloading cost of MTJD system

4r: energy cost per temperature range r box

Fr: energy cost per temperature range r cabinet

CE: energy cost of MTJD system

Vs: capacity of unit cold box

Qijt: amount of food i needed by retailer j at period t

gi: temperature range in which food i needs to be stored

VG: capacity of unit cold cabinet CT: transportation cost of MTJD system

CI: inventory cost of MTJD system

yﬁjt: time when food i, needed by retailer j at period t, arrives at distribution center bi: inventory cost per unit of time, item of food i

uijt: lower bound of the soft time-window speciﬁed by retailer j who needs food i at

period t

sijt: upper bound of the soft time-window speciﬁed by retailer j who needs food i at

period t

Uijt: the earliest acceptable time for early arrival of food i, needed by retailer j at

period t

Sijt: the latest acceptable time for late arrival, of food i, needed by retailer j at period t

ci: penalty cost due to violating the earliest and latest acceptable times in the

time-window for each item of food i

rj: expected travel time from distribution center to retailer j’s location

Hi: value of unit item of food i

di: ratio of penalty to food i value

vi: exponent parameter of penalty cost function of food i

CP1: penalty cost for food i needed by retailer j at period t

CP: penalty cost of MTJD system

Mr,n: time interval during which the carrier accumulates temperature range r food to

dispatch at nDr

m: number of periods

3ijt¼

0 if penalty for food i; needed by retailer j at period t; is Qijtci

1 otherwise :

Ntr: number of normal containers used for temperature range r food at period t of

TMVD system

VN: capacity of one normal container

d: loading/unloading cost per normal container CT0: transportation cost of TMVD system

kr

t: number of temperature range r vehicles dispatched at period t

fr:ﬁxed cost for dispatching a temperature range r vehicle Fr: energy cost for using a temperature range r vehicle tN: time duration to unload a container from vehicle

ar: cost for loss of energy per unit of time, temperature range r vehicle

C0

E: energy cost of TMVD system

GðQijtÞ: probability that Qijtitems of food i, ordered by retailer j at period t, perished

due to opening the cargo hold per unit of time CI0: inventory cost of TMVD system

[: number of temperature ranges

Xr: average duration between two shipping demands for temperature range r

6X

r: average shipping demand during Xr

n’: smallest natural number such that n’Xr> Vs

Z0: initial temperature of the SA algorithm

A: feasible solution z: objective function

Zx: temperature of the SA algorithm at the xth move

p: random variable for determining direction of adjusting delivery cycle A’: adjacent solution

p1: random variable for determining whether A should be replaced by A’