Vol.4, No.9B, 70-75 (2013) Agricultu ral Sciences
http://dx.doi.org/10.4236/as.2013.49B012
Safety reliability optimal allocation of food cold chain
Yifeng Zou*, Ruhe Xie, Guanghai Liu
Business School of Guangzhou University, Guangzhou, China; *Corresponding Author: [e-mail]
Received August 2013
ABSTRACT
This paper applied the safety reliability of food
cold chain logistics to establish reliability allo-
cation model for cold chain systems, designed
optimization algorithms, and made a case anal-
ysis. By applying system reliability allocation
principle, this article firstly built saf ety reliabilit y
allocation model of food cold chain logistics
system without cost constraint based on the
safety reliability model of food cold chain logis-
tics system, and then it set up optimal decision-
making model of food cold chain logistics sys-
tem with cost constraint using the functional
relationship between the time, temperature of
cold chain logistics and logistics costs. Next,
according to the characteristics of the model, a
heuristic algorithm was proposed to allocate
safety reliability of the system to each cold
chain unit so as to achieve the goal of operating
costs optimization subject to assurance of o ver-
all safety reliability of the cold chain system.
Taking the safety impact factor of food cold
chain unit as a weight, the article also deduced
the equation of reallocation of safety reliability
of food cold chain system. In the end, these
models were used to optimize the allocation of
safety reliability in an example of Sushi cold
chain process. It provided a new thought and
method to optimally plan the unit safety of food
cold chain system as well as reduce the cost of
food cold chain.
Keywords: Cold Ch ain; Safety Reliabilit y; Food
Logistics System; Allocation Optimi zat ion
1. INTRODUCTION
Food cold chain is a system engineering which can
keep perishable or fresh foods under prescriptive tem-
perature across whole process of logistics such as pro-
duction, storage, transportation, deliver y and sale, so that
to keep the food safety, reduce loss and prevent pollution
[1]. As a result of lack of index reflecting continuous
changes of food safety in logistics process, few of essays
thus far quantitatively studied the safety in the process of
food logistics from the system point of view, instead,
most of them focused on food quality loss and risk man-
agement in food supply chain [2-6]. Although microbial
kinetics model may be used to predict the state of food
safety [7-9], to be viewed from the safety analysis and
optimization of logistics system, it has three aspects of
insufficiencies to represent food safety by total bacterial
count.
1) It is not intuitive. The safety degree of cold chain
can not be determined by means of total bacterial count
directly. For example, knowing that the total bacterial
count of certain food is 10,000 cfu/ml is not enough for
logistics operators and consumers to make exact judg-
ment of safety or not, one also has to understand whether
it exceeds health standards before making conclusion.
2) It is not a linear relationship between pathogens
number and degree of harm to human body. For such
food that its total bacterial co unt not ex ceed ing the health
standards, even people can tell it is safe, they are not
informed of the specific safety status and level, hence
unable to determine its shelf life.
3) The microbial growth model is precisely in an ex-
ponential form. It is quite hard to solve an optimization
model with exponential representation.
In view of the above-mentioned facts, [10] modified
predictive microbiology model by applying system relia-
bility theory, and established safety reliability model of
cold chain that was intuitive and simple to reflect the
continuing impact of temperature on the safety of the
food cold chain. He also defined the term of the safety
reliability of food logistics unit as the probability of food
logistics unit controlling the safety of food in an intended
scope for a stated period of time under specified operat-
ing conditions.
Reliability allocation is a procedure during planning of
system that, according to the general requirements of
system reliability, to gradually break it up from global to
local and allocate to various sub-systems and facilities
[11]. It is in fact to solve following basic inequalities:
*
12
(, ,,)
s ins
RRRR RR≥
(1)

Y. F. Zou et al. / Agricultur al Sciences 4 (2013) 70-75
71
*
12
(, ,,)
s ins
gRRR Rg≤
(2)
where:
*
s
R
is system reliability output constraint,
*
s
g
is synthesis constraint conditions to system planning,
including expenses, weights, power loss, etc.
For a cascade system, Eq.1 may be converted to
*
12
() ()()()()
i ns
Rt RtRtRtRt⋅ ⋅⋅⋅≥
. (3)
To achieve safety and high efficiency of food cold
chain system, it shall take into account of two aspects of
restraints when allocate safety reliability: First, the safety
constraint, i.e. under the condition of given initial safety
reliability, it must comply with the stipulated require-
ments after food passes through the cold chain system;
Second, the cost re straint. Food cold chain system is also
an economic system that seeks for minimum logistics
cost while satisfying safety demands. Therefore, safety
reliability allocation of cold logistics system is how to
allocate the total safety reliability to each unit so as to
minimize total logistics cost. Yet there is no related re-
search at present. This article applies the safety reliability
of food cold chain logistics to establish reliability alloca-
tion model for cold chain logistics systems, design opti-
mization algorithms, and make a case analysis and dis-
cussion.
2. ALLOCATION OF SAFETY
RELIABI-LITY WITHOUT COST
CONSTRAINT
For a cold chain logistics system composed of m cold
chain units, according to [10], its safety reliability Ri af-
ter food continuously passes through the ith (i ≤ m)unit
can be expressed as
( )
2
01
01
i
ijj i
j
RR dTtR
=
=− ∆≤≤
∑
, (4)
where: R0 is initial safety reliability of food cold chain
system,
minjj
TTT∆=−
(Tj is the temperature of cold
chain unit j, Tmin is the temperature at which no growth
of microbial), tj is logistics time of cold chain unit j, d is
a parameter related to food variety).
As
2
0
jj
Tt∆≥
, Ri is non-increasing function which
means in food cold chain system, safety reliability grad-
ually decreases from initial point to destination . Normal-
ly lower temperature has higher demand for equipment
conditions. Yet it is impossible for facilities to reach inf i-
nite low temperature, there have to be certain low limits
(for instance, some refrigerated trucks’ lowest tempera-
ture can only reach down to 0˚C), which we indicate by
*
j
T
. Moreover, it usually has re st ric ti ons on t he processing
time of logistics unit. For example, the quickest for-
warding time under existing conditions of food transpor-
tation and distribution; the shortest time for producing
and handling. Here let
*
j
t
represent the time-limiting
requirement. We now have safety reliability optimization
allocation model of cold chain system as follows:
2
01
Max
m
m jj
j
RRdTt
=
=−∆
∑
(5)
s.t.
20
1
1
m
jj
j
Tt R
d
=
∆≤
∑
(6)
( )
*
01, 2,
jj
TT jm∆≥ ≥=…
(7)
( )
*
01, 2,
jj
tt jm≥≥=…
(8)
Obviously when
*
j
T
and
*
j
t
, the lower limits of both
the temperature and time of each cold chain unit are ap-
plied to the above equations,
2
1
m
jjj
Tt
=
∆
∑
is the smallest
and the objective function reaches its maximum. Mean-
time, the cold chain system has highest safety reliability.
3. ALLOCATION OF SAFETY
RELIABILITY WITH COST
CONSTRAINT
3.1. Model Establishing
Suppose a cold chain system composed of m cold
chain units, after food passed unit j(j=1, 2,… m), its lo-
gistics cost is Cj, logistics service volume provided is Vj
and safety reliability is Rj, then the function of cost of th e
unit is
(, )
jjj j
C CVR=
. (9)
Logistics service provided by food cold chain unit has
typical scale effect (e.g. the higher inventory level in
cold store, the lower unit operation cost is; the bigger
delivery amount of refrigerated truck, the lower unit dis-
tribution cost), that is to say, logistics cost and logistics
service volume has the relation of
0
j
j
C
V
∂>
∂
and
2
2
0
j
j
C
V
∂<
∂
.
Given that cold chain system is a cascade system
whose processing capacity is determined by the unit of
least processing capacity, when system processing capac-
ity is fixed, it can be represented by the processing time
of the logistics system, i.e. for food logistics of certain
quantity, processing time shortens as processing capacity
increases. Therefore under the condition of fixed process-
ing capacity of the system, the processing volume of
food cold chain system is proportional to its processing
time, while the above equation can be written as
0
j
j
C
t
∂>
∂
and
2
2
0
j
j
C
t
∂<
∂
(10)
To maintain low temperature in the process of food
cold chain, it needs special refrigerate equipment and
consumes energy. Moreover, along with the cold chain

Y. F. Zou et al. / Agricultur al Sciences 4 (2013) 70-75
72
unit temperature drops, much more energy is consumed
and equipment of higher technical conditions is required.
Consequently, logistics cost increases quickly [12]. The-
reupon, cold chain logistics cost and temperature has
following relationship:
0
j
j
C
T
∂>
∂∆
and
2
2
0
j
j
C
T
∂<
∂∆
(11)
At present food cold chain usually extends to selling
section only. For the sake of consumption safety, it is
necessary to reserve certain safety reliability at selling
unit. Suppose the requisite final safety reliability of lo-
gistics system is no less than RE, then we obtain from
Eq.6 that
20
1
1[]
m
jj E
j
TtR R
d
=
∆≤ −
∑
. (12)
In summary, for food cold chain system having con-
straints of processing capability and safety reliability, its
cost optimal decision-making model is as below:
11
Min(,)(, )
mm
j jjjjj
jj
CCVRC Tt
= =
== ∆
∑∑
(13)
s.t.
20
1
1[]
m
jj E
j
TtR R
d
=
∆≤ −
∑
0
j
j
C
t
∂>
∂
and
2
20
j
j
C
t
∂<
∂
0
j
j
C
T
∂>
∂∆
and
2
2
0
j
j
C
T
∂<
∂∆
*
0
jj
TT∆≥ ≥
*
0
jj
tt≥≥
.
3.2. Heuristic Algorit hm
Considering that every cold chain unit has different
cost function, plus the expression is complicated, it is
rather difficult to solve the model directly. However, we
can derive a quite straightforward heuristic algorithm
from the special relationship of cold chain cost with
temperature and time.
Note from Eq.4 that the decrease of safety reliability
is proportional to logistics time whereas it has a square
relation with temperature in logistics unit. Hence it is
more effective to regulate the temperature in logistics
unit rather than logistic time for acquiring specific safety
reliability. And the smaller temperature difference be-
tween the various aspects of cold chains, it is more fa-
vorable for food quality maintenance [13,14]. That is to
say, under the premise of system’s overall safety reliabil-
ity being no less than end-point safety reliability RE,
starting from the logistics unit of lowest temperature,
advance its temperature to the same value of the unit of
most adjacent temperature; If there are a number of lo-
gistics units of the same temperature, then give priority
to the unit with longest logistics time; Thus keep adjust-
ing until system’s overall safety reliability reaches the
requirement RE. Following is the algorithm.
Step 1: Arrange
*
j
T
in ascending order to obtain a set
S(s1, s2,……sm).
Step 2: Let
*
jj
tt=
,
*
jj
TT∆=
(j=1, 2,… m).
Step 3: If
20
1
1[]
m
jj E
j
TtR R
d
=
∆> −
∑
, then no solution
and stop; Otherwise go to step 4.
Step 4: Tak e the smallest element si from the remaind-
er of set S, its corresponding logistics unit j, if there are
more than one element of the lowest value, then take the
one with greatest
*
j
t
.
Let
1
()
jji i
T Tss
+
∆=∆+−
, and temperatures of the
rest units remain invariable.
If
20
1
1[] m
jj E
j
TtR R
d
=
∆= −
∑
, the optimal solution de-
clared.
If
20
1
1[] m
jj E
j
TtR R
d
=
∆> −
∑
, then gradually narrow
down
j
T∆
till it satisfies
20
1
1[] m
jj E
j
TtR R
d
=
∆= −
∑
, and we
obtain the optimal solution.
Step5: If
20
1
1[] m
jj E
j
TtR R
d
=
∆< −
∑
, then remove si
from the set S, rearrange set S per non-descending order
of
*
j
T
and return to Step 4.
4. REALLOCATION OF SAFETY
RELIABILITY
As aforementioned, there is no solution to the optimi-
zation model if
20
1
1[] m
jj E
j
TtR R
d
=
∆> −
∑
. Such situa-
tion indicates that the existing food cold chain system
can not meet the minimum safety reliability requirements
and further modification of o rig inal planning is necessary
to enhance its safety reliability, i.e. need to reallocate
safety reliability to each logistics unit.
The principle of reliability reallocation is that the unit
of lower reliability is easier to improve, otherwise, more
difficult. This is actually to reallocate reliability accord-
ing to the weight of unit in the system. In food cold chain
system, the cold chain unit of bigger
2
Tt∆
has lower
safety reliability whereas the one with smaller
2
Tt∆
has bigger safety reliability. Therefore, reallocation of
safety reliabilit y of food cold chain syste m can be carried
on in accordance w ith
2
Tt∆
of cold chain unit.
Define
2
Tt∆
of food cold chain unit j as its safety
impact factor
j
f
.

Y. F. Zou et al. / Agricultur al Sciences 4 (2013) 70-75
73
2
j jj
f Tt= ∆
(14)
Assume food cold chain system has m units, then wj,
the weight of unit j in the system is:
1
j
jm
j
j
f
wf
=
=
∑
(15)
Safety reliability of system before optimization is:
01
m
mj
j
R Rdf
=
= −
∑
(16)
Let
s
m
R
(
s
m
R
>m R) the optimized safety reliability of
cold chain system,
*
j
f
the safety impact factor of each
cold chain unit. Apply to Eq.4 to obtain
*
01
m
s
mj
j
R Rdf
=
= −
∑
. (17)
Thus, according to the principle of reliability realloca-
tion, the equation of optimized safety reliability of each
cold chain unit
*
j
R
can be written as follows:
*
()
s
jjjm m
RR wRR=+−
(18)
5. CASE ANALYSIS
As made by hand, Sushi is apt to be infected by Sta-
phylococcus aureus to cause food safety problem. Table
1 shows the data of temperature and time restrictions in the
process of Sushi cold chain. Suppose the initial safety
reliability is 1.0, expected safety reliability is 0.8 so as to
ensure edibility after consumer purchased sushi for a
period of time; moreover, d is 0.0003 and no-microbial-
growth temperature is 5˚C [9].
According to Eq.4, we apply the lower limits of both
the temperature and time of each cold chain unit to the
equation, and obtain the end-point safety reliability 0.851
(refer to Table 1 for solving details) which is higher than
the expectation of 0.8. From a cost perspective, the logis-
tics system still needs to be optimized under the premise
of ensuring safety.
In the light of solution method, first is to optimize the
unit of the lowest temperature. It is selling unit (11˚C) in
this system. Use previous heuristic algorithm to advance
the temperature of selling unit to the most adjacent value
13˚C of delivering unit, and solve again to acquire safety
reliability of the system, which is by then still higher
than the expectation of 0.8. Thus after several loops of
solving steps, the algorithm finds optimal solution to the
temperature and time of each cold chain unit (as shown
in Table 2), i.e. under the premise of full assurance of
food safety, it can save logistics cost as far as possible by
an appropriate increase in the temperature of delivering
unit and selling unit (2˚C and 2.5˚C respectively).
If the end-point safety reliability requirement is raised
to 0.9, when choosing the lower limits of both the tem-
perature and time of each cold chain unit, the end-point
safety reliability is 0.851 (see Table 1) which is lower
than the expectation, the system cannot satisfy the re-
quirement and must reallocate the safety reliability.
The results of safety weight of each cold chain unit wj
are shown in Table 3, wherein the delivering unit has the
highest degree of weight that reaches 0.353, and then
followed by the selling unit, 0.221, the loading unit is the
least, only 0.044 .
After reallocation, the system output of safety reliabi l-
ity is 0.901 that satisfies the safety reliability requirement
of no less than 0.9. By comparing
j
R
and
*
j
R
we can
find that the safety reliability of delivering unit, the one
with highest degree of weight needs to be improved
Table 1. Data of Cold chain Units before Optimization.
No. j Cold Chain Unit Temperature Tj (˚C) Time t j (h)
2
jj
Tt∆
2
1
i
jj
j
Tt
=
∆
∑
d
2
1
i
jj
j
Tt
=
∆
∑
i
R
1 Producing 15.0 0.8 80.0 80.0 0.024 0.976
2 Packing 17.0 0.5 72.0 152.0 0.046 0.954
3 Loading 22.0 0.1 28.9 180.9 0.054 0.946
4 Delivering 13.0 2.3 147.2 328.1 0.098 0.902
5 Unloading 21.0 0.2 51.2 379.3 0.114 0.886
6 Handling 20.0 0.2 45.0 424.3 0.127 0.873
7 Selling 11 .0 2.0 72.0 496.3 0.149 0.851
Table 2. Data of cold chain units after optimization.
No. j Cold Chain Unit Temperature Tj (˚C) Time t j (h)
2
jj
Tt∆
2
1
i
jj
j
Tt
=
∆
∑
d
2
1
i
jj
j
Tt
=
∆
∑
i
R
1 Producing 15.0 0.8 80.0 80.0 0.024 0.976
2 Packing 17.0 0.5 72.0 152.0 0.046 0.954
3 Loading 22.0 0.1 28.9 180.9 0.054 0.946
4 Delivering 15.0 2.3 230.0 410.9 0.123 0.877
5 Unloading 21.0 0.2 51.2 462.1 0.138 0.862
6 Handling 20.0 0.2 45.0 507.1 0.152 0.848
7 Selling 13.5 2.0 144.5 651.6 0.196 0.804

Y. F. Zou et al. / Agricultur al Sciences 4 (2013) 70-75
74
the most by 0.048; while handling unit of the least
weight needs to be improved the least by only 0.005.
6. CONCLUSION AND DISCUSSION
Safety reliability provides a new quantitative method
for the analysis and optimization of food cold chain sys-
tem. For food cold chain system, once knowing ambient
temperatures and logistics time, it is convenient to cal-
culate the safety reliability of any cold chain unit and
cold chain system, which effectively solves the problem
of lack of quantitative tools in the study of cold chain
system. However as a new indicator, the validity and
scope of application is naturally an issue of concern. We
re-calculated the shelf life in some liter ature according to
the safety reliability model and then compared with the
original data. The results indicated that the deviation fell
in acceptable range (see Table 4). That proves that safety
reliability model can be applied to reliability optimally
allocate of food cold chain system under certain condi-
tions.
Certainly any model has its applicable conditions.
Safety reliability model of food cold chain is built on the
basis of kinetics model of microbial growth whose ap-
plicable conditions also apply to safety reliability model.
To be specific, there are following three aspects:
1) Sealed package. The sealed package is mainly to
avoid secondary or multiple infections by microorgan-
isms in food logistics process; it also prevents the physi-
cal and chemical hazards. Without seal ed pack agin g, safety
reliability model is unable to calculate and assess the
final result because of uncertainty of the infected micro-
be species and their initial values, plus possible conti-
nuous new pathogenic invasion in logistics process.
2) Protein-rich food. Protein-r ich food (milk, meat and
poultry, etc.) is suitable for microbial growth and micro-
be hazard is the most important factor affecting its logis-
tics safety, which can be expressed by safety reliability
model. Yet for low-protein-content food such as fruits
and vegetables, microbe hazards are not the main safety
factors; moreover, if not to be eaten as it is (e.g. fruits,
vegetables, etc.), instead, cleaned and even cooked for
consumption, then the propagation of pathogenic bacteria
may not cause actual harm to the human body. Thus the
model has better effect for the safety assessment of fresh
milk, chilled meat, frozen food and cooked food, etc.
3) Minimum logistics ambient temperature. When food
logistics ambient temperature is below the minimum
microbial growth temperature, the number of microbes
will not increase, food safety remains unchanged. Mean-
while in safety reliability model,
2
min
( )0
i
TT t−≥
indi-
cates that safety reliability is reduced. Therefore safety
reliability is only suitable for the situation when logistics
ambient temperature is higher than the minimum micro-
bial growth temperature. For the case of logistics am-
bient temperature being lower than the minimum micro-
bial growth temperature, as the microbes in a dormant
state, pathogenic bacteria number in the logistics process
will not increase, consequently at this moment there is no
need to consider the safety changes in logistics proc ess.
7. ACKNOWLEDGEMENTS
We thanks the National Nature Science Fund of China because our
research was supp orted by NSFC “Safety Reliability of Fresh Agricul-
tural Products Cold Chain Logistics and Its Dynamic Optimization”
Table 3. Safety Reliability of Each Cold Chain Unit after Relocation.
No. j Cold Chain Unit
j
f
j
w
*
j
f
*
j
R
1 Producing 80.0 0.123 59.9 0.982
2 Packing 72.0 0.110 54.0 0.965
3 Loading 28.9 0.044 21.7 0.959
4 Delivering 230.0 0.353 172.4 0.938
5 Unloading 51.2 0.078 38.5 0.926
6 Handling 45.0 0.069 33.7 0.916
7 Selling 144.5 0.221 36.1 0.901
Table 4. Comparison of shelf life results predicted by safety reliability model and original data in literature.
Food Type T (˚C) Shelf Life (h) Reference Model Calculation (h) Deviation
Sushi 17.4 4.7 [9] 4.26 9.3%
Sushi 21.8 3.1 [9] 3.49 12.6%
Chicken 8.0 168 [15] 187 11.3%
Pork 4.0 11 6 [16] 124 6.9%
Pork 10.0 60 [16] 55 9.2%

Y. F. Zou et al. / Agricultur al Sciences 4 (2013) 70-75
75
(NO.71172077) and “The study on control mechanism and key para-
meters optimization of perishable food multi-temperature refrigerated
transport” (NO. 51008087).
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