Application of Cold Chain Logistics Safety Reliability in Fresh Food Distribution Optimization

In view of the nature of fresh food’s continuous decrease of safety during distribution process, this study applied safety reliability of food cold chain logistics to establish fresh food distribution routing optimization model with time windows, and solved the model using MAX-MIN Ant System (MMAS) with case analysis. Studies have shown that the mentioned model and algorithm can better solve the problem of fresh food distribution routing optimization with time windows.


INTRODUCTION
In the current studies of fresh food distribution system, most of them just considered physical loss or time window restraints during food distribution process on the basis of TSP problem. For example, Sexton and Choi (1986) evaluated single vehicle routing and scheduling problem with time windows by Bender's Decomposition method. Taillard et al. (1997) adopted the concept of Tabu Search for solving vehicle routing problem with soft windows. Koskosidis et al. (1992), based on general assignment mode proposed by Fisher and Jaikumar (1981), offered an optimal-solution-based heuristic method. Malandraki and Daskin (1992) explored the vehicle routing problem with time windows and location requirements. Li et al. (2006) optimized the distribution system considering loss of goods. However, physical loss of food and food safety is not the same concept. Even with no significant physical changes, some food such as milk, sushi, etc., may have not met the sanitary standards. Aiming at this situation, this study introduced safety reliability of food cold chain logistics to the vehicle routing optimization problem, set up a model and used MAX-MIN Ant System to solve it.

SAFETY RELIABILITY OF FOOD COLD CHAIN
The reliability is the probability of a process or a product performing its intended function for a stated period of time under specified operating conditions. So, the term of the safety reliability of food logistics unit is defined as the probability of food logistics unit retaining the safety of food within an intended scope for a stated period of time under specified operating conditions (Zou et al., 2010a). This probability is determined by the number of microorganism in foods, because bacterial hazard is the most serious factor influencing food's safety and microbial number in food varies dynamically in the course of logistics. Though food also suffers from physical hazard and chemical hazard during logistics, the probabilities of those hazards are generally considered as tiny statistical constants for a practical food logistics system. Hence, the greater the number of microorganism presented in food, the smaller the safety reliability of food is, vice versa.
According to "T. T. T" (Time-Temperature-Tolerance) theory of food logistics, the number of microorganism in specific food during logistics is mainly controlled by temperature and time. Many models can be applied for modeling the growth of microorganism in food. Zwietering et al. (1996) believed that the following exponential model had best prediction effect after he researched several microbial growth models: where, N t = The concentration of microorganism at time t (CFU/g) N 0 = The initial concentration of microorganism (CFU/g) b = A parameter in experiment T = The temperature during logistics (°C) T min = The temperature at which no growth will occur (°C) λ = The lag time of microorganism growth Let ΔT = T -T min , the above equation can be converted as: To eat putrid food is liable to be sick. The illness probability is proportional to the common logarithm of microorganism number in food. Therefore, safety reliability of food can be defined as follows:  The value of safety reliability plus illness probability is 1.  The safety reliability of food is 1 if the concentration of microorganism is equal to or less than 1 CFU/g. The reason is: if the concentration of microorganism in food is 1 CFU/g, then log 10 N = 0, therefore the safety reliability of food is 1; if the concentration of microorganism in food is less than 1 CFU/g, food safety is better, its safety reliability is also 1.  If the concentration of microorganism in food reaches N D (the minimum concentration to cause food borne disease), the safety reliability of food is zero. It means that at this point the food cannot be eaten and the safety reliability of food is the least.
For a cold chain logistics system composed of m cold chain units, its safety reliability R i after food continuously passes through the i th (i≤m) unit can be expressed as (Zou et al., 2010b): where, R 0 = The initial safety reliability of food cold chain system ΔT j = T j -T min (T j is the temperature in logistics unit j) t j = Logistics time of cold chain unit j d = A parameter related to food variety

MODELING
As illustrated in Fig. 1, food distribution network G (V, A) generally consists of distribution center, refrigerated vehicles and retail sites. The problem can be described as follows: the distribution center has m vehicles of same model and deadweights Q 0 ; these vehicles set off from distribution center (V 0 ), pass through all retail sites (V i : I = 1, 2, … n) and then return to distribution center; the food distributed are of single type and food safety continuously change; the demand quantity of retailer i is q i and bounded with delivery time, otherwise rejected; the requirement is to arrange the dispatch sequence, dispatch time and routing of vehicles starting from fresh food distribution center in such a way as to minimize the total cost and maximize the safety of the distribution system subject Note that we set the lowest comprehensive distribution cost and the highest safety reliability as subjective function. The comprehensive distribution cost includes vehicle transport cost and loss cost of food safety reliability.
Here, we consider the duration of vehicle k departing from distribution site i until it leaving the next distribution site j to be a cold chain logistics unit U ij , whose distribution time includes the travel time from distribution site i to j t d ijk and the process time at site j (containing loading and unloading time, receiving time and meal time) t h jk : Assume the temperature in the vehicle remains constant during the distribution process, then the loss of food safety reliability after it passes cold chain logistics unit U ij is: L represents the set of sites following site i in route k. Because demand quantity of retailers are not identical, total loss of safety reliability is not only associated with delivery time but also with the sequence of retail sites in the route. Let n k the number of retail sites committed to vehicle k, R k the set of routes that vehicle k has passed, q k i the demand quantity of retail site of sequence i in route k, then total loss of food safety reliability from retail site of order i-1 to retail site of order i in route k is: and loss of safety reliability in the entire distribution process is: Assume variable cost of the vehicle C d is proportional to its travel time, then: , 0 The sale of fresh food features high timeliness that have to limit its distribution with time windows. We apply hard time window constraints because of the characteristics of high timeliness and coordination in food cold chain logistics, consequently the constraint m j ≤e j ≤n j (where e j is the arrival time of vehicle at the customer j) is appended to the optimization model.
To sum up, the fresh food distribution routing optimization model with time windows and load capacity is as follows:

SOLUTION
Fresh food distribution routing optimization problem with time windows is an extension of VRPTW problem, and hence a NP-hard problem without effective polynomial algorithms as of yet. Ant Colony Optimization (ACO) algorithms are heuristic methods proved effective to find approximation to large-scale TSP problem. MAX-MIN Ant System (MMAS) is the modification of ACO algorithm that can effectively prevent search stagnation by putting limits on maximum and minimum value of pheromone trails (τ max , τ min ) on each path. The algorithm steps are as follows: Step 1: Initialization of correlated variables: Set the initial time Δτ ij = 0, Pheromone trail on each path τ ij = 1, iteration number nc ← 0, k ← 1, vehicle driving time T solu = 0, vehicle remaining load Q net = Q 0 , set of retail sites in outstanding demand V net = {V 1 , V 2 , …, V n }, Z best = M, M is a relatively large positive number.
Step 2: Define the set of allowable candidate moves by ants V allow via the restraints of vehicle load capacity and time windows. Judge whether V allow is an empty set, if it is, set k ← k + 1, T solu = 0, Q net = Q, V allow = V net .
Step 3: Calculate the move probability of ants' candidate nodes: If j ∈ V allow , generate a random number and choose the next node ' V , ants are to be positioned according to the mentioned random number and probability, then update Q net , T solu and V net .    Step 4: Judge whether V net is an empty set. If not, return to step 2; if it is an empty set, means The deliveries to all sites have accomplished, then take note of number of ants m ← k.
Step 5: Update pheromone trails on each side (i, j): Step 6: Define and update the higher bound and lower bound on values of pheromone trails: Otherwise; 0<k<8, k>8 Step 7: Concerning each side (i, j): Set Δτ ij ← 0; nc ← nc + 1. Calculate objective function value and judge whether the value has improved. Take note of the solution so far if the target function value is improved.
Step 8: If nc<N C (the maximum iteration number with no change of best-so-far solution), re-iterated; otherwise abort.
Case: A distribution center of certain fresh milk producer covers 15 retail sites demanding for distribution service. The load capacity of vehicle is no more than 2 tons; the demand quantity of 15 sites, the arrival time windows of conveyances and the processing time after arrival are shown in attached Table 1; Assume distribution temperature is 10°C, fresh milk's experimental constant d = 10 -5 , traffic conditions among the distribution sites are same, the vehicle speed is 50 km/h, unit transportation cost is RMB Yuan 1.00/km, unit loss of fresh milk's safety is RMB Yuan 10,000/ton. The problem is how should the distribution center arrange its distribution route and priority order so as to obtain the lowest logistics distribution costs (distribution cost and loss cost of safety) satisfying the constraints of arrival time windows. We use MATLAB 7.0 for programming and substitute with the parameters and distance matrix, α = 1, β = 5, ρ = 0.5, number-of-ants = 15, NC = 10. The results show that it requires 6 vehicles for the distribution service and the routing is: 0-9-2-0, 0-8-6-3-0, 0-12-4-0, 0-13-10-11-0, 0-5-15-14-0, 0-1-7-0. The load capacity and distribution costs of each vehicle are shown as Table 2. For this case, the total distribution cost is RMB Yuan 1,718.80.

CONCLUSION
It is necessary to consider loss of food safety reliability when evaluate fresh food distribution routing optimization problem, consequently it will be more difficult to solve the distribution routing optimization model. The example demonstrates that using MMAS can better solve fresh food delivery routing optimization problem with time windows.