LIRP optimization of cold chain logistics in satellite warehouse mode of supermarket chains

Abstract

Aiming at the problem of high cost in cold chain logistics of fresh products home-delivery in supermarket chain in the new retail era, the paper constructs the model of Location Inventory Routing Problem (LIRP) optimization in Satellite Warehouse mode in view of customer satisfaction with the broken line soft time windows. The model minimizes the total cost of the cold chain logistics system of supermarket chain through the location allocation, inventory optimization, the determination of distribution service relationship between Satellite Warehouse and customer, and the constraint of time penalty cost. Then, the paper designed an improved ant colony optimization to solve the LIRP model of supermarket chain. Finally, the simulation in MATLAB verifies and analyzes the validity of the model and algorithm. Therefore, LIRP optimization model in Satellite Warehouse mode can effectively improve the operational efficiency of fresh products home-delivery in the supermarket chain and thus reduce the logistics cost.

Keywords

Supermarket chain Satellite Warehouse cold chain logistics LIRP

1 Introduction

With higher levels of consumption and the pursuit of a healthier lifestyle, people are paying increasing attention to the quality and freshness of food in recent years. The advance of Internet technologies has enabled online shopping to reshape the consumer experience in a brand-new way, thus giving rise to new retail. New retail refers to a new way of retail combining online and offline with logistics. The new retail mainly relies on big data of the Internet to upgrade the circulation and delivery of goods, so that consumers can get new consumption experience. Compared with online retail, which benefits from the convenience and low operating costs, the traditional brick-and-mortar retailing system is facing unprecedented challenges. Under these threatens, some domestic supermarket chains have attempted online grocery, including delivery service. Profiting from the stable supply chain, the well-established procurement network, and the geological proximity to their customers, supermarket chains are expected further to eliminate the boundary between online and offline services as well as improve customer satisfaction. As a typical representative of the traditional brick-and-mortar retailer, the operating cost of supermarket chains has been rising in recent years. It is facing unprecedented challenges in its development with the heavy blow of changes in market demand and the impact of online retail. In China, relied on the developed purchasing network and good consumption experience, some supermarket chains with physical brands and stable supply have settled in home-delivery market of fresh products (fresh products home-delivery is a form of service in which fresh and healthy fresh products are delivered to the place designated by consumers without the need for consumers to purchase them personally at the place of sale). They have infused fresh blood to the development of supermarket chains.

However, currently supermarket chains seldom succeed in developing the fresh products home-delivery service. High cost of cold chain logistics is the main restriction to the development. Fresh food has a higher requirement for storage temperature and time. On the other hand, consumers have higher requirements for the quality of products, and they also have higher requirements for logistics delivery speed, especially for the fresh products. These factors promote the supermarket chains to approach a more efficient distribution mode of cold chain logistics system.

Satellite Warehouse refers to a small warehouse with a service radius of 3–5 km, which is relatively close to consumers. It calls Satellite Warehouse because of the small area. The primitive advantage of Satellite Warehouse distribution mode lies in reducing the logistics cost with the large-scale benefit. In fact, the mode replaces the direct delivery from the distribution center to customers with the large-scale transportation from distribution center to Satellite Warehouse. The second advantage of the mode is low requirements for cold chain equipment during the delivery. The distance is short from Satellite Warehouse to customers, and a short-time delivery will not have much influence on deterioration of the fresh products. Satellite Warehouse distribution mode was first adopted by the fresh food e-commerce because of a short distance, fast delivery and low operating cost. Different from the mode of fresh food e-commerce, supermarket chains have to reconcile the online part with offline stores. For site selection, the construction of new dark warehouse and the reconstruction of offline stores are both adopted to achieve mutual development online and offline, and also to expand business for the fresh products. However, system optimization of location allocation, inventory control and routing decision become relatively complex, because the supermarket chains juggle online and offline business and different modes of cost accounting are involved. The difficulty has an increase in LIRP optimization of cold chain Logistics. Therefore, supermarket chains’ LIRP optimization in Satellite Warehouse mode has immediate significance.

LIRP of the logistics system refers to determination of the quantity, size and location of the warehouse according to customers’ need, and optimizing the inventory strategy and vehicle routing after the determination. At the beginning, relative researches focused on the combination of two kinds of decisions, such as location inventory decision, location routing decision and inventory routing decision. Later, scholars noticed that there is an obvious trade-off in the optimization of location-inventory-routing. For example, the increase in distribution centers will lead to a rise in inventory costs, but it will shorten the vehicle routing, thus reduce transportation costs. Therefore, research from a practical and global perspective is needed to minimize the total cost of location-inventory-routing decision.

On the basis of on multi-depot location routing problem (MDLRP), Liu and Lee (2003) first proposed the LIRP model for single product multi-depot location routing in regard of inventory control, and they designed a two-phase heuristic algorithm for the solution [6]. Then, Liu and Lin (2005) constructed the model of a combined location routing and inventory problem (CLRIP) based on customer demands, and solved the model with hybrid heuristic algorithm, and verified the model and algorithm [7]. Diverse perspective is appeared in further researches on location-inventory-routing optimization. Panagiotis et al. (2019) designed a Location-Inventory-Routing Problem (LIRP) optimization model of the complex supply chain from the perspective of distribution outsourcing. Due to the complicated calculation, they designed a general variable neighborhood meta-heuristic algorithm to solve the model. Finally, they verified the fit of the model and algorithm by solving large-scale cases, however, the proposed models and algorithms are inappropriate for small and medium-sized problem instances [5]. Ahmad Sayed et al. (2019) studied the inventory location routing problem optimization model from the perspective of a vendor managed inventory, and designed an improved genetic algorithm for solution. They verified the model and algorithm by combining two examples (10 and 30 customers) and found that the model and algorithm have obvious advantages in the solution to the inventory location routing problem with a high number of customers [10]. Compared with ordinary temperature commodities, fresh products have strict requirements on storage time and temperature, thus it is more complex for the research on LIRP optimization of fresh products. Abdelhalim et al. (2017) focused on the LIRP model for perishable products. They added location decision of alternative warehouse to inventory path optimization, and achieved the integration of strategic decision, tactical decision and operation level decision. In addition, they devoted a new approach of chromosome coding and local search heuristics to solve the model [3]. Faced with the uncertainty of customer demands and the diversity of distribution vehicles, Zahra et al. (2018) established a LIRP model of perishable products with multiple periods and varieties, and designed a heuristic algorithm based on Lagrange to solve the model. In their research, the effectiveness of the model and algorithm was verified by simulation experiments [9]. The perishability of fresh food causes high frequency of logistics distribution, uncertain demands and frequent occurrence of customers chargeback. Aiming at the problems, Yang Xiaohua et al. (2019) proposed a multi-period closed-loop logistics network for fresh products with fuzzy variables, and improved Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm for the solution to model [19].

In summary, the greater part of the literature focuses on LIRP optimization problem of the logistics system. However, few published studies noticed the LIRP problem of the cold chain logistics system of fresh food.

The increasing demand for fresh products in the consumer market ask for adding some constraints in the future research, such as time window restrictions, vehicle models and parking lots, to meet customer needs and market changes. Then, specific objectives of optimization will be achieved, such as the minimum total cost of logistics, low carbon emissions. Current window constraints research on customer time often defines the penalty cost and the time deviation as a simple linear relationship. In fact, for the deviation of the best service time window, there are tolerant customers and the intolerants. Therefore, in view of customers’ tolerance, the paper established a broken line soft time window for construction of penalty function, and LIRP optimized in Satellite Warehouse system of supermarket chains for the solution to practical problems.

2 Problem analysis and description

2.1 Problem description

The paper examined a three-echelon logistics network consisting of a single distribution center, multiple Satellite Warehouse and customers. Satellite Warehouse periodically sends out order requests to the distribution center according to the inventory level. Then the supermarket chain matches the orders to each Satellite Warehouse, and the staff of the Satellite Warehouse directly sorts the goods and arranges the delivery vehicle to transport the goods to the designated location within the time period requested by the customer. In the case of satisfying the following assumptions, the location and inventory optimization of the Satellite Warehouse new construction and reconstruction process are realized, and the allocation service relationship between the Satellite Warehouse and the customer is optimized.

2.2 Model assumptions

a) The fresh products delivered by the Satellite Warehouse are all of the same kind, and the deterioration rate of the fresh products is constant.

b) The distribution center can satisfy the order requirements of the Satellite Warehouse.

c) The number, location and capacity limits of the alternative Satellite Warehouse are given.

d) Satellite Warehouse implements (Q, R) inventory control mode.

e) A Satellite Warehouse can be equipped with multiple similar distribution vehicles. Each vehicle belongs to only one Satellite Warehouse, and can serve multiple customers and return to the Satellite Warehouse.

f) Define customers with closer physical distances as the same customer, the location of the customer is known, the demand of customer is a random variable, subject to normal distribution and independent of each other.

g) The distribution task must be completed within the time period requested by the customer. Otherwise penalty costs will be incurred.

2.3 Symbols and models

2.3.1 Symbol description

I: represents the collection of all Satellite Warehouse, I ={ i|i = 1, 2, ·· · , m };

J: Represents a collection of customers, J ={ j|j = 1, 2, ·· · , n };

K: Represents a collection of vehicles, K ={ k|k = 1, 2, ·· · , l };

Decision variables include

x_i: 1 If the Satellite Warehouse i is opened, 0 otherwise;

y_ij: 1 If the Satellite Warehouse i provides distribution services for customer j, 0 otherwise;

L_ghk: 1 If the vehicle k passes from node g to node h, 0 otherwise;

Intermediate variables include:

μ_i: Expected value of daily demand for Satellite Warehouse i, $μ_{i} = \sum_{j = 1}^{n} y_{ij} μ_{j}$ ;

$σ_{i}^{2}$ : Variance of daily demand for Satellite Warehouse i, $σ_{i}^{2} = \sum_{j = 1}^{n} y_{ij} σ_{j}^{2}$ ;

2.3.2 Determination of the objective function

(1) The location cost of Satellite Warehouse.

The expression of Satellite Warehouse location cost is Z₁, illustrated in Equation (1). $Z_{1} = \sum_{i = 1}^{m} F_{i} x_{i} + \sum_{i = 1}^{m} H_{i} x_{i}$ (1)

F_i is the opening and operating cost of “new Satellite Warehouse” i (unit/year), H_i is the transformation and operation cost of “transformed Satellite Warehouse” i.

(2) Total inventory cost of Satellite Warehouse.

① Order cost is illustrated in Equation (2). ${WC}_{i} = \frac{η μ_{i}}{Q_{i}} Q_{i} P_{1} = η μ_{i} P_{1}$ (2)

η is the annual working days of chain supermarkets; μ_i is the expected value of daily demand for Satellite Warehouse i, Q_i is the single order batch of Satellite Warehouse i, P₁ is the unit transportation cost from the distribution center to the Satellite Warehouse, which includes the costs incurred by the cold chain equipment.

② Inventory holding cost is illustrated in Equation (3).

$\begin{matrix} {HC}_{i} = G_{i} P_{2} \int_{0}^{T_{i}} R_{i} (t) dt = \frac{P_{2}}{θ} \\ [G_{i} (e^{θ T_{i}} - 1) \cdot (z_{α} σ_{i} \sqrt{L} + \frac{μ_{i}}{θ}) - μ_{i} η] \end{matrix}$ (3)

T_i is the order period of Satellite Warehouse, P₂ is the cost incurred by the Satellite Warehouse for storing fresh food per unit weight, R_i (t) is the inventory level at time t, θ is the deterioration rate of fresh products, z_α is the safety stock factor, L is the order lead time, G_i is the number of annual orders for the Satellite Warehouse.

③ Loss cost is illustrated in Equation (4).

PC_i = G_iP₃ (Q_i - μ_iT_i) is the loss cost, $Q_{i} = (e^{θ T_{i}} - 1) ({ss}_{i} + \frac{μ_{i}}{θ})$ , ${ss}_{i} = z_{α} σ_{i} \sqrt{L}$ , put Q_i into PC_i, Then the annual loss cost is as follows,

${PC}_{i} = G_{i} P_{3} (z_{α} σ_{i} \sqrt{L} (e^{θ T_{i}} - 1) + μ_{i} (\frac{e^{θ T_{i}} - θ T_{i} - 1}{θ}))$ (4)

P₃ is the unit loss cost of the product, Q_i is the single order batch of Satellite Warehouse i, ss_i is the safety stock quantity of Satellite Warehouse.

④ Out-of-stock cost

The out-of-stock cost of Satellite Warehouse refers to the loss caused by the supply interruption due to insufficient inventory. The calculation formula is illustrated in Equation (5). ${SC}_{i} = G_{i} P_{4} σ_{i} [α + {ss}_{i} \cdot (α - 1)]$ (5)

P₄ is the unit out-of-stock cost of the Warehouse i, α is the inventory service level, so the total inventory cost of the Satellite Warehouse is illustrated in Equation (6) [1]. $Z_{2} = \sum_{i = 1}^{m} ({WC}_{i} + {HC}_{i} + {PC}_{i} + {SC}_{i})$ (6)

(3) Distribution cost

Satellite Warehouse distribution costs Z₃ include transportation costs and vehicle start-up costs, the distribution cost is illustrated in Equation (7). $Z_{3} = \sum_{i = 1}^{n} \sum_{g = 1}^{n + m} \sum_{h = 1}^{n + m} (η P_{5} d_{gh} L_{ghk} + P_{6} L_{ghk}) x_{i}$ (7)

P₅ is the unit transportation cost of the distribution vehicle, d_gh is the transportation distance from node g to node h, and P₆ is the starting cost of the vehicle.

(4) Time penalty cost and broken line soft time window constraint.

The paper adds the penalty cost of distribution to the objective function, and studies the LIRP multi-objective decision optimization in the Satellite Warehouse mode constricted by the broken line soft time window. In the traditional soft time window, regardless of whether the vehicle arrives before ET_j or after LT_j, The penalty cost and the degree of time deviation are generally simple linear relationships, and the penalty cost function is shown in Fig. 1. In fact, for the deviation of the best service time window, customers have the difference in tolerance. If within the tolerance range of the customer, the deviation of the unit time window only needs to pay less penalty cost; if it exceeds the tolerance range of the customer, and more penalty cost will be paid. Based on the traditional soft time window, the paper proposed a broken line soft time window in view of customers’ tolerance, and the penalty cost function is shown in Fig. 2. According to the customer’s tolerance level ɛ and service duration s_j, a tolerance time window[eT_j, lT_j]can be obtained on the basis of the best time window, eT_j = ET_j - ɛS_j, lT_j = LT_j + ɛS_j. If the vehicle provides services within the optimal time window [ET_j, LT_j], the customer’s satisfaction is the highest and there is no penalty cost; if the vehicle provides services within the interval [eT_j, ET_j] or [LT_j, lT_j], the deviation of the unit time window will only be paid less penalty cost; if the vehicle starts service time earlier than eT_j or later than lT_j, the deviation of the unit time window will have to pay more penalty cost [4]. Compared with the traditional soft time window, the broken line soft time window takes into account the actual experience of customers, which is conducive to the chain supermarkets to better meet customer needs, while rationally allocating and optimizing resources.

Fig. 1

Penalty costs under traditional soft time window.

Fig. 2

Penalty costs under broken line soft time window.

The corresponding expression is illustrated in Equation (8).

$Z_{4} = f (a_{j}) = {\begin{matrix} C_{1} ({eT}_{j} - a_{j}) + C_{2} ({ET}_{j} - {eT}_{j}) a_{j} ⩽ {eT}_{j} \\ C_{2} ({ET}_{j} - a_{j}) {eT}_{j} < a_{j} ⩽ {ET}_{j} \\ 0 {ET}_{j} < a_{j} ⩽ {LT}_{j} \\ C_{3} (a_{j} - {LT}_{j}) {LT}_{j} < a_{j} ⩽ {lT}_{j} \\ C_{3} ({lT}_{j} - {LT}_{j}) + C_{4} (a_{j} - {lT}_{j}) a_{j} > {lT}_{j} \end{matrix}$ (8)

ɛ is the customer’s tolerance level, S_j is the duration of service received at customer j, [ET_j, LT_j] is the best service time window for customer j, eT_j is the upper limit of the tolerance time window for customer j, lT_j is the lower limit of the tolerance time window for customer j, eT_j = ET_j - ɛS_j, lT_j = LT_j + ɛS_j, if vehicle k starts serving customers before eT_j, the unit time penalty cost for violating the time window is C₁; if vehicle k starts serving customers within [eT_j, ET_j], it violates the unit of the time window penalty cost is C₂; if the vehicle k starts serving customers within [ET_j, LT_j], the penalty cost it should pay is 0; if the vehicle k starts serving customers within [LT_j, lT_j], its unit time penalty cost for violating the time window is C₃; if the vehicle k starts to serve customers after lT_j, the unit time penalty cost for violating the time window is C₄. Among them, the setting of parameter C should meet C₁ > C₂ and C₄ > C₃.

(5) Overall objective function and constraints $min Z = Z_{1} + Z_{2} + Z_{3} + Z_{4}$ (9) $\sum_{g = 1}^{n + m} \sum_{k = 1}^{l} L_{ghk} = 1 h = 1, 2, \cdot \cdot \cdot n$ (10) $\sum_{g = 1}^{m} \sum_{h = 1}^{m} L_{ghk} ⩽ 1 k = 1, 2, \cdot \cdot \cdot, l$ (11) $\sum_{g, h = 1}^{n + m} L_{ghk} - \sum_{h, g = 1}^{n + m} L_{ghk} = 0 k = 1, 2, \cdot \cdot \cdot, l$ (12) $\sum_{i = 1}^{m} x_{i} ⩾ \sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{ij}$ (13) $μ_{i} ⩽ C_{Q} i = 1, 2, \cdot \cdot \cdot, m$ (14) $\sum_{g = 1}^{n + m} \sum_{h = 1}^{n + m} L_{ghk} ⩽ 6$ (15) $w_{g} = max {a_{g} - {ET}_{g}, 0} g = 1, 2, \cdot \cdot \cdot, n + m$ (16) $\begin{matrix} a_{h} = a_{g} + w_{g} + {dis}_{gh} + s_{a}, g \neq h, \\ g = 1, 2, \cdot \cdot \cdot, m + n, h = 1, 2, \cdot \cdot \cdot, n [6] \end{matrix}$ (17)

In the above model, the objective function indicates that the total cost of the cold chain logistics LIRP system of the chain supermarket is the smallest; the meaning contained in the constraint conditions is: Equation (10) indicates that the demand of each customer can only be satisfied by one warehouse and one distribution vehicle. Equations (11) and (12) indicate that the start node and end node of each delivery vehicle can only be a Satellite Warehouse, and Equation (13) indicates that the customer’s needs can only be met by the selected Satellite Warehouse, Equation (14) indicates the capacity limitation of the Satellite Warehouse, Equation (15) indicates that each Satellite Warehouse corresponds to a maximum of 6 delivery vehicles, and Equation (16) indicates the waiting time of the delivery vehicle, a_g is the arrival of the vehicle the time of customer g, ET_g is the start time of the best delivery. Equation (17) indicates the time when the vehicle arrives at the customer h = the time the vehicle arrives at the last customer g⁺ the waiting time of the vehicle at the customer g⁺ the time the vehicle provides services to customer g⁺ the time spent from customer g to customer h.

3 Algorithm design

Since the decision variables within the cold chain logistics system of supermarket chain are closely related and many customers involved, the calculation process is with heavy workload. Therefore, it is necessary to design the corresponding algorithm based on the characteristics of the Satellite Warehouse model. This paper combines the ant colony algorithm with genetic algorithm, and make use of genetic algorithm global search to make up for long search time by ant colony algorithm, and to avoid local optimization of the resulting solution. Practical steps include: first add the solution obtained by the ant colony algorithm to the initial population of the genetic algorithm, and then use the genetic algorithm selection, crossover, mutation and other operational processes to obtain the offspring population of better performance. Individuals in the top 20% fit are directly copied to the next generation. In the initial stage of the improved optimization algorithm, the genetic algorithm can quickly converge around the optimal solution, and then use the advantages of the local optimization of the ant colony algorithm to find a more accurate optimal solution. Finally, an example verifies the effectiveness of the algorithm.

3.1 Algorithm flow

(1) Chromosome coding method

This algorithm employes the natural number coding method. Each Satellite Warehouse is equipped with k vehicles for customer distribution. If there are m Satellite Warehouse and n customers, a chromosome of length (n + km - 1) can be formed. Each distribution vehicle has a corresponding Satellite Warehouse. {1, 2, 3, ·· · , n } represents the customer, {n+ 1, n + 2, ·· · , (n + km - 1) } represents the vehicle, and only the data is transferred to the kmth delivery vehicle without coding.

For example, suppose there are 3 Satellite Warehouse and 15 customers, each Satellite Warehouse is configured with 2 vehicles, then 1 to 15 are customer numbers, 16 to 20 are delivery vehicle numbers, vehicles 16 and 17 belong to the first Satellite Warehouse, vehicles 18 and 19 belong to the second Satellite Warehouse, vehicles 20 and 21 belong to the third Satellite Warehouse, and only data is transferred to the vehicle 21 without coding. Suppose a chromosome numbered {16,1,2,3,17,4,5,6,18,19,7,8,9,10,11,20,12,13,14,15} is generated,Then, the chromosome can be decomposed to generate distribution routes, that is, route one 16, 1, 2, 3, route two 17, 4, 5, 6,, route three 18, 19, 7, 8, 9, 10,11, route four 20,12,13,14,15, where route one represents customers 1, 2, and 3 being served by vehicle No. 16, and route two represents customers 4, 5, and 6 being served by vehicle No. 17, and route 3 has two cars next to each other, indicating that customers 7, 8, 9, 10, and 11 are provided by No. 19 car, No. 18 car is not activated, and route 4 indicates customers 12, 13, 14, 15 are served by the 20th car, then this chromosome indicates that the three alternative Satellite Warehouse are open. Assuming a certain chromosome {1,2,3,16,17,18,19,4,5,6,7,8,9,10,11,20,12,13,14,15}, then route one 1, 2, 3, route two 16, 17, 18, 19, 4, 5, 6, 7, 8, 9, 10, 11, route three 20, 12, 13, 14, 15, because there is no delivery vehicle code on route one, the delivery vehicle with the number 21 will provide delivery service for customers 1,2; route two means that customers 4, 5, 6, 7, 8, 9, 10, and 11 will be served by vehicle No. 19, route three indicates that customers 12, 13, 14, 15 are served by car 20, the second and third Satellite Warehouse in this chromosome are open, and the delivery vehicle corresponding to the first Satellite Warehouse is not activated, indicating that the Satellite Warehouse has not been opened.

(2) State transition rules

In this paper, the ant’s state transition is defined as the distribution vehicle’s choice of the next node. The deterministic selection is combined with the pseudo-random ratio to obtain the distribution vehicle’s choice of the next node. In other words, when the distribution vehicle is located at customer i, it will use the deterministic selection rule with probability q₀ or the pseudo-random proportional selection rule with probability 1 - q₀ to select the next customer j.

Deterministic selection rules is illustrated in Equation (18).

$P_{j} = {\begin{matrix} max_{j \in J_{k} (i)} [τ_{ij} (t)]^{α} [η_{ij} (t)]^{β} [γ_{j}]^{ɛ} [ω_{j}]^{λ} q ⩽ q_{0} \\ P_{ij}^{k} otherwise \end{matrix}$ (18)

Pseudo-random ratio rules is illustrated in Equation (19).

$P_{ij}^{k} = {\begin{matrix} \frac{[τ_{ij} (t)]^{α} [η_{ij} (t)]^{β} [γ_{j}]^{ɛ} [ω_{j}]^{λ}}{\sum_{j \in J_{k} (i)} [τ_{ij} (t)]^{α} [η_{ij} (t)]^{β} [γ_{j}]^{ɛ} [ω_{j}]^{λ}} j \in J_{k} (i) \\ 0 otherwise \end{matrix}$ (19)

In the formula, τ_ij (t) is the pheromone concentration on the connection path between node i and node j at time t, η_ij (t) =1/d_ij is the heuristic function, which represents the ant’s expectation to move from node i to node j, γ_j represents the time difference between the time of ants from i to j and the upper limit of the broken line soft time window (the latest service start time that the customer can accept), ω_j represents the waiting time at j after the vehicle goes from i to j, and α determines the importance of τ_ij (t), and β determines the η_ijt importance, ɛ, λ determine the importance of γ_j and ω_j respectively, q₀ ∈ (0, 1) is a constant, q is a random number in the interval (0, 1), J_k (i) is the set of the next node when ant k is at the node, $P_{ij}^{k}$ is when ant k is at node i. At this time, the closer to the ant k, the greater the pheromone concentration, and the shorter the time difference and the shorter the waiting time, the easier the node j to be selected.

(3) Update pheromone

The pheromone on the path will gradually disappear as each ant releases the pheromone, so after the end of a cycle, the pheromone concentration needs to be updated in time. The pheromone update formula is illustrated in Equation (20). $\begin{matrix} {\begin{matrix} τ_{ij} (t + 1) = (1 - ρ) τ_{ij} (t) + Δ τ_{ij} \\ Δ τ_{ij} = \sum_{k = 1}^{n} τ_{ij}^{k} \end{matrix} 0 < ρ < 1 \end{matrix}$ (20)

$Δ τ_{ij}^{k}$ represents the concentration of pheromone released when ant k walks between nodes i and j; Δτ_ij represents the total concentration of pheromone released by all ants walking between nodes i and j; ρ is the pheromone volatilization factor.

In this paper, the ant cycle system model is used to calculate Δτ_ij, that is, when ant k is transferred from node i to node j, the released pheromone concentration is Δτ_ij = Q/L_k, otherwise it is Δτ_ij = 0. Among them, Q is a constant, which means all the pheromones produced by the ant in one cycle; L_k is the total length of the path that the ant k traverses.

(4) Cross operation

The crossover operation refers to randomly selecting two individuals, swapping and combining some genes of two chromosomes under the set crossover probability P_c, and inheriting the excellent characteristics of the parent to the offspring, so as to achieve the recombination of the chromosome genes. The two chromosomes to be crossed are X and Y. After the crossover operation, new chromosomes X₁ and Y₁ are obtained. In this study, the sequential crossover method is used for crossover operation. The schematic diagram is shown in Fig. 3.

Fig. 3

Schematic diagram of cross operation.

(5) Mutation operation

Chromosome variation is to prevent the reduction of population diversity and premature phenomenon. The operation process includes randomly selecting a chromosome, randomly selecting two gene points to exchange positions in the selected chromosome, calculating the chromosomal adaptation value after mutation, if better, it is retained.

The steps of algorithm design are as follows, the flow chart is shown in Fig. 4

Fig. 4

Algorithm flowchart.

1) Initialize the parameters. Let NC = 0, set the number of ants m, the maximum number of ant colony algorithm cycles NC_max, and the number of genetic algorithm iterations NG.

2) Place all ants randomly on the starting point.

3) Determine whether the ant transfer process has formed a complete path. If yes, proceed to step 5. Otherwise, follow step 4 to select the next customer.

4) The kth ant selects the next customer based on the combination of deterministic selection and pseudo-random ratio and updates the pheromone.

5) Put the solution of ant colony algorithm in the initial population of genetic algorithm.

6) Select, cross and mutate to generate new offspring.

7) Determine whether the maximum number of iterations of the genetic algorithm is reached, if so, proceed to step 9, otherwise proceed to step 6.

8) Update the pheromone matrix according to τ_ij (t + 1) = (1 - ρ) τ_ij (t) + Δτ_ij.

9) Determine whether the maximum cycle number NC_max of the ant colony algorithm is reached, and if so, output the optimal path and optimal solution, otherwise proceed to step 2.

4 Example verification

4.1 Case introduction

A case study of a chain supermarket operation project in Tangshan City was utilized as an example to analyze the model and algorithm using Matlab2019b software. The supermarket chain plans to build and renovate the Satellite Warehouse in Lubei District, Tangshan City. Chain supermarket’s working days are 360 days/year. Distribution of alternative Satellite Warehouse and customers is shown in Figure 5, marked by continuous natural numbers. Selection of positions and customers, in which figure marks 36 customers, indicated by numbers 1 to 36, figure marks 13 alternative Satellite Warehouse, indicated by numbers 1 to 13. The distance between the alternative Satellite Warehouse and each customer is measured by baidu map ranging tool. The distance between some nodes is shown in Table 1.

Fig. 5

Distribution of alternative Satellite Warehouses and customer points.

Table 1

The distance between some nodes

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	0.0	1.8	5.1	6.8	2.5	1.2	1.6	5.3	1.7	7.4	3.6	8.1	7.8	8.1	4.6	4.3	0.8	1.3
2	1.8	0.0	3.9	5.4	1.4	1.5	1.1	3.8	1.5	5.7	2.9	6.3	6.0	6.5	4.2	3.4	2.3	1.2
3	5.1	3.9	0.0	1.8	2.7	5.2	3.6	0.9	5.3	3.2	1.9	4.3	3.8	3.4	2.3	1.2	5.9	3.9
4	6.8	5.4	1.8	0.0	4.3	6.8	5.2	1.5	6.8	1.9	3.7	3.0	2.5	1.6	3.9	3.0	7.6	5.6
5	2.5	1.4	2.7	4.3	0.0	2.7	0.9	2.8	2.8	5.0	1.6	5.7	5.4	5.5	3.0	2.1	3.3	1.3
6	1.2	1.5	5.2	6.8	2.7	0.0	2.0	5.3	0.5	7.2	4.1	7.7	7.4	8.0	5.4	4.8	1.1	1.9
7	1.6	1.1	3.6	5.2	0.9	2.0	0.0	3.6	2.2	5.8	2.1	6.6	6.2	6.4	3.4	2.8	2.4	0.4
8	5.3	3.8	0.9	1.5	2.8	5.3	3.6	0.0	5.3	2.5	2.5	3.4	3.0	2.8	3.1	2.1	6.0	4.1
9	1.7	1.5	5.3	6.8	2.8	0.5	2.2	5.3	0.0	6.9	4.4	7.4	7.2	7.8	5.7	5.0	1.6	2.2
10	7.4	5.7	3.2	1.9	5.0	7.2	5.8	2.5	6.9	0.0	5.0	1.0	0.6	1.1	5.6	4.5	8.1	6.3
11	3.6	2.9	1.9	3.7	1.6	4.1	2.1	2.5	4.4	5.0	0.0	5.9	5.5	5.2	1.4	0.7	4.5	2.3
12	8.1	6.3	4.3	3.0	5.7	7.7	6.6	3.4	7.4	1.0	5.9	0.0	0.5	1.9	6.5	5.5	8.6	7.0
13	7.8	6.0	3.8	2.5	5.4	7.4	6.2	3.0	7.2	0.6	5.5	0.5	0.0	1.5	6.1	5.1	8.3	6.7
14	8.1	6.5	3.4	1.6	5.5	8.0	6.4	2.8	7.8	1.1	5.2	1.9	1.5	0.0	5.6	4.7	8.8	6.9
15	4.6	4.2	2.3	3.9	3.0	5.4	3.4	3.1	5.7	5.6	1.4	6.5	6.1	5.6	0.0	1.2	5.5	3.5
16	4.3	3.4	1.2	3.0	2.1	4.8	2.8	2.1	5.0	4.5	0.7	5.5	5.1	4.7	1.2	0.0	5.1	3.0
17	0.8	2.3	5.9	7.6	3.3	1.1	2.4	6.0	1.6	8.1	4.5	8.6	8.3	8.8	5.5	5.1	0.0	2.1
18	1.3	1.2	3.9	5.6	1.3	1.9	0.4	4.1	2.2	6.3	2.3	7.0	6.7	6.9	3.5	3.0	2.1	0.0
19	2.6	2.2	2.7	4.5	1.1	3.2	1.2	3.2	3.5	5.6	1.1	6.5	6.1	6.1	2.2	1.7	3.5	1.4
20	3.5	2.0	1.9	3.4	1.0	3.5	1.9	1.8	3.5	4.0	1.7	4.8	4.4	4.6	3.0	1.9	4.2	2.3

Figure 5 Distribution of alternative Satellite Warehouses and customer points.

4.2 Basic data

Some of the parameters in the model are: Satellite Warehouse capacity limit C_Q is 1200 kg, the freight for fresh products from the distribution center to the Satellite Warehouse P₁ is 1 yuan / kg, the distribution vehicle model is the same, the distribution vehicle startup cost P₆ is 35000 yuan / year, each vehicle has a load capacity of 150kg, and the penalty factor set by each delivery vehicle for not meeting the time limit of the customer is C₁ = 1/12, C₂ = 1/24, C₃ = 1/3, C₄ = 1/12. The upper limit of the number of Satellite Warehouse delivery vehicles is 6, and the unit transportation cost P₅ of fresh products delivered by distribution vehicles is 1 yuan /km. The deterioration rate θ of the fresh products stored in the Satellite Warehouse is 0.03, the inventory holding cost P₂ is 4 yuan /kg, the loss cost P₃of the fresh products is 5 yuan /kg, and the out-of-stock cost due to the supply interruptionP₄ is 1 yuan /kg. Satellite Warehouse inventory service levelα is 90%, order lead time L is 1 day. The construction cost and management cost of the newly built Satellite Warehouse are shown in Table 2, and the construction cost and operation cost of Satellite Warehouse that may be transformed are shown in Table 3. The relevant parameters of the customer are known, as shown in Table 4.

Table 2
Cost Parameters of Newly Built Satellite Warehouse

Warehouse number Coding number Fixed cost Management fees Rental area Monthly rent Rental cost Total cost

1 1 20000 100000 350 5250 63000 183000

2 2 20000 100000 350 7350 88200 208200

3 3 20000 100000 350 7000 84000 204000

4 4 20000 100000 400 7600 91200 211200

9 9 20000 100000 400 7600 91200 211200

10 10 20000 100000 350 6650 79800 199800

11 11 20000 100000 400 8400 100800 220800

12 12 20000 100000 350 6650 79800 199800

Warehouse number	Coding number	Fixed cost	Management fees	Rental area	Monthly rent	Rental cost	Total cost
1	1	20000	100000	350	5250	63000	183000
2	2	20000	100000	350	7350	88200	208200
3	3	20000	100000	350	7000	84000	204000
4	4	20000	100000	400	7600	91200	211200
9	9	20000	100000	400	7600	91200	211200
10	10	20000	100000	350	6650	79800	199800
11	11	20000	100000	400	8400	100800	220800
12	12	20000	100000	350	6650	79800	199800

Table 3

Cost parameters of Satellite Warehouse that may be transformed

Warehouse number	Coding number	Warehouse type	Warehouse area	Fixed cost	Operating cost	Retrofit cost	Total cost
5	5	Large warehouse	200	20000	100000	113400	233400
6	6	Medium warehouse	70	20000	100000	52800	172800
7	7	Large warehouse	150	20000	100000	105600	225600
8	8	Medium warehouse	90	20000	100000	91200	211200
13	13	Small warehouse	50	20000	10000	57600	177600

Table 4

Customers Parameter values

Customer	Daily demand expectation μj (kg)	Daily demand variance σj2 (kg)	Service time window [eTj, ETj, LTj, lTj]	Operating time time
1	37	3	[10.5, 11, 17, 17.5]	1
2	42	4	[24.5, 25, 31, 31.5]	1
3	35	5	[28.5, 29, 35, 35.5]	1
4	43	4	[6.5, 6.5, 19, 19.5]	1
5	27	5	[10.5, 11, 17, 17.5]	1
6	33	3	[30.5, 31, 37, 37.5]	1
7	52	2	[18, 19, 31, 32]	2
8	26	6	[13.5, 14, 20, 20.5]	1
9	34	4	[12.5, 13, 25, 25.5]	1
10	42	6	[26, 27, 33, 34]	2
11	51	3	[20, 21, 27, 28]	2
12	67	6	[14, 15, 21, 22]	2
13	65	2	[6, 7, 19, 20]	2
14	46	6	[15, 16, 22, 23]	2
15	35	4	[21.5, 22, 28, 28.5]	1
16	46	6	[10, 11, 17, 18]	2
17	58	2	[22, 23, 29, 30]	2
18	61	5	[24, 25, 31, 32]	2
19	47	6	[14, 15, 23, 24]	2
20	49	3	[1, 2, 13, 14]	2
21	37	6	[15.5, 16, 21, 21.5]	1
22	29	3	[12.5, 13, 21, 21.5]	1
23	38	4	[10.5, 11, 19, 19.5]	1
24	37	4	[6.5, 7, 19, 19.5]	1
25	44	2	[12, 13, 27, 28]	2
26	31	5	[21.5, 22, 31, 31.5]	1
27	52	6	[28, 29, 37, 38]	2
28	39	3	[18.5, 19, 25, 25.5]	1
29	33	5	[26.5, 27, 35, 35.5]	1
30	26	2	[12.5, 13, 19, 19.5]	1
31	38	5	[10.5, 11, 21, 21.5]	1
32	52	2	[14, 15, 27, 28]	2
33	46	2	[24, 25, 37, 38]	2
34	35	5	[6.5, 7, 13, 13.5]	1
35	27	3	[14.5, 15, 21, 21.5]	1
36	55	4	[10, 11, 19, 20]	2

4.3 Results analysis

The algorithm parameters are as follows: α =1, β =1.5, ρ =0.5, q₀ =0.3, Q = 1, the crossover probability is 0.6, the mutation probability is 0.1, and the maximum number of iterations is 1000. Use Matlab R2019b software to run. The iteration curve of the algorithm is shown in Fig. 8. It can be seen that the algorithm starts to converge at the 200th iteration, close to the optimal value, and obtains the optimal solution at the 550th iteration. The cost of the optimal solution is shown in Table 5. The total cost of the cold chain logistics system for fresh food in supermarket chains is 2,279,300 yuan. The total location cost is 637,200 yuan, accounting for 28% of the total cost of the logistics system; the total inventory cost is 125,2436 yuan, accounting for 55% of the total cost of the logistics system; the total distribution cost is 370,584 yuan, accounting for 16% of the total cost of the logistics system; and the total time penalty cost is 19,080 yuan, accounting for 1% of the total cost of the logistics system. It can be seen that the inventory cost accounts for the largest proportion of the total cost of the logistics system. Since the location cost of the warehouse is relatively high, and the location cost of each warehouse varies greatly, it is necessary to consider whether the distribution cost and time penalty cost can effectively reduce the total cost of the logistics system when choosing a warehouse with high location cost.The optimal solution of the model is shown in Table 6. Satellite Warehouse are established at 1, 5, and 11, respectively, of whichSatellite Warehouse1 and 11 are newly-built “ Satellite Warehouses”, and “ Satellite Warehouses” 5 are reconstruction “ Satellite Warehouses”, resulting in 9 distribution routes; three routes starting from the newly built Satellite Warehouse 1, covering 10 customer sites, with a total cost of 728256 yuan. The transformation of No. 5 Satellite Warehouse produced four routes with a total cost of 965994 yuan, which is logistics the highest total cost of the system, but the transformed Satellite Warehouse has the most customers covered by geographical advantages, accounting for half of the total customers. The new Satellite Warehouse 11 generates two routes, the total cost it is 585050 yuan, which is the smallest total cost in the open Satellite Warehouse, but it covers only 7 customer sites. Programming of main parameters is shown in Fig. 6 and the main implementation process of the algorithm is shown in Fig. 7.

Table 5
Costs corresponding to the optimal solution

Location cost Inventory cost Transportation cost Penalty cost Total cost

637200 1252436 370584 19080 2279300

Location cost	Inventory cost	Transportation cost	Penalty cost	Total cost
637200	1252436	370584	19080	2279300

Table 6

The optimal solution of the Satellite Warehouse logistics system model

Satellite Warehouse	Optimal cost (yuan)		Delivery route
1	Location cost	183000	37-11-4-25-37 37-27-15-18-37 37-19-21-29-28-37
	Inventory cost	417542
	Transportation cost	121236
	Penalty cos	6478
	Total cost	728256
5	Location cost	233400	41-36-10-32-41 41-16-6-33–20-41 41-9-31-12-14-26-30-41 41-23-22-34-3-5-7-41
	Inventory cost	555466
	Transportation cost	168656
	Penalty cos	8472
	Total cost	965994
11	Location cost	220800	47-8-24-2-1-47 47-13-35-17-47
	Inventory cost	279428
	Transportation cost	80692
	Penalty cos	4130
	Total cost	585050

Fig. 6

Programming of main parameters.

Fig. 7

Implementation process of the algorithm.

Fig. 8

Iteration curve.

In the research case, the chain supermarket plans to use self-support logistics to establish a fresh food cold chain logistics distribution system. In order to verify the effectiveness of the Satellite Warehouse mode, this paper compares the total cost generated by the self-operated logistics program with the total cost generated by the third-party logistics program, so as to explore the practical significance of the Satellite Warehouse mode of cold-chain supermarket logistics program. Carry out logistics cost calculation for the above cases with reference to the quotation of Shenxue Cold Transportation, a subsidiary of Shentong Express. Since the cost of cold chain packaging per order in the third-party cold chain logistics distribution process cannot be ignored, in the Satellite Warehouse case study, for ease of calculation, some customers with closer physical distances are set as the same customer. Therefore, in order to make a more meaningful comparison between the two schemes, based on the network survey and the actual situation of third-party logistics distribution, the order of each customer is divided into 2 - 5kg orders with uneven weight. The order quotation is shown in Table 7.

Table 7

Third-party logistics quotation

First weight (1kg)	Continued weight	Packaging costs
8 yuan	2 yuan/kg	6 yuan

Therefore, the total cost of third-party cold chain logistics can be expressed as $W = k \sum_{i = 1}^{n} (δ + b * ω + λ)$ . k is the number of working days in a year, δ is the cost of the first weight, b is the weight of the continued weight, ω is the unit continued weight, and λ is the packaging cost. The comparison results of the two methods are as follows.

It can be seen from Table 8 that the total cost of logistics generated by the Satellite Warehouse operation project is reduced by 920,000 yuan compared to the third-party logistics program, accounting for 71.14% of the total cost of the third-party logistics program. This shows that the scale efficiency generated by the operation mode of the Satellite Warehouse is sufficient to offset its increased cost of location, inventory and routing. This operation mode not only reduces the total cost of cold chain logistics system of chain supermarkets, but also improves the customer satisfaction and enterprise competitiveness of fresh delivery business of chain supermarkets, so the LIRP optimization model of the Satellite Warehouse model of chain supermarkets has practical significance.

Table 8

Satellite Warehouse logistics costs and third-party logistics costs comparison

Satellite Warehouse (yuan)	Third-party logistics (yuan)	Proportion	Decrease value (yuan)
2279300	3204000	71.14%	924700

5 Conclusion

The paper starts from the practical problems of the cold chain logistics system of fresh food in chain supermarkets, and applies the Satellite Warehouse model to the LIRP optimization problem of chain supermarkets. The specific work is reflected in the following aspects:

Supermarket chains already have stores in some areas of the city, but the distribution capacity is not enough to cover all customers, thus the two methods of site selection in new Satellite Warehouse and transformation of original stores are adopted. The existing stores are transformed into a compound warehouse with integrated transfer and service functions. It can meet the needs of customers both online and offline, and effectively reduce the total cost of the cold chain logistics system.

According to the reality of cold chain logistics in supermarket chains, the paper proposed the location-inventory-routing optimization model in the Satellite Warehouse mode to achieve the goal of the smallest total cost of the logistics system. Since customers have different tolerance limits for the deviation of delivery time, the paper constructed the time penalty cost in the broken line soft time window in view of customer satisfaction. Compared with the traditional soft time window, the broken line soft time window takes into account the actual feelings of customers, and the supermarket chains will meet the customer demands and allocate resources rationally with optimization.

For complexity of the model, the paper designed a corresponding ant colony genetic algorithm for the solution, and implemented a simulation experiment on a supermarket chain enterprise in Tangshan City. By comparison, the results showed that the system optimization model in the Satellite Warehouse mode effectively reduces the total cost of the cold chain logistics system and verifies the effectiveness of the model and algorithm.

The paper proposed a model of the cold chain logistics in chain supermarkets from the perspective of LIRP system optimization. For convenience, the fresh products in the research are simply abstracted into similar products; while in reality, chain supermarkets engage in a variety of fresh products. Therefore, the distribution of multiple varieties of fresh products in the Satellite Warehouse mode will be the new tendency for further research.

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	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	0.0	1.8	5.1	6.8	2.5	1.2	1.6	5.3	1.7	7.4	3.6	8.1	7.8	8.1	4.6	4.3	0.8	1.3
2	1.8	0.0	3.9	5.4	1.4	1.5	1.1	3.8	1.5	5.7	2.9	6.3	6.0	6.5	4.2	3.4	2.3	1.2
3	5.1	3.9	0.0	1.8	2.7	5.2	3.6	0.9	5.3	3.2	1.9	4.3	3.8	3.4	2.3	1.2	5.9	3.9
4	6.8	5.4	1.8	0.0	4.3	6.8	5.2	1.5	6.8	1.9	3.7	3.0	2.5	1.6	3.9	3.0	7.6	5.6
5	2.5	1.4	2.7	4.3	0.0	2.7	0.9	2.8	2.8	5.0	1.6	5.7	5.4	5.5	3.0	2.1	3.3	1.3
6	1.2	1.5	5.2	6.8	2.7	0.0	2.0	5.3	0.5	7.2	4.1	7.7	7.4	8.0	5.4	4.8	1.1	1.9
7	1.6	1.1	3.6	5.2	0.9	2.0	0.0	3.6	2.2	5.8	2.1	6.6	6.2	6.4	3.4	2.8	2.4	0.4
8	5.3	3.8	0.9	1.5	2.8	5.3	3.6	0.0	5.3	2.5	2.5	3.4	3.0	2.8	3.1	2.1	6.0	4.1
9	1.7	1.5	5.3	6.8	2.8	0.5	2.2	5.3	0.0	6.9	4.4	7.4	7.2	7.8	5.7	5.0	1.6	2.2
10	7.4	5.7	3.2	1.9	5.0	7.2	5.8	2.5	6.9	0.0	5.0	1.0	0.6	1.1	5.6	4.5	8.1	6.3
11	3.6	2.9	1.9	3.7	1.6	4.1	2.1	2.5	4.4	5.0	0.0	5.9	5.5	5.2	1.4	0.7	4.5	2.3
12	8.1	6.3	4.3	3.0	5.7	7.7	6.6	3.4	7.4	1.0	5.9	0.0	0.5	1.9	6.5	5.5	8.6	7.0
13	7.8	6.0	3.8	2.5	5.4	7.4	6.2	3.0	7.2	0.6	5.5	0.5	0.0	1.5	6.1	5.1	8.3	6.7
14	8.1	6.5	3.4	1.6	5.5	8.0	6.4	2.8	7.8	1.1	5.2	1.9	1.5	0.0	5.6	4.7	8.8	6.9
15	4.6	4.2	2.3	3.9	3.0	5.4	3.4	3.1	5.7	5.6	1.4	6.5	6.1	5.6	0.0	1.2	5.5	3.5
16	4.3	3.4	1.2	3.0	2.1	4.8	2.8	2.1	5.0	4.5	0.7	5.5	5.1	4.7	1.2	0.0	5.1	3.0
17	0.8	2.3	5.9	7.6	3.3	1.1	2.4	6.0	1.6	8.1	4.5	8.6	8.3	8.8	5.5	5.1	0.0	2.1
18	1.3	1.2	3.9	5.6	1.3	1.9	0.4	4.1	2.2	6.3	2.3	7.0	6.7	6.9	3.5	3.0	2.1	0.0
19	2.6	2.2	2.7	4.5	1.1	3.2	1.2	3.2	3.5	5.6	1.1	6.5	6.1	6.1	2.2	1.7	3.5	1.4
20	3.5	2.0	1.9	3.4	1.0	3.5	1.9	1.8	3.5	4.0	1.7	4.8	4.4	4.6	3.0	1.9	4.2	2.3

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	0.0	1.8	5.1	6.8	2.5	1.2	1.6	5.3	1.7	7.4	3.6	8.1	7.8	8.1	4.6	4.3	0.8	1.3
2	1.8	0.0	3.9	5.4	1.4	1.5	1.1	3.8	1.5	5.7	2.9	6.3	6.0	6.5	4.2	3.4	2.3	1.2
3	5.1	3.9	0.0	1.8	2.7	5.2	3.6	0.9	5.3	3.2	1.9	4.3	3.8	3.4	2.3	1.2	5.9	3.9
4	6.8	5.4	1.8	0.0	4.3	6.8	5.2	1.5	6.8	1.9	3.7	3.0	2.5	1.6	3.9	3.0	7.6	5.6
5	2.5	1.4	2.7	4.3	0.0	2.7	0.9	2.8	2.8	5.0	1.6	5.7	5.4	5.5	3.0	2.1	3.3	1.3
6	1.2	1.5	5.2	6.8	2.7	0.0	2.0	5.3	0.5	7.2	4.1	7.7	7.4	8.0	5.4	4.8	1.1	1.9
7	1.6	1.1	3.6	5.2	0.9	2.0	0.0	3.6	2.2	5.8	2.1	6.6	6.2	6.4	3.4	2.8	2.4	0.4
8	5.3	3.8	0.9	1.5	2.8	5.3	3.6	0.0	5.3	2.5	2.5	3.4	3.0	2.8	3.1	2.1	6.0	4.1
9	1.7	1.5	5.3	6.8	2.8	0.5	2.2	5.3	0.0	6.9	4.4	7.4	7.2	7.8	5.7	5.0	1.6	2.2
10	7.4	5.7	3.2	1.9	5.0	7.2	5.8	2.5	6.9	0.0	5.0	1.0	0.6	1.1	5.6	4.5	8.1	6.3
11	3.6	2.9	1.9	3.7	1.6	4.1	2.1	2.5	4.4	5.0	0.0	5.9	5.5	5.2	1.4	0.7	4.5	2.3
12	8.1	6.3	4.3	3.0	5.7	7.7	6.6	3.4	7.4	1.0	5.9	0.0	0.5	1.9	6.5	5.5	8.6	7.0
13	7.8	6.0	3.8	2.5	5.4	7.4	6.2	3.0	7.2	0.6	5.5	0.5	0.0	1.5	6.1	5.1	8.3	6.7
14	8.1	6.5	3.4	1.6	5.5	8.0	6.4	2.8	7.8	1.1	5.2	1.9	1.5	0.0	5.6	4.7	8.8	6.9
15	4.6	4.2	2.3	3.9	3.0	5.4	3.4	3.1	5.7	5.6	1.4	6.5	6.1	5.6	0.0	1.2	5.5	3.5
16	4.3	3.4	1.2	3.0	2.1	4.8	2.8	2.1	5.0	4.5	0.7	5.5	5.1	4.7	1.2	0.0	5.1	3.0
17	0.8	2.3	5.9	7.6	3.3	1.1	2.4	6.0	1.6	8.1	4.5	8.6	8.3	8.8	5.5	5.1	0.0	2.1
18	1.3	1.2	3.9	5.6	1.3	1.9	0.4	4.1	2.2	6.3	2.3	7.0	6.7	6.9	3.5	3.0	2.1	0.0
19	2.6	2.2	2.7	4.5	1.1	3.2	1.2	3.2	3.5	5.6	1.1	6.5	6.1	6.1	2.2	1.7	3.5	1.4
20	3.5	2.0	1.9	3.4	1.0	3.5	1.9	1.8	3.5	4.0	1.7	4.8	4.4	4.6	3.0	1.9	4.2	2.3

LIRP optimization of cold chain logistics in satellite warehouse mode of supermarket chains

Abstract

Keywords

1 Introduction

2 Problem analysis and description

2.1 Problem description

2.2 Model assumptions

2.3 Symbols and models

2.3.1 Symbol description

2.3.2 Determination of the objective function

3.1 Algorithm flow

4.1 Case introduction

Table 5 Costs corresponding to the optimal solution Location cost Inventory cost Transportation cost Penalty cost Total cost 637200 1252436 370584 19080 2279300

References

Table 5
Costs corresponding to the optimal solution

Location cost Inventory cost Transportation cost Penalty cost Total cost

637200 1252436 370584 19080 2279300

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	0.0	1.8	5.1	6.8	2.5	1.2	1.6	5.3	1.7	7.4	3.6	8.1	7.8	8.1	4.6	4.3	0.8	1.3
2	1.8	0.0	3.9	5.4	1.4	1.5	1.1	3.8	1.5	5.7	2.9	6.3	6.0	6.5	4.2	3.4	2.3	1.2
3	5.1	3.9	0.0	1.8	2.7	5.2	3.6	0.9	5.3	3.2	1.9	4.3	3.8	3.4	2.3	1.2	5.9	3.9
4	6.8	5.4	1.8	0.0	4.3	6.8	5.2	1.5	6.8	1.9	3.7	3.0	2.5	1.6	3.9	3.0	7.6	5.6
5	2.5	1.4	2.7	4.3	0.0	2.7	0.9	2.8	2.8	5.0	1.6	5.7	5.4	5.5	3.0	2.1	3.3	1.3
6	1.2	1.5	5.2	6.8	2.7	0.0	2.0	5.3	0.5	7.2	4.1	7.7	7.4	8.0	5.4	4.8	1.1	1.9
7	1.6	1.1	3.6	5.2	0.9	2.0	0.0	3.6	2.2	5.8	2.1	6.6	6.2	6.4	3.4	2.8	2.4	0.4
8	5.3	3.8	0.9	1.5	2.8	5.3	3.6	0.0	5.3	2.5	2.5	3.4	3.0	2.8	3.1	2.1	6.0	4.1
9	1.7	1.5	5.3	6.8	2.8	0.5	2.2	5.3	0.0	6.9	4.4	7.4	7.2	7.8	5.7	5.0	1.6	2.2
10	7.4	5.7	3.2	1.9	5.0	7.2	5.8	2.5	6.9	0.0	5.0	1.0	0.6	1.1	5.6	4.5	8.1	6.3
11	3.6	2.9	1.9	3.7	1.6	4.1	2.1	2.5	4.4	5.0	0.0	5.9	5.5	5.2	1.4	0.7	4.5	2.3
12	8.1	6.3	4.3	3.0	5.7	7.7	6.6	3.4	7.4	1.0	5.9	0.0	0.5	1.9	6.5	5.5	8.6	7.0
13	7.8	6.0	3.8	2.5	5.4	7.4	6.2	3.0	7.2	0.6	5.5	0.5	0.0	1.5	6.1	5.1	8.3	6.7
14	8.1	6.5	3.4	1.6	5.5	8.0	6.4	2.8	7.8	1.1	5.2	1.9	1.5	0.0	5.6	4.7	8.8	6.9
15	4.6	4.2	2.3	3.9	3.0	5.4	3.4	3.1	5.7	5.6	1.4	6.5	6.1	5.6	0.0	1.2	5.5	3.5
16	4.3	3.4	1.2	3.0	2.1	4.8	2.8	2.1	5.0	4.5	0.7	5.5	5.1	4.7	1.2	0.0	5.1	3.0
17	0.8	2.3	5.9	7.6	3.3	1.1	2.4	6.0	1.6	8.1	4.5	8.6	8.3	8.8	5.5	5.1	0.0	2.1
18	1.3	1.2	3.9	5.6	1.3	1.9	0.4	4.1	2.2	6.3	2.3	7.0	6.7	6.9	3.5	3.0	2.1	0.0
19	2.6	2.2	2.7	4.5	1.1	3.2	1.2	3.2	3.5	5.6	1.1	6.5	6.1	6.1	2.2	1.7	3.5	1.4
20	3.5	2.0	1.9	3.4	1.0	3.5	1.9	1.8	3.5	4.0	1.7	4.8	4.4	4.6	3.0	1.9	4.2	2.3