Multi-objective Optimization for Location-Routing-Inventory Problem in Cold Chain Logistics Network with Soft Time Window Constraint

Cold chain is a special supply chain with strict temperature requirements in the handling, holding, and transportation of products. Owing to the time-sensitivity of the quality of perishable products, it is important for cold chain logistics to meet the requirement on service time. To reduce supply chain cost, the cold chain logistics enterprise must satisfy the service time requirement by optimizing its location, routing, and inventory strategy. The service time can be divided into fuzzy time window (FTW), hard time window (HTW), and soft time window (STW).

Taking STW for example, every cold chain logistics enterprise must deliver the products in the STW required by customers. Early or late arrival will be penalized by customers according to the violation time, pushing up the total cost. To meet customer demand, perishable products need to be distributed within the prescribed time window constraint (TW). This helps to reduce the corruption rate of perishable products, and guarantee product quality.

This paper models the location-inventory-routing problem (LIRP) of cold chain logistics network (CCLN) under soft time window constraint (STW). In the model, customers have the right to penalize the cold chain logistics enterprise according to the length of violation time, forcing the enterprise to provide cold chain logistics services within the STW specified by customers.

Since the quality of perishable products declines with time, the corruption cost was considered to optimize the service cost. Then, the multi-objective ant colony optimization (MACO) was improved to solve the LIRPSTW. Simulation results show that the improved MACO can effectively minimize the total cost of the cold chain logistics enterprise in LIRPSTW.

3.1 Problem description

To meet the STW required by customers, the penalty cost was introduced to the LRIPSTW optimization model. As shown in Figure 1, the two-echelon CCLN consists of a manufacturer, several retailers, and multiple distribution centers (DCs).

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Figure 1. The structure of the CCLN

This paper aims to fulfil three objectives: select the best DCs to minimize the location cost; determine the inventory quantity based on the quantity of ordered perishable products to minimize the inventory cost; optimize the distribution route to meet the STW of retailers to minimize the transportation cost and penalty cost.

3.2 Assumptions

(1) The CCLN has only one manufacturer, and each retailer is only served by on DC.

(2) The customer demand obeys random distribution.

(3) Each vehicle must leave from and return to the same DC, and serve only one retailer.

(4) All vehicles are of the same type with the same capacity.

(5) The loss of perishable products in the producing area is not considered.

(6) The temperature throughout the distribution process meets the quality and safety standards for perishable products.

(7) The corruption rate of perishable products remains constant.

(8) The penalty cost is imposed whenever a vehicle exceeds the STW.

3.3 Notations

(1) Sets

I is the set of DCs;

J is the set of retailers;

K is the set of vehicles.

(2) Parameters

D_jis the annual demand of retailer J;

Q_i is the order quantity of DC_i;

f_iis the fixed construction cost of DC_i;

O_i is the cost for DC_i to order products from the manufacturer;

h_iis the unit inventory holding cost of DC_i;

L_iis the lead time of DC_i;

α is the service level of DC;

P is the unit price of products;

θ is the corruption rate of products during transportation;

d_i is the travel distance from the manufacturer to DC_i;

d_giis the travel distance from node (customer) g to node (customer) i;

p_t is the transportation cost per unit distance of each product;

V_iis the mean vehicle speed from the manufacturer to DC_i;

V_gj is the mean vehicle speed from node g to node j;

N_iis the maximum capacity of DC_i;

b is the maximum capacity of each vehicle.

(3) Decision variables

X_i is the quantity of products transported the manufacturer to DC_i;

Y_ij is the quantity of products transported from node i to node j;

$U_{i}=\left\{\begin{array}{cc}1 & \text { If } i \text { was chosen to be } \mathrm{DC} \\ 0 & \text { otherwise }\end{array} i \in I\right.$;

$R R_{i j}=\left\{\begin{array}{lc}1 & \text { If } \mathrm{DC}_{i} \text { serves retailer } J \\ 0 & \text { otherwise }\end{array} i \in I, j \in J\right.$;

$q_{g j k}=\left\{\begin{array}{lr}1 & \text { If vehicle travels from node } g \text { to node } j \\ 0 & \text { otherwise }\end{array}\right.$

$\forall g \in(I \cup J), \quad j \in J, k \in K$

3.4 Mathematical model

$\min {{Z}_{1}}=\sum\limits_{i\in I}^{{}}{{{f}_{i}}}{{U}_{i}}$    (1)

$\min {{Z}_{2}}={{p}_{t}}(\sum\limits_{i\in I}^{{}}{{{X}_{i}}}{{d}_{i}}{{U}_{i}}+\sum\limits_{k\in K}{\sum\limits_{g\in (I\bigcup J)}{\sum\limits_{j\in J}{{{Y}_{gj}}{{d}_{gj}}{{q}_{gjk}})+}}}p[\sum\limits_{i\in I}^{{}}{{{X}_{i}}}\left( 1-{{e}^{{-\theta {{d}_{i}}}/{{{v}_{i}}}\;}} \right)+\sum\limits_{k\in K}{\sum\limits_{g\in (I\bigcup J)}{\sum\limits_{j\in J}{{{Y}_{gj}}\left( 1-{{e}^{{-\theta {{d}_{gj}}}/{{{v}_{gj}}}\;}} \right){{q}_{gjk}}]}}}$    (2)

$\min Z_{3}=\sum_{i \in I}\left(O_{i} \sum_{j \in J} D_{j} R R_{i j} / Q_{i}+h_{i} Q_{i} / 2+h_{i} z_{\alpha_{0}} \sqrt{L_{i} \sum_{j \in J} \sigma_{j}^{2} R R_{i j}}\right)$    (3)

$\min Z_{4}=P_{d e} \sum_{j \in J} \max \left(E_{j}-T_{j k}, 0\right)+P_{d L} \sum_{j \in J} \max \left(T_{j k}-L_{j}, 0\right)$    (4)

Subject to:

$Q_{i}+z_{\alpha_{0}} \sqrt{L_{i} \sum_{j \in J} \sigma_{j}^{2} R R_{i j}} \leq N_{i}$    (5)

$\sum\limits_{j\in J}^{{}}{{{D}_{j}}}\sum\limits_{j\in J}^{{}}{{{q}_{gjk}}}\le b$    (6)

$\sum\limits_{k\in K}^{{}}{\sum\limits_{g\in (I\bigcup J)}^{{}}{{{q}_{gjk}}}}=1$    (7)

$\sum\limits_{j\in J}^{{}}{\sum\limits_{g\in (I\bigcup J)}^{{}}{{{q}_{gjk}}}}\le 1$    (8)

$\sum\limits_{k\in (I\bigcup J)}^{{}}{{{q}_{gjk}}}-\sum\limits_{k\in (I\bigcup J)}^{{}}{{{q}_{jgk}}}=0$    (9)

$\sum\limits_{i\in I}^{{}}{{{X}_{i}}}\ge \sum\limits_{j\in J}^{{}}{{{Y}_{ij}}}$    (10)

$\sum\limits_{i\in I}^{{}}{{{Y}_{ij}}}={{D}_{j}}$    (11)

${{T}_{jk}}=({{T}_{gk}}+S{{T}_{gk}}+{{d}_{gj}}/\text{ }{{V}_{gj}})\times {{q}_{gjk}}$,

$\forall g\in (I\bigcup J),j\in J,k\in K$    (12)

${{U}_{i}}=\left\{ 0,1 \right\},i\in I$    (13)

$R{{R}_{ij}}=\left\{ 0,1 \right\},i\in I,j\in J$    (14)

${{q}_{gjk}}=\left\{ 0,1 \right\},\forall g\in (I\bigcup J),j\in J$    (15)

Objective function (1) aims to minimize the location cost of DC in CCLN. Objective function (2) aims to minimize the transportation cost. Objective function (3) aims to minimize the inventory cost. Objective function (4) aims to minimize the penalty cost for the violation of the STW. Constraint (5) limits the capacity of each DC. Constraint (6) limits the capacity of each vehicle. Constraint (7) means each retailer is only served by one DC. Constraint (8) means each vehicle only serves one DC. Constraint (9) means no vehicle can remain on any node. Constraint (10) means the quantity of products transported to DCs is greater than that transported to the retailers. Constraint (11) means the demand of each retailer can be fulfilled. Constraint (12) means the arrival time of vehicle k at retailer j. Constraints (13)-(15) defines the possible values.

Table 3. The simulation results

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Figure 5. The cost change with the number of iterations

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Figure 6. The comparison of convergence curves

Figure 5 displays the variations in the minimum values of location cost, inventory cost, transportation cost, penalty cost and the total cost with the number of iterations. It can be seen that every cost item of the enterprise continued to decrease with the growing number of iterations, which demonstrates the effectiveness of the improved MACO. The minimum values of location cost and inventory cost fluctuated greatly between the 5^th and 30^th iterations, suggesting that the improved MACO converges to the global optimal solution at the expense of an increase in an objective function. Any cost reduction will push up other costs. Therefore, there is no unique solution to the multi-objective problem, which optimizes different objectives simultaneously. Instead, a Pareto optimal solution set was formed for the LRIPSWT. The cold chain logistics enterprise can choose the best solution from the set based on the actual situation and requirements.

To verify its feasibility and effectiveness, the improved MACO was compared with the original MACO, using the same initial parameters. Figure 6 compares the convergence curves of the two algorithms. It can be seen that the improved MACO converged to the optimal solution in 62 iterations, while the original MACO converged in 130 iterations. Thus, the improved MACO clearly outperforms the original MACO in solving the LIRPSWT.