Entropy-Maximization-Based Customer Order Allocation of Clothing Production Enterprises in the Sharing Economy

With the rapid development of the sharing economy, more and more platform operators apply the sharing concept in manufacturing, which increases the efficiency of assets utilization. Considering the apparel industry, clothing enterprises or manufacturers may share their excess orders between each other via a manufacturing cloud platform. Under the traditional production mode, manufacturers focus on processing their individual orders. There may be a coexistence of insufficient and surplus production capabilities. Some manufacturers cannot meet their customer demands due to limited capabilities and some orders have to be rejected, while some other manufacturers may have excess capacities with insufficient demands. It results in loss of revenue, and it is not conducive to maintaining a good customer relationship. In this paper, we consider a shared system with multiple manufacturers that produce homogeneous products, and the manufacturers in the shared system can share customer orders with each other. Once any manufacturer cannot fulfill all of its orders, the unsatisfied ones will be shared with other manufacturers that have surplus capacities with the aim of improving the balance of resource utilization and risk resistance of all manufacturers on the platform. The entropy maximization theory is mainly adopted to facilitate the formulation of the objective function. We apply a Taylor expansion to reformulate the objective function and construct a mixed-integer quadratic programming (MIQP) model. We employ off-the-shelf solvers to solve small-scale problems, and also propose a two-stage constructive heuristic algorithm to solve large-scale problems. Numerical experiments are conducted to demonstrate the efficiency of the algorithm.


Introduction
With the growth and maturity of the e-commerce system in China, there has been a commensurate growth in the textile and apparel e-commerce industry, which has continued to grow and develop.In 2020, despite the negative impact of the COVID-19 epidemic, the volume of textile and apparel e-commerce transactions still showed steady growth, reaching CNY 7.29 trillion, an increase of 8.97% year-on-year.This tends to illustrate that we can expect a further broad development in the apparel industry [1].In the traditional production mode in the apparel industry, customers submit production orders to their respective clothing manufacturers, and the manufacturers process the orders by themselves and then transport the finished products to the corresponding customers.All the relevant manufacturing and delivery activities for each production order occur between the customer and the corresponding manufacturer, and the information about the production order, traditionally regarded as a kind of trade secret, cannot be shared among manufacturers.Therefore, it is quite possible to see a rise in the problem of workload imbalance among manufacturers.Contrary to the traditional processing mode, the sharing economy mode can well solve the issue of the simultaneous existence of resource shortage and resource surplus among manufacturers.In a shared economy mode, each manufacturer receives customer orders via a sharing platform.If there are manufacturers failing to meet all their orders, the sharing platform will transfer their surplus orders to other manufacturers with excess capacities.
In practice, Alibaba's 1688 Tao factory platform for e-commerce is a typical application where over ten thousand small apparel manufacturers share their manufacturing abilities and satisfy customer demands with cooperation.Compared with thousands of offline orders, the number of orders placed by e-commerce customers each time is relatively small, only a few hundred or even dozens.However, many of these customers are highquality merchants of Taobao and Tmall.They have strong ability to deliver goods, and often reorder after a week.One Alibaba 1688 clothing supply chain expert said "frankly speaking, although the order quantity is small, the profit is high".As another example, Luyou Fiber Co., Ltd., established in 1966, is one of the major silk stocking manufacturers in Taiwan, China, and is also a hosiery copacker of Chanel, Armani, Miu Miu and other brands.After 2000, with the rise of skirts and sandals, consumers began to regard silk stockings as popular goods rather than necessities.Diversified demand has become an inevitable trend of development.In order to survive and develop, Luyou has become an order receiving platform, connecting 50 small factories to manufacture orders together.Each manufacturer on the platform obtains a different number of production qualifications.If the specified manufacturer cannot meet its orders due to capacity limitations, Luyou allocates the remaining orders to other manufacturers.Today Luyou is no longer limited to the production of silk stockings and underwear, but also engaged in the customized production of leather clothing, vest and other clothing [2].
Compared with the standardized customization and economics of scale of large factories, the small-batch and diversified productions of small factories are more in line with consumer needs.However, small factories are more vulnerable to unexpected accidents because of their small scale of operations.At the same time, unexpected incidents influenced our world more intensely than then yeas ago (Liu et al. [3]), especially in recent years, as all kinds of unexpected failures occur more frequently from the impact of COVID-19.Therefore, how to improve the ability to cope with the risk of force majeure for small factories in the current economy is a meaningful and important question.
The studies on customer order allocation mainly focus on the manufacturing industry, and most of them aim to maximize the total profitability and the customer satisfaction (Mafakheri et al. [4], Zhang et al. [5]), minimize procurement cost (Ruengsa and Van Nguyen [6]), minimize total cost (Liu et al. [7]) and so on.In this article, our objective is to maintain customer relations by improving the stability of the shared system, in other words, we attempt to minimize the overall losses when there are manufacturers undergoing failure.In this work, we propose a new framework for sharing customer orders among enterprises such that it can minimize the losses caused by the unexpected failure of several manufacturers with the purpose to keep cooperation relationships with as many customers as possible.
In order to quantify the above objective, we employ the entropy theory and enlarge its applications.The concept of entropy originates from physics and is used to measure the degree of disorder of a thermodynamic system.The most disordered system contributes to the maximum entropy and will reach the most stable or antirisk state.The key strength of entropy maximization (EM) is that it is easy to minimize the risk of something unexpected happening when all uncertainties are considered.In relation to our customer order allocation problem mentioned above, we aim to make a better customer order sharing scheme via minimizing the risk of failure of manufacturing, from the viewpoint of the platform operator.
Similar to Teye et al. [8] and Xie et al. [9], we use the measure of entropy to evaluate the antirisk level of the manufacturing system, which includes all manufacturers on the sharing platform.That is, the order allocation scheme with maximum entropy corresponds to the minimum profit loss, provided that some manufacturers experience failure.Mean-while, feasible sharing schemes in our problem are subject to practical constraints that manufacturers are of limited processing capacities.
The main contributions of this work are threefold: (i) For the problem of customer order allocation in the sharing economy, we introduce the criterion of entropy maximization to optimize the order sharing scheme which minimizes the profit loss of the manufacturing system in case several manufacturers fail to produce.
(ii) We establish a mathematical model with the objective function of entropy maximization and approximately transform the model into a mixed-integer quadratic programming model by means of a second-order Taylor expansion.
(iii) We propose an efficient two-stage constructive heuristic to handle large-scale instances of the considered problem.
The remainder of this work is organized as follows.First, a brief literature review of relevant studies is given in Section 2. Section 3 describes the considered problem and presents several fundamental assumptions.In Section 4, we propose the mathematical model, transform the objective function of entropy maximization into a minimization objective and approximate the original model with a mixed-integer quadratic program.We further devise a two-stage constructive heuristic in Section 5. Numerical experiments are conducted in Section 6 to validate its efficiency.Finally, Section 7 concludes this work and suggests future research directions.

Literature Review
In this work, we focus on the production order sharing among manufacturers in the sharing economy and evaluate the allocation schemes of manufacturing resources.Below we mainly review relevant studies on the utilization of the metric of entropy maximization and the order allocation problems.

Applications of Maximum Entropy Metric
Nowadays the idea of entropy maximization has been applied in various areas, such as traffic assignment equilibrium (Lu et al. [10]; Xie and Nie [11]), network detection (Banerjee and Maity [12]), and computer field (Chen et al. [13]).
Ref. [14] discussed the origin of information entropy, the difference between information entropy and thermodynamic entropy, the role of information entropy in complexity theories.Ref. [13] introduced the concept of entropy into the multiprocessor and multicomputer area to find even distribution strategies to improve the computing performance of whole cluster systems.Christodoulou et al. [15] proposed a method of resource allocation and scheduling for resource constrained projects by using entropy metrics to make projects more orderly and stable and more manageable.
The combination of the entropy maximization principle and the location problem has been the focus of many scholars.The mixed-integer linear programming method (Arnold et al. [16]; Lin et al. [17]) is commonly used for locating facilities, such as in the intermodal facility location problem, especially under uncompetitive environments.However, it is unreasonable because this method leads to an "all-or-nothing" assignment of flows.Therefore, Teye et al. [8] employed the method of entropy maximization to combine the choice model with a facility location model, where users had options not to use the intermodal terminals.The proposed model could be decomposed into a location problem and a mode choice problem, where terminal operators minimized the fixed cost of an IMT location while users minimized the transport cost of using the facility.Teye et al. [18] further enhanced this model, by including the cost of the IMT location in the overall costs.Furthermore, a complete enumeration algorithm was employed to solve the instance of the problem.Finally, a case study implemented in New South Wales, Australia, was given to verify the effectiveness of the enhanced model.Teye et al. [19] proposed a model based on the principle of entropy maximization, which could also be decomposed into a facility location problem and a mode choice problem.The mode choice problem was cast as a three-level nested probability model used to estimate shipper demand for a given location.It was shown that for terminals with sufficiently large handling capacities, the objective function of the mode choice problem was the same as that of the overall problem, and the result could be used to locate facilities to maximize the shippers' expected utility.An efficient algorithm was also proposed for solving large instances.
Christodoulou [20] studied an integrated traffic modeling and college bus routing problem.They presented a method by which traffic flow estimation between known origins and destinations could be evaluated based on a modified entropy model, and bus routing optimization could be performed.Li et al. [21] addressed the task partitioning and scheduling problem in a multithreaded execution environment.They developed a race-condition-aware and hardwareoriented task partitioning and scheduling algorithm, using the entropy maximization model so as to decrease the risk of computational errors and system hazards.
Xie et al. [9] presented a joint maximum entropy-least squares estimator for the subnetwork origin-destination trip matrix estimation problem.

Order Allocation Problems
The studies on customer order allocation in the current literature are often associated with supplier selection and supply chain optimization.Several scholars have investigated the integrated multiobjective decision-making method, which considers both supplier selection and order allocation problems (Azadeh et al. [22], Mafakheri et al. [4]).In addition, there are several scholars, such as Hamdan et al. [23], Yeh and Chuang [24], who have focused on order allocation under the green criteria.
The growth of the logistics service supply chain (LSSC) has directed considerable research attention on its order allocation.Zhang et al. [5] built an LSSC under a cloud environment, established an optimization model with an aim to maximize customer satisfaction and viewed the order allocation process as the cloud service matching process.In addition, several studies have examined the order allocation problem in a two-echelon LSSC (Liu et al. [3], Liu et al. [25]).Liu et al. [3] presented a multiobjective planning model for the emergency order allocation of a two-echelon LSSC.Liu et al. [25] discussed the order allocation model in a two-echelon LSSC when the FLSP has pre-estimated behavior.Liu et al. [26] conducted a study on the order allocation of a three-echelon LSSC.
The above research studies show that the order allocation is a scientific problem worth studying.To the best of our knowledge, however, few of them have considered the robustness of order allocation in the sharing economy.Based on this, this paper introduces the measure of entropy to evaluate the antirisk level of the manufacturing system.Specifically, we attempt to generate a clothing order allocation solution on a sharing platform using the method of maximum entropy, against the loss of profit due to the failures of any manufacturers in order to keep customers.

Problem Description and Fundamental Assumptions
In this section, we offer the description and fundamental assumptions of the considered problem.

Problem Description
On an apparel production order sharing platform, there are |N| clothing manufacturers to process production orders demanded by |K| customers in a given time period, where N denotes the set of manufacturers and K the set of customers.Each manufacturer i (i ∈ N) with a processing capacity of b i receives an order with d ik units from customer k (k ∈ K).We also use d ik to denote that order, for simplicity.Each manufacturer i gives a high priority of its manufacturing capacity to satisfy corresponding customer orders.If there is not enough processing capacity, i.e., b i < ∑ k∈K d ik , then the sharing platform helps to transfer the unsatisfied units to other manufacturers.On the contrary, if b i > ∑ k∈K d ik , implying that manufacturer i has a surplus capacity, then the platform may help to process the orders that are transferred from other manufacturers on the platform with insufficient capacities.
Given a set of customer orders, before optimizing the order scheme among the manufacturers, we first make a preprocessing allocation, because in practice a high priority is usually given to the manufacturer's own orders.Let z ik , a decision variable, be the quantity of products processed by manufacturer i for order d ik ; clearly, we need to allocate surplus customer order units for manufacturer i to other manufacturers on the sharing platform.

Fundamental Assumptions
For the considered problem, we give the following basic assumptions.

1.
To ensure that any manufacturer is capable of processing any order, assume that all the customer orders are homogeneous in product requirement.Each order can be arbitrarily split and shared among manufacturers, if necessary.

2.
Each manufacturer has the priority to process its own orders.It shares its order units with other manufacturers via the sharing platform automatically, provided that the amount of the order units is strictly larger than its capacity and its own capacity has been used up in processing the orders.This case happens to the manufacturers associated with Luyou; only when the capacity is insufficient will Luyou share the unsatisfied orders with other manufacturers.

3.
We consider the case with unbounded capacity of transportation, and the transportation cost is out of consideration, as we focus on the customer production order allocation plans.

Methodology
In this section, we first give the definitions of the problem parameters and decision variables in Section 4.1, and then the mathematical model is described in Section 4.2.Furthermore, in Section 4.3, we propose a mathematical model transformation.

Notation
The notations used in the model are defined as follows: Indices: i, j: indices of manufacturers; -k: index of customers.

Parameters:
-N: set of manufacturers; |N| denotes the number of manufacturers; -K: set of customers; |K| denotes the number of customers; -q: the total quantity of products in all the orders demanded by the |K| customers; -b i : the processing capacity of manufacturer i; -d ik : the quantity of products customer k demands for manufacturer i (∑ i∈N ∑ k∈K d ik = q); α i = 1, if the capacity of manufacturer i is greater than or equal to the amount of its corresponding customers, i.e., b i ≥ ∑ k∈K d ik ; 0, otherwise; -M: a sufficiently large integer.

Decisions variables:
z ik : a positive integer variable representing the quantity of products for d ik that is processed by manufacturer i itself, where i ∈ N, k ∈ K; -y ijk : a positive integer variable representing the quantity of products for d ik that is shared with manufacturer j by manufacturer i, and y iik = 0, where i ∈ N, j ∈ N, k ∈ K.

Mathematical Formulation
By employing the principle of entropy maximization, the sharing platform aims to make a best production order allocation scheme so as to minimize the profit loss caused by unexpected failures of some manufacturers.In the remaining part of this subsection, we first present the objective function and then give the necessary constraints of the model.
The formula on the right-hand side of Equation (1) defines the objective function or the entropy of any order allocation (Teye et al. [8]).
EM makes no assumption about the unknown information but rather considers all possible states of the variable.A higher entropy value means that there is a higher probability of the event occurring with a lower level of uncertainty.In terms of this work, it means a stronger antirisk ability.Mathematically, the number of possible ways that a state of the system can occur is given as: The formula on the right-hand side of Equation ( 1) defines the objective function or the entropy of any order allocation (we refer to Teye et al. [8]).
The goal is to maximize E subject to all the information we have about the sharing economy system.
q is the total quantity of demands in all the orders placed by the |K| customers, and ∑ i∈N b i is the total production capacity of all the manufacturers.Moreover, Since ln E is a monotonic function of E and is easier to work with, maximizing ln E implies the maximization of E ((we refer to Teye et al. [8]).Thus, taking the natural logarithm, we have ( According to Stirling's approximation, we further simplify Equation (2) to Equation ( 3) as (we refer to Teye et al. [8]) In the above expression, the values of ln q! and ln(∑ i∈N b i )! are constants for any given set of customer orders and manufacturers.Thus, ignoring the constant terms, we formulate the objective function as in Equation ( 4), because maximizing Equation ( 3) is equivalent to maximizing the following formula (we refer to Teye et al. [8]).
Making a transformation from maximization to minimization of the objective function, we have the objective function below (we refer to Teye et al. [8]).
We only work with the situation where all the customer orders are satisfied, i.e., ∑ i∈N ∑ k∈K d ik ≤ ∑ i∈N b i (For the other case where the total amount of clothing orders demanded by all the customers is greater than the total processing capacity of all the manufacturers, the model is shown in Appendix A).The constraints of the model are formulated and explained below.
Each order d ik must be satisfied by all the manufacturers, as shown in Constraint (6), where the first and the second items on the left-hand side of the equation denote the quantity of the products processed by the manufacturer i and those shared to the other manufacturers, respectively.
Constraint (7) indicates that the total quantity of the orders completed by manufacturer j cannot exceed its processing capacity.
Constraints ( 8) and ( 9) ensure that each manufacturer gives a higher priority to using its manufacturing capacity to satisfy its own customer orders.That is, if Constraints ( 10) and (11) state that for manufacturers with capacity limitations, they need to use up all their capacities in production (by Assumption 2).That is, if Constraint (12) indicates the ranges of the decision variables.

Mathematical Model Transformation
We observe that it is hard to directly solve the above model with integer variables and the nonlinear objective function (5).Therefore, in this subsection we make a further transformation of the objective function.We approximate the previous mathematical model via a mixed-integer quadratic programming (MIQP) model, which can be well solved by the CPLEX solver.

Second-Order Taylor Expansion of the Objective Function
In view of Equation ( 5), we take advantage of the Taylor expansion to construct a polynomial to approximate the value of the function in the neighborhood of some point x 0 .Consider the function f (x) which is defined as We give its second-order Taylor expansion as follows: in which we have with algebra that Taking the second-order Taylor expansion of Equation ( 13) at point x 0 = 1, we have the approximation of function f (x), which is a polynomial, as follows: By the polynomial approximation function given in Equation ( 14), the following two formulae are straightforward: Therefore, the above mathematical model is finally approximated as follows: Subject to: Constraints ( 6)-( 12).We shall mention that the above MIQP model can be solved by the CPLEX solver.

Solution Approach
For solving problem instances on a large scale, we propose a two-stage constructive heuristic, which adopts a balancing strategy.That is, it evenly distributes the capacities of the manufacturers with b i < ∑ k∈K d ik among their customers, and averagely allocates the remaining orders to the manufacturers with b i > ∑ k∈K d ik .
Based on the above idea, we develop a constructive heuristic consisting of two stages, namely DE-APP.In the first stage we use a differential evolution (DE) algorithm to calculate z ik and in the second stage, y ijk is obtained by the average assignment principle (AAP).Let N 1 be the set of manufacturer i with b i < ∑ k∈K d ik , N 2 be the set of manufacturers with b i ≥ ∑ k∈K d ik and |N 1 | and |N 2 | denote the number of manufacturers in the set of N 1 and N 2 , respectively (i.e., |N 1 | + |N 2 | = |N|).It is easy to allocate capacities for any manufacturer i with b i ≥ ∑ k∈K d ik , that is, z ik = d ik (by Assumption 2).Next, we mainly generate an allocation scheme for manufacturers in the set of N 1 .
The framework of the DE-AAP algorithm is given in Figure 2, and the main steps of the algorithm are as follows.
Step 1 Take two parameters, customer order d ik and manufacturer capacity b i , as input; Step 2 Randomly generate N p individuals x 1 , x 2 . . .x NP forming the initial population {x 1 , x 2 . . .x NP }, where individual x 1 is generated by the method shown in Section 5.1 and the rest of the individuals are generated by the method shown in Section 5.2; Step 3 Adopt Algorithm 2 to perform gene repair for each individual x i obtained in Step 2; Step 4 Calculate the allocation of the remaining orders in the second stage using the method shown in Section 5.3; Step 5 Calculate the fitness of each individual x i in the population using Formula (17); Step 6 If the termination condition is satisfied, the individual with the smallest fitness function is taken as the desired result and the algorithm ends; Step 7 Randomly select 3 different chromosomes from the initial population for mutation operation according to the mutation rate and replace the chromosome undergoing mutation operation with the resulting new chromosome; Step 8 Swap the gene fragments on the chromosomes by the method shown in Section 5. 5 and replace the initial chromosome with the resulting new chromosome.

Initial Solutions of DE
In this subsection, a heuristic algorithm is devised, known as Algorithm 1, for generating one of the initial solutions of the DE algorithm so as to enhance the efficiency of its performance.For calculation purposes, we define a parameter, customer order sequence Cs, which represents the row arrangement of all customer orders belonging to the set N 1 .A corresponding example is given in Figure 3, where N 1 (1) is exemplified by the customer order {4, 9, 8, 10, 30} (|K| = 5).b i = b i (% b i is the manufacturing capacity of manufacturer i); z ik = b mean (% allocate capacity to customers having orders that are greater than the average capacity.);

In Algorithm 1,
Step 1: allocates capacity to manufacturers with b i ≥ ∑ k∈K d ik , that is z ik = d ik (since this step is so simple, it is not depicted in Figure 4).* Step 2 and Step 3 symbolize the allocation process of manufacturers with b i < ∑ k∈K d ik based on the average allocation idea put forth above, with the specific steps explained as follows: Step 2.1: Calculate the average capacity of manufacturers in the set N 1 (b i < ∑ k∈K d ik ), let z ik = d ik for customers whose orders are less than the average capacity; Step 2.2: Update the average manufacturing capacity and repeat Step 2.1 until all the orders that are less than the manufacturer's average capacity are allocated; Step 3.1: For all unassigned customers, let z ik = average manu f acturer s capacity; Step 3.2: Round up noninteger genes (the average manufacturer's capacity might not be an integer); Step 4: If the initial allocation scheme is greater in comparison to the manufacturing capacity, randomly select s i customers with nonzero orders (s i = ∑ k∈K z ik − b i ), each one minus one unit.Step2.1:The number of unallocated customers is 5, and hence the average capacity is 8, and neither the first nor third genes exceed the average capacity.
Step2.2: 3 is the number of unallocated customers, the remaining capacity is 28, so the average capacity is 9.3.The second gene is not more than the average capacity.
Step3.1: Unallocated customers number is 2, the remaining capacity is 19, and hence the average capacity is 9.5.All unallocated genes exceed the average capacity.

Step3.2: Round up non-integer genes
Step4: The total allocated capacities 41 are one more than the manufacturing capacity 40.Choose one gene at random between the fourth and the fifth gene, and minus one value.

Algorithm 2 Gene repair strategy
Require: Population N p, initial chromosome, customer order sequence Cs, manufacturing capacity b i 1: x i ← round up gene values for initial chromosome; 2: Assign 0 to the gene with a value less than 0 (% there may be genes smaller than 0 owing to the subsequent mutation operation); 3: for i = 1 to N p do 4: is a number that symbolizes the gap between manufacturing capacity as well as its total allocated capacities); 5: g i = D i − x i (% g i is a sequence that denotes the gap between the customer order sequence and the chromosome x i .);6: Adopt the method of step 4 in the algorithm 1 to repair the gene; 9: end if 10: end for 11: return feasible solution

Population Initialization
The DE algorithm uses N p real-valued vectors with dimension D as the population of each generation.Each individual is represented as x D i,G (i = 1, . . ., N p), where: i : the sequence index of individuals in the population; G : the current evolutionary generation; N p : the size of the population; N p remains unchanged during the evolution process; D : the dimension of a chromosome; it also denotes the length of a chromosome, that is, D = |N 1 | × |K|, and all consecutive |K| genes represent an allocation sequence of the manufacturer i in the set N 1 .
The gene value is calculated as follows.rand(0, 1) denotes generating a random number between 0 and 1.
x D = rand(0, 1) * Cs.Given this gene generation method, it is possible that for manufacturer i, the capacity allocated to each of its customers does not match its actual capacity.Therefore, we put forward a gene repair strategy in Algorithm 2 so as to correct the error.In Algorithm 2, the gene repair strategy is proposed in order to eliminate infeasible solutions so that for any manufacturer the total of capacities allocated to each of its corresponding customers are equal to its capacity that is, ∑ i∈N ∑ k∈K z ik = b i .Taking the customer order sequence as well as the initial chromosome in Figure 6 as an instance, Figure 7 presents a detailed diagram of the gene repair strategy.

Heuristic-AAP
In stage 2, we present a heuristic called AAP to optimize the remaining orders allocation.The main steps of the AAP are as follows: • Step 1: Sort the remaining orders placed by the |K| customers to the |N 1 | manufacturers in the nonincreasing order of their unassigned quantities.Denote the order sequence by Λ = (R 1 , R 2 , . . ., R H ), where H ≤ |K| × |N 1 | is the number of orders with nonzero unsatisfied demands, R i also represents the amount of unassigned quantity in the order; • Step 2: Sort the remaining capacities of the set of N 2 manufacturers in the nondecreasing order of their remaining capacities, labeled as Γ = (S 1 , S 2 , . . ., S L ), , where S i denotes the remaining capacities of the manufacturers with b i > ∑ k∈K d ik ; • Step 3: Allocate the remaining orders one by one evenly among |L| manufacturers, i.e., the percentage is R i |L| , and for each order R i (i = 1, 2, . . ., H), the allocation |L| , . . ., < S i (In other words, there is a manufacturer that does not have enough remaining capacity to cover the current allocation of orders.),give priority to the manufacturer with the minimum remaining capacity.

Mutation
For any chromosome x D i,G undergoing mutation, the DE algorithm generates offspring chromosomes v D i,G+1 as follows: in which i = r 1 = r 2 = r 3 , (i = 1, . . ., N p).r 1 , r 2 and r 3 are all random integers in the interval [1, N p].F is the adaptive mutation operator.In order to avoid a premature convergence of the algorithm, the adaptive mutation operator F is designed as: . G m : maximum number of iterations, set as 500; G : the current iteration.This mutation operation method can ensure that for any manufacturer in the set N 1 , the total capacity allocated to all its customers is still equal to its capacity following the mutation.Figure 8 illustrates an example of a mutation operation.After mutation, the offspring chromosomes may be infeasible, as indicated in the gray cell in Figure 1, the allocated capacity is less than 0 or not an integer.Therefore, the gene repair strategy is applied again in order to eliminate the error.

Crossover
To guarantee the diversification of the population in the new generation, the following fragment crossover operation is applied to renew the offspring chromosome obtained by the above mutation operation: , otherwise in which C r represents the crossover operator and was set to 0.7.rand denotes a random number between 0 and 1 generated in each iteration.Superscript (1 indicates different gene fragments, where j = 1, . . ., |N 1 |.Every consecutive |K| genes represent a fragment.A number between 0 and 1 is randomly generated.If this number is greater than the crossover coefficient 0.7, a fragment is randomly selected in the target chromosome x D i to replace the fragment at the same position of the chromosome obtained by the above mutation operation v D i .Otherwise, the chromosome remains unchanged (obtained by a mutation operation).Figure 9 shows an example of the crossover operation, where we take the segment with j = 2 as the crossover segment.

Selection
Equation ( 17) serves as the fitness function.In order to evaluate the fitness of the offspring chromosomes, the DE algorithm compares the objective value of the offspring chromosome u D i with that of the current target chromosome x D i .The chromosome with a smaller objective value is retained for the next generation.That is, where in the above equation, i = 1, 2, . . ., N p.

Termination Conditions
When the number of iterations reaches a maximum of 500, or the fitness value of the best solution stays unchanged in 100 consecutive iterations, it terminates and outputs the best solution.

Computational Experiments
In this section, we carry out computational experiments to evaluate the performance of the proposed algorithm.The numerical calculations were conducted on a PC with Intel Core i5, 3.1 GHz, processors and 4 GB RAM under a Windows 10 operating System, and the computation time was calculated in CPU seconds.The mathematical model proposed in Section 4 was solved via CPLEX 12.6.All algorithms were implemented using Matlab R2014b.The time limit of CPLEX was set to 7200 s.

Instance Generation
It is estimated that in the future, market requirements of the garment industry will consist of small and micro-orders, averaging one hundred pieces [27].To further evaluate this, and bearing in mind the characteristics of the garment industry which in the era of fast consumption can deal with "small batches" and "multiple batches" of orders, we set up four scales of experiments, i.e., extrasmall size, small size, middle size and large size, which were distinguished by customer order volume as well as the numbers of manufacturers and customers.

Generation of Order d ik
Each module in Table 1 gives the range of customer orders d ik of each instance size as well as the corresponding numbers of manufacturers and customers.U[a, b] represents the uniform distribution in interval [a, b].For example, d ik was generated randomly between 0 and 10 for the extrasmall size, and for the small size it ranged from 0 to 50.In addition, the numbers |N| and |K| for the four instance sizes are also given in Table 1, and the values vary from 5 to 60.
The manufacturing capacity interval of manufacturers in the set . This method can ensure that the manufacturing capacity of all the manufacturers is within a reasonable interval and satisfies all the customer demands as well.

Computational Result
In the experiments, there were a total of 64 combinations (16 combinations per size × 4 sizes).Specifically, for each instance size, four different ratios of N 1 : N 2 were tested, and four various combinations of manufacturers and customers were tested for each ratio.Furthermore, for each combination, five instances were generated randomly.In total, 320 benchmark instances were generated.For most extrasmall combinations, CPLEX could output optimal solutions of the MIQP model presented in Section 4.3.In these cases, we measured the performance of the proposed algorithm in comparison with optimal solutions.For relatively large instances, CPLEX could not output exact solutions within 7200 s.Thus, we relaxed the integer constraints of z ik and y ijk to obtain the MIQP model's lower bound by CPLEX, for evaluating the performance of the proposed algorithm.
Table 2 shows the computational results of the extrasmall size in all four ratios.Every row in the table reports the average results of five instances for each combination.The leftmost column represents the serial number of the combination.The second column "N 1 : N 2 " indicates the ratio of N 1 to N 2 .Columns "|N|" and "|K|" give the number of manufacturers and customers, respectively.The "LB" column reports a lower bound of an optimal solution by relaxing the integer constraints of decisions variables.The sixth and eighth columns ("Obj")report the average objective function values obtained by CPLEX and DE-AAP, respectively.The seventh and ninth columns ("Time (s)") give the average running time of the two methods, respectively."--" in sixth column indicates that there was no optimal solution obtained by CPLEX in 7200s.Column "Gap 1 (%)" gives the relative error compared to LB for each combination, which was calculated by (Obj DE−AAP − Obj LB )/Obj LB × 100%, where Obj x denotes the objective function value of algorithm x.Column "Gap 2 (%)" gives the gap in comparison with CPLEX, that is, Gap 2 = (Obj DE−AAP − Obj CPLEX )/Obj CPLEX × 100%."--" in the "Gap 2 (%)" column means that the error could not be obtained because the combinations could not get the optimal solutions within the limited time."Average" in the last row represents the average results for all the instances with optimal solutions obtained in 7200 s.The underlined digit in the last row indicates the average value over eight combinations with optimal solutions.
According to Table 2, the running time of CPLEX increased rapidly from less than 1 s to more than 7200 s as the numbers of manufacturers and customers increased, while DE-AAP solved all the combinations in less than 13 s.It means that the running time of CPLEX was much more sensitive to the combination size than that of DE-AAP.
There were several advantages of the algorithm used.Firstly, in terms of the average running time over all the combinations, DE-AAP apparently outperformed CPLEX.That is, the average running times were about 5.8 s and 3609.4 s for the two approaches.Thus, DE-AAP saved a lot of computational time.For some relatively large size combinations, the effectiveness of DE-AAP was even more obvious.Secondly, in terms of solution quality, DE-AAP could output optimal solutions for some extrasmall combinations, and the "Average Gap" was 0.93%.Furthermore, to evaluate the effectiveness of the lower bound, we reprocessed the experiments for the same instances while omitting the integer constraint.The "Average Gap" between DE-AAP and LB was 5.03%, slightly weaker than 0.93%.Therefore, it is believed that the method of acquiring the lower bound is effective.
The numerical results of small-sized instances are given in Table 3.As previously mentioned, CPLEX could not output optimal solutions for small-size combinations within the given time limit.Therefore, we compared the solution of the proposed algorithm with the lower bound.In Table 3, the column "Gap" denotes the relative error between the lower bound and the objective value by DE-AAP, i.e., the same as Gap 1 (%) in Table 2.The rest of the notations are also the same as in Table 2.The gap for DE-AAP varied from 0.46% to 7.99% with an average error of 3.23%.More specifically, with the increase of the proportion of N 1 in N, the Gap increased, upper-bounded by 8%.With regard to the running time, DE-AAP output solutions in about 80.3s on average.
The computational results of medium-size instances are given in Table 4.In terms of solution quality, the relative error of DE-AAP compared with the lower bound was averagely 2.70%, and the average running time was about 130.9 s.Table 5 reports the results of large-sized instances.DE-AAP had a relative error of 3.68% on average and a maximum one of 8.28% with regard to solution quality, and it consumed 625.0 s on average.
By the above experimental results, DE-AAP was relatively robust in performance.It could output optimal solutions for some extrasmall-size instances, and for other size instances, the fluctuation of its average relative error compared with the lower bound was small and less than 4%.The maximal relative error of the algorithm for all the instances tested was less than 9%.The numerical results also gave some managerial insights.By the intuitive average allocation scheme of surplus customer demands among manufacturers, it could achieve a good balance of manufacturing resource utilization.Moreover, such allocation strategy is easy to implement for the sharing platform, and it can improve the antirisk ability of the whole manufacturing system on the failure of some manufacturer's processing resource as well.

Conclusions
This work studied the apparel production order allocation problem in the sharing economy.We adopted the principle of entropy maximization to represent the objective requirement and established the MIQP model for the considered problem.A two-stage algorithm was proposed to solve the problem.We validated the efficiency of the proposed algorithm by numerical experiments.The entropy target order distribution method gave us certain management insights, especially in the development of the e-commerce economy of "small batches" and "multiple batches".The entropy target model provided a meaningful idea of average distribution for customer orders so as to reduce the loss caused by force majeure and keep customers as much as possible.
However, there exist some limitations to this work.Firstly, we assumed the customer orders were homogeneous so that any manufacturer was capable to produce the orders.Secondly, we supposed manufacturers gave priority to their own orders.Future research

Figure 1 Figure 1 .
Figure 1.Order processing under sharing economy mode.

10 :
b mean = b i |K| (% the average capacity of manufacturer i for all |K| customers); 11: while b i > 0 do 12: for k = 1 to |K| do 13: if d ik ≤ b mean then 14: z ik = d ik ; 15: b i = b i − z ik (% recalculate the capacity b i ); 16: |K| = |K| − 1 (% renew the number of unallocated customers); = b i |K| (% renew the mean remaining capacity of manufacturers); 20: if b mean == b mean then for k = 1 to |K| do 28: if d ik ≥ b mean then 29:

Figure 5 Figure 5 .
Figure 5 gives the flow chart of step 2 in Algorithm 2.

Figure 6
Figure 6 gives an example to show the generation method of an initial chromosome.

Figure 6 .
Figure 6.Example of initial chromosome generation.
N 2 ← find the set of manufacturers with b i ≥ ∑ k∈K d ik ; 3: for i ∈ N 2 do N 1 ← find the set of manufacturers with b i < ∑ k∈K d ik ; 8: for i ∈ N 1 do 9:

Table 1 .
Values of |N| and |K|.In order to evaluate the performance of the algorithm proposed above more comprehensively, various ratios between the numbers of |N 1 | and |N 2 | were tested.Specifically, the rations of |N 1 | : |N 2 | for different combinations were set as 1:4, 2:3, 3:2 and 4:1.The generation rule of the manufacturing capacity b i was as follows: • Randomly select |N 1 | manufacturers among the total |N| manufacturers.For each selected manufacturer, the manufacturing capacity is randomly generated between zero and the total received orders, that is, b i = rand(0, ∑ k∈K d ik ), (i ∈ N 1 ); • Calculate the sum of all the orders that cannot be satisfied by the |N 1 | manufacturers, that is ∑ i (∑ k∈K d ik − b i ) (i ∈ N 1 ), labeled as Ω 1 , and randomly divide Ω 1 into |N 2 | parts, marked as I 1 1 , I 1 2 , . . ., I 1 |N2| (∑ i∈N 2 I 1 i = Ω 1 ); • Calculate the total orders received by the |N 1 | manufacturers, that is, ∑ k∈K d ik , (i ∈ N 1 ) labeled as Ω 2 , and randomly divide it into |N 2 | parts, marked as I 2 Calculate all the orders of the manufacturers in the set N 2 , that is,

Table 2 .
Computational results of extrasmall combination size.

Table 3 .
Computational results of small combination size.

Table 4 .
Computational results of large combination size.

Table 5 .
Computational results of extra-large combination size.