Designing a resilient-sustainable supply chain network for perishable goods considering inter-chain competition

3.2.1 Objective functions

In this problem, we have investigated the closed-loop supply chain network design. The first objective function (OF1) of the network design problem is to maximize the profit from selling final products in forward logistics and recycled products in reverse logistics. In Equation (1), we have considered the revenue from selling products to end customers and the revenue from recycled materials. To calculate the supply chain cost, we have taken into account the cost of purchasing raw materials, the cost of producing each product at production centers, the cost of activating each of the existing facilities in the supply chain, the cost of storing products at distribution centers, the cost of dealing with shortages to meet customer demand, the cost of purchasing buy from customers, the cost of increasing capacity at production centers and distribution centers to improve resilience and deal with potential disruptions, the cost of recycling waste products, and the cost of moving products between each of thefacilities.

In order to establish a green supply chain, we have also investigated the environmental index alongside the economic objective of the supply chain network design problem. To evaluate the environmental impacts of supply chain activities, such as establishing facilities, transporting products between different members of the supply chain, and producing and recycling products, we have considered the amount of pollution and energy consumption resulting from these activities. Thus, the environmental objective function (OF2) in the problem is calculated according to Equation (2). max OF 1 = ∑ n ∑ t ∑ d ∑ k ∑ p p k , p n , t x d , k , p n , t + ∑ n ∑ t ∑ k ∑ c ∑ p sv c , p x k , c , p n , t - [ ∑ n ∑ t ∑ s ∑ m ∑ , p ′ x s , m , p ′ n , t c s , p ′ + ∑ n ∑ t ∑ m ∑ p cf m , p q m , p n , t + ∑ n ∑ m foc m n ym m n + ∑ n ∑ d foc d n yd d n + ∑ n ∑ c foc c n yc c n + ∑ n ∑ r foc r n yr r n + ∑ n ∑ v foc v n yv v n + ∑ n ∑ t ∑ d ∑ p hc d , p * ( h d , p n , t + 1 2 ∑ m x m , d , p n , t n m , d ) + ∑ n ∑ t ∑ k ∑ p st k , p w k , p n , t + ∑ n ∑ t ∑ k ∑ c ∑ p rep k , p n , t x k , c , p n , t + ∑ n ∑ t ∑ m ∑ p inc m , p n , t cap m + ∑ n ∑ t ∑ d ∑ p inc d , p n , t cap d + ∑ n ∑ t ∑ c ∑ r ∑ p φ r , p x c , r , p n , t + ∑ n ∑ t ∑ s ∑ m ∑ p ′ tc s , m , p ′ x s , m , p ′ n , t + ∑ n ∑ t ∑ m ∑ d ∑ p tc m , d , p x m , d , p n , t + ∑ n ∑ t ∑ d ∑ k ∑ p tc d , k , p x d , k , p n , t + ∑ n ∑ t ∑ k ∑ c ∑ p tc k , c , p x k , c , p n , t + ∑ n ∑ t ∑ c ∑ v ∑ p tc c , v , p x c , v , p n , t + ∑ n ∑ t ∑ c ∑ r ∑ p tc c , r , p x c , r , p n , t + ∑ n ∑ t ∑ r ∑ v ∑ p tc r , v , p x r , v , p n , t + ∑ n ∑ t ∑ r ∑ m ∑ p tc r , m , p x r , m , p n , t ](1) max OF 2 = ∑ m e m ym m n + ∑ d e d yd d n + ∑ c e c yc c n + ∑ r e r yr r n + ∑ v e v yv v n + ∑ n ∑ t ∑ m ∑ p ep m , p q m , p n , t + ∑ n ∑ t ∑ d ∑ p es d , p h d , p n , t + ∑ n ∑ t ∑ k ∑ c ∑ p ec c , p x k , c , p n , t + ∑ n ∑ t ∑ c ∑ r ∑ p er r , p x c , r , p n , t + ∑ n ∑ t ∑ s ∑ m ∑ p ′ et s , m , p ′ x s , m , p ′ n , t + ∑ n ∑ t ∑ m ∑ d ∑ p et m , d , p x m , d , p n , t + ∑ n ∑ t ∑ d ∑ k ∑ p et d , k , p x d , k , p n , t + ∑ n ∑ t ∑ k ∑ c ∑ p et k , c , p x k , c , p n , t + ∑ n ∑ t ∑ c ∑ r ∑ p et c , r , p x c , r , p n , t + ∑ n ∑ t ∑ r ∑ m ∑ p et r , m , p x r , m , p n , t + ∑ n ∑ t ∑ c ∑ v ∑ p et c , v , p x c , v , p n , t + ∑ n ∑ t ∑ r ∑ v ∑ p et r , v , p x r , v , p n , t(2)

The value and social justice are evaluated based on the number of job opportunities created in a specific area, and creating job opportunities in an area is deemed to have a higher social value. The objective of this problem is, therefore, to maximize the level of social responsibility and commitments in the supply chain. By activating each facility, more job opportunities can be created for individuals. In this model, we have considered the job opportunities created as a result of the operation or construction of each of the suppliers, production, distribution, collection, recycling, and disposal centers. This objective is the social objective function (OF3) and can be calculated according to Equation (3).

max OF 3 = ∑ s job s ys s n + ∑ m job m ym m n + ∑ d job d yd d n + ∑ c job c yc c n + ∑ r job r yr r n + ∑ v job v yv v n(3) In this problem, we assume suppliers may face disruptions due to various natural disasters or human-made incidents, which may prevent them from serving their customers on time. These disruptions are of the partial disruption type, meaning that the supplier’s capacity decreases but does not disappear entirely. To address this issue and fill the literature gap, several resilience strategies have been considered for this supply chain network design problem. These strategies include increasing the capacity of production and distribution centers. The concept of resilience has been incorporated into the closed-loop supply chain network design using Equation (4) as the resilience objective function (OF4). max OF 4 = ∑ m r m cap m + ∑ d r d cap d(4)

Equation (5) shows that each production center purchases the necessary raw materials for producing the final product from the supplier center.

∑ s x s , m , p ′ n , t = ∑ p ρ p , p ′ q m , p n , t(5)

Equation (6) indicates that all the products produced by each production center, along with the recycled products imported from the recycling centers to the production center, are immediately sent to the distribution centers.

∑ d x m , d , p n , t = ∑ r x r , m , p n , t + q m , p n , t(6)

Equations (7) and (8) indicate that the inflow to a distribution center is balanced with its outflow so that the incoming goods to the distribution centers are transferred to the customers.

h d , p n , t - 1 + ∑ m x m , d , p n , t = h d , p n , t + ∑ k x d , k , p n , t(7) h d , p n - 1 , t + ∑ m x m , d , p n , t = h d , p n , t + ∑ k x d , k , p n , t(8) Equation (9) indicates that a portion of the goods sold by distribution centers to customers are repurchased by the customers and transferred to collection centers for evaluation and decision-making regarding their recycling or disposal.

∑ k ∑ c x k , c , p n , t ≤ ∑ d ∑ k x d , k , p n , t(9)

Equation (10) shows that the sum of the supplied demand and the shortage of products for customers is equal to the total market share of the supply chain.

∑ d x d , k , p n , t + w k , p n , t ≤ ∑ k ( d k , p n , t - α k , p p k , p n , t + α k , p ′ p k , p n , t )(10)

Equation (11) indicates the balance of the flow of consumed and returned goods at collection centers so that a portion of the collected goods is transferred to recycling centers. In contrast, the portion that cannot be recycled is sent to disposal centers for disposal.

∑ k x k , c , p n , t = ∑ r x c , r , p n , t + ∑ v x c , v , p n , t(11)

Equations (12) and (13) show the number of recycled goods that are transferred to production centers so that the portion of goods that cannot be recycled or lose their functionality during the recycling process is sent to disposal centers for disposal.

∑ c x c , r , p n , t - 1 = ∑ m x r , m , p n , t + ∑ v x r , v , p n , t(12) ∑ c x c , r , p n - 1 , t = ∑ m x r , m , p n , t + ∑ v x r , v , p n , t(13)

Equation (14) indicates that the number of goods transferred from collection centers to recycling centers equals the number of returned customer products.

∑ k x k , c , p n , t ≤ e i , k , p n , t + β k , p rep k , p n , t - β k , p ′ rep k , p n , t(14)

Equations (15) to (18) show that the activated facilities must remain active until the end of the active planning horizon. ym m n ≥ ym m n - 1(15) yd d n ≥ yd d n - 1(16) yc c n ≥ yc c n - 1(17) yr r n ≥ yr r n - 1(18)

Equations (19) and (20) make the performance of each producer and distribution center subject to the maximum capacity of the facilities. ∑ p ur m , p h d , p n , t + ∑ m ∑ p ur m , p x m , d , p n , t n m , d ≤ yd d n max cap d(19) ur m , p cap m ≤ ym m n max cap m(20)

Equation (21) shows that the purchased raw materials from suppliers must be transformed into finished products within their lifespan, and the system must be planned so that there is no waste or loss of products. ∑ t min { t + E x p ' , T } ∑ m ∑ p ρ p , p ' q m , p n , t − ∑ t ∑ s ∑ m x s , m , p ' n , t ≥ 0(21) Equation (22) shows that the produced finished products must be transferred to distribution centers at most l _t periods before the end of their consumption cycle. So that they can be transferred to the customer at the appropriate time. ∑ t min { t + Ex p - l p , T } ∑ m ∑ d ∑ p x m , d , p n , t - ∑ t ∑ m ∑ p q m , p n , t ≥ 0(22)

Equations (23) and (24) indicates the maximum capacity of production and distribution centers.

cap m ≤ max cap m(23) cap d ≤ max cap d(24)

Equations (25) and (26) show that the number of goods sent from production and distribution centers exceeds their maximum capacity.

∑ n ∑ t ∑ p ∑ d x m , d , p n , t ≤ max cap m(25) ∑ n ∑ t ∑ p ∑ k x d , k , p n , t ≤ max cap d(26)

Equation (27) specifies the type of variables in the problem. x s , m , p ′ n , t , x i , j , p n , t , q m , p n , t , h d , p n , t , w k , p n , t , p k , p n , t , rep k , p n , t , cap i ≥ 0 ys s n , ym m n , yd d n , yc c n , yr r n , yv v n = { 0 , 1 }(27)

4.2.1 Non-dominated sorting genetic algorithm

The genetic algorithm is a problem-solving algorithm inspired by biological models of populations. It improves the characteristics of a population by generating new populations through the combination of previous ones, with the values of objective functions representing these characteristics. The Non-dominated Sorting Genetic Algorithm (NSGA-II) is a multi-objective variant of the genetic algorithm. It assigns ranks to individuals based on their non-dominance and proceeds to re-rank the remaining individuals without considering the effect of ranking on the population.

In the initial stage of NSGA-II, lower-ranking individuals replace their parental counterparts and undergo sorting based on their crowding distance. The population resulting from crossover and mutation, along with the initial population, is categorized by rank, and individuals with lower ranks are removed. In the subsequent stage, the remaining population is sorted within each front based on their crowding distance. Figure 3 provides an illustration of this process.

OPEN IN VIEWER

Fig. 3

Pseudo code of non-dominated sorting genetic algorithm.

The Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is a dynamic optimization technique utilized for solving multi-objective optimization problems. Each particle in the algorithm represents a point in a multi-dimensional space. The algorithm aims to generate the optimal set of parameters by evaluating metrics such as Euclidean distance, polynomial distance, and Hawking distance between points. These metrics help determine the optimality of a point in the objective space, which is the multi-dimensional space representing the objectives. The algorithm calculates the optimality score for each point based on the defined metrics for different objectives. Parameters such as population size, number of generations, memory size, and particle movement direction are optimized within the MOPSO algorithm to search for optimal solutions. The steps of this algorithm are as follows [22]:

Generate an initial population randomly and evaluate them.

The termination conditions are:

Reaching an acceptable level of performance and checking a specific number of iterations.

The constraint handling policy in both proposed algorithms involves generating feasible solutions, and if an infeasible solution is generated, a new solution is produced and checked for feasibility.