A Closed-loop Manufacturing Production Decision Optimization Method Based on Benders Iterative Decomposition

Abstract

The present invention discloses a method for optimizing closed-loop manufacturing production decisions based on Benders iterative decomposition. Step S1: Construct an optimization model for closed-loop manufacturing production decisions, as well as the feasibility constraints and optimality constraints of the optimization model for closed-loop manufacturing production decisions. Step S2: Perform Benders iterative decomposition on the optimization model for closed-loop manufacturing production decisions to obtain a master problem and a sub-problem. Step S3: Perform a dual operation on the sub-problem to form a new dual sub-problem and a new primal problem. Step S4: Solve the new dual sub-problem, and add feasibility constraints or optimality constraints to the new primal problem according to the type of solution to solve for the upper and lower bounds of the optimization model for the range of closed-loop manufacturing production decisions. Step S5: Compare the upper bound with the lower bound and perform iterative processing. Step S6: If the upper bound and the lower bound are not close, repeat Steps S4 - S6. Step S7: If the upper bound and the lower bound are close, obtain the optimal solution of the optimization model for closed-loop manufacturing production decisions.

Description

Technical Field

[0001] The present invention relates to the technical fields of geographic information systems and spatial optimization technologies, and particularly to a closed-loop manufacturing production decision optimization method based on Benders iterative decomposition. Background Art

[0002] The manufacturing industry incorporates the recycling manufacturing process into the supply chain to form a closed-loop supply chain, which is conducive to the recycling of resources. However, the specific production decision optimization problem of the closed-loop manufacturing system in the closed-loop supply chain has been lacking consideration. The closed-loop manufacturing system of electronic assembly is a common closed-loop manufacturing system in the closed-loop supply chain. As shown in the appendix Figure 1 This system consists of 7 different manufacturing units, 5 warehouses, and 4 recycling units. The recycling units in the recycling process are distributed in a decentralized manner. This system involves 6 different types of materials (3 types of components, 1 type of module, 1 type of assembly, 1 type of semi-finished product). The 3 types of components and assemblies are purchased from 3 different suppliers, and the purchase unit prices are also different. There is a product demand for 3 cycles; different materials are processed by different manufacturing units and inventories, and finally processed into the required three types of products; thus, it can be seen that the overall closed-loop manufacturing system is complex and variable. Therefore, the overall decision-making of the closed-loop manufacturing system is beneficial for decision-makers to plan the overall operating costs of enterprise production and manufacturing and maintain enterprise efficiency; on the other hand, if specific observations and recycling of waste units can be carried out in the closed-loop manufacturing system, the operability of resource reuse can be achieved, which is beneficial for the enterprise to move towards the direction of green environmental protection. Summary of the Invention

[0003] (1) Technical Problems to be Solved

[0004] The present invention provides a closed-loop manufacturing production decision optimization method based on Benders iterative decomposition to solve the above problems.

[0005] (2) Technical Solutions

[0006] To achieve the above object, the present invention provides the following technical solutions: A closed-loop manufacturing production decision optimization method based on Benders iterative decomposition, comprising the following steps:

[0007] Step S1: Construct a closed-loop manufacturing production decision optimization model, as well as the feasibility constraints and optimality constraints of the closed-loop manufacturing production decision optimization model;

[0008] Step S2: Perform Benders iterative decomposition on the closed-loop manufacturing production decision optimization model to obtain a master problem and a sub-problem;

[0009] Step S3: Perform a dual operation on the sub-problem to form a new dual sub-problem and a new primal problem;

[0010] Step S4: Solve the new dual sub-problem, and add feasibility constraints or optimality constraints to the new primal problem according to the type of solution to obtain the upper and lower bounds of the optimization model for the closed-loop manufacturing production decision-making scope;

[0011] Step S5: Compare the upper bound with the lower bound and perform iterative processing;

[0012] Step S6: If the upper bound and the lower bound are not close, repeat Steps S4 - S6;

[0013] Step S7: If the upper bound and the lower bound are close, obtain the optimal solution of the closed-loop manufacturing production decision optimization model;

[0014] The closed-loop manufacturing production decision optimization model is:

[0015]

[0016] In the formula: The startup capital required to purchase the nth material from the s-th supplier in the t-th period is: When the supplier is selected, SP i,s,t is 1, otherwise it is 0; ST i,s,t is the corresponding startup capital when the supplier is selected; j mc is the manufacturing unit of material i; v is the inventory unit of material i; y is the recycling unit of material i; A i,s,t , B i , C i , D i , E i are the coefficients of the cost function; P i,s,t , MM i,t , IV i,t , MY i,t , QT i,t are the procurement cost, manufacturing cost, warehouse cost, recycling cost, and transportation cost respectively; The transportation cost includes:

[0017]

[0018] Among them, i ∈ N, t ∈ T are respectively the cost of transporting the material from the manufacturing unit j mc to the warehouse j wh , the cost of transporting from the warehouse j wh to the manufacturing unit j mc , the cost of transporting from the manufacturing unit j mc to the recycling unit j rc , the cost of transporting from the recycling unit j rc to the warehouse j wh and the cost of transportation between the recycling units j rc ;

[0019] The feasibility constraints include: production constraints in the production and manufacturing process, recycling constraints in the recycling process, production demand constraints, and inventory balance constraints in the warehouse; the optimality constraints include supplier selection constraints;

[0020] The master problem and sub-problem in step S3 are respectively:

[0021]

[0022] Step S4 performs a dual operation on the sub-problem, simplifies the closed-loop manufacturing production decision optimization model, and forms

[0023] A new primal problem:

[0024]

[0025] A new dual sub-problem:

[0026]

[0027] where w is a decision variable, and (BB - SP i,s,t ) is the coefficient of the decision variable w; BB is a constant; G, H, K, L represent dual cost coefficients; B i , C i , D i , E i are constants;

[0028] The solution of the new dual sub-problem has multiple types:

[0029] The new dual sub-problem g(w) has no solution:

[0030]

[0031] where the corresponding new primal problem also has no solution;

[0032] The new dual sub-problem g(w) has a bounded solution:

[0033]

[0034] where is the initial solution;

[0035] The new dual sub-problem g(w) finds a bounded solution g(w * ), adds optimality constraints to the new primal problem, and obtains the following formula (22):

[0036]

[0037] where w *As the decision solution of the new dual sub-problem of the closed-loop manufacturing production decision, the solution of the new primal problem obtained by the constraint is used as or to update the upper bound up: g(w * );

[0038] If the new dual sub-problem g(w) has an unbounded solution, add a feasibility constraint to the new primal problem to obtain the following formula (23):

[0039] 0≥(BB - SP i,s,t )w * (23)

[0040] Furthermore, update the lower bound of the range of the closed-loop manufacturing production decision optimization model

[0041]

[0042] where lb is the lower bound, w * is the production decision obtained at this time;

[0043] The step S5: Compare the upper bound and the lower bound, and perform iterative processing:

[0044]

[0045] where up and lb represent the upper bound and the lower bound of the range of the closed-loop manufacturing production decision optimization model;

[0046] In the step S7, the upper bound is close to the lower bound, that is

[0047] up≈lb(25)

[0048] where the fact that the upper bound is close to the lower bound indicates that the optimal solution satisfying the closed-loop manufacturing production decision optimization model is calculated through Benders decomposition;

[0049] In the step S6, the upper bound is not close to the lower bound, that is

[0050] up>lb (26)

[0051] where the fact that the upper bound and the lower bound are not close indicates that: the optimal solution satisfying the closed-loop manufacturing production decision optimization model is not calculated through Benders decomposition; by bringing the initial solution back into the new dual sub-problem, start the iteration, repeat steps S4 - S6, and through the continuous iteration and approaching of the upper bound and the lower bound of the closed-loop manufacturing production decision optimization model, obtain the optimal production decision.

[0052] (III) Beneficial Effects

[0053] Compared with the prior art, the beneficial effects of the present invention are:

[0054] The present invention establishes a closed-loop manufacturing system that combines forward production and reverse recycling in the manufacturing industry, achieving the full reuse of resources. While conducting production, it also indirectly realizes environmental protection and sustainable development; the present invention introduces the idea of Benders iterative decomposition into the production planning of closed-loop manufacturing, achieving the rapid customization of production decision-making schemes, and the decision-making planning schemes are more reasonable and effective; the present invention considers the closed-loop production and manufacturing processes of multiple cycles, multiple products, and multiple links. In large-scale production plans, it improves the efficiency of enterprises in different production and manufacturing stages and significantly reduces production costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] The drawings are used to provide a further understanding of the present invention and constitute a part of the specification. They are used together with the embodiments of the present invention to explain the present invention and do not constitute a limitation to the present invention. In the drawings:

[0056] Figure 1 Shows a schematic diagram of the closed-loop manufacturing system for electronic assembly according to an embodiment of the present invention;

[0057] Figure 2 Shows the overall flowchart according to an embodiment of the present invention;

[0058] Figure 3 Shows the calculation logic diagram of the closed-loop manufacturing production decision optimization model according to an embodiment of the present invention;

[0059] Figure 4 Shows the demand curve graph of products in different cycles according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0060] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without making creative efforts shall fall within the protection scope of the present invention.

[0061] Refer to the attached Figure 1 - attached Figure 4 , the present invention discloses a closed-loop manufacturing production decision optimization method based on Benders iterative decomposition, including the following steps:

[0062] Step S1: Construct a closed-loop manufacturing production decision optimization model, as well as the feasibility constraints and optimality constraints of the closed-loop manufacturing production decision optimization model;

[0063] Step S2: Perform Benders iterative decomposition on the closed-loop manufacturing production decision optimization model to obtain the master problem and the sub-problem;

[0064] Step S3: Perform a dual operation on the sub-problem to form a new dual sub-problem and a new primal problem;

[0065] Step S4: Solve the new dual sub-problem, and add feasibility constraints or optimality constraints to the new primal problem according to the type of solution to obtain the upper and lower bounds of the optimization model for the closed-loop manufacturing production decision range;

[0066] Step S5: Compare the upper bound and the lower bound and perform iterative processing;

[0067] Step S6: If the upper bound and the lower bound are not close, repeat Steps S4 - S6;

[0068] Step S7: If the upper bound and the lower bound are close, obtain the optimal solution of the optimization model for the closed-loop manufacturing production decision.

[0069] Specifically, the optimization model for the closed-loop manufacturing production decision is:

[0070]

[0071] In the formula: The startup capital required to purchase the nth material from the s supplier in the t period is: When the supplier is selected, SP i,s,t is 1, otherwise it is 0; ST i,s,t is the corresponding startup capital when the supplier is selected; j mc is the manufacturing unit of material i; v is the inventory unit of material i; y is the recycling unit of material i; A i,s,t , B i , C i , D i , E i are the coefficients of the cost function; P i,s,t , MM i,t , IV i,t , MY i,t , QT i,t are the procurement cost, manufacturing cost, warehouse cost, recycling cost and transportation cost respectively; The transportation cost includes:

[0072]

[0073] Among them, i ∈ N, t ∈ T are respectively the cost of transporting the material from the manufacturing unit j mc to the warehouse j wh , the cost of transporting from the warehouse j wh to the manufacturing unit j mc , the cost of transporting from the manufacturing unit j mc to the recycling unit j rc and the cost of transporting from the recycling unit jrc Cost of transportation to warehouse j wh and the cost of the recycling unit j rc Cost of transportation between them

[0074] Furthermore, the optimality constraints include:

[0075] Supplier selection constraints

[0076]

[0077] Among them, and represent the minimum and maximum quantities of material i purchased from merchant s

[0078] The feasibility constraints include production constraints in the manufacturing process, recycling constraints in the recycling process, production demand constraints, and inventory balance constraints in the warehouse. Specifically:

[0079] Production constraints in the manufacturing process

[0080]

[0081] Among them, represents the upper limit of the production capacity of manufacturing unit j mc during manufacturing; λ i,t represents the unqualified rate of material i during manufacturing in the manufacturing unit. The unqualified processed materials are transported to the recycling unit through ; the qualified materials are transported to the warehouse through ; β i',i represents the material loss rate from warehouse W(i') storing material i' to manufacturing unit j mc

[0082] Recycling constraints in the recycling process

[0083]

[0084] Among them, represents the upper limit of the maximum capacity of material i processed by the recycling unit in recycling unit Y(j rc )

[0085]

[0086] Among them, represents the process of transporting from manufacturing unit j mc or recycling unit j' rc to another recycling unit j rc ; represents the recycled material i transported from recycling unit j rc to the waste unit j ro or warehouse j​wh process

[0087]

[0088] wherein, δ i is the rejection rate of material i when being processed in recycling unit j rc The rejected materials are transported to the waste unit, and the materials that can be reused are transported to the corresponding warehouse.

[0089] Production demand constraint

[0090]

[0091] wherein, pd i,t is the demand for the product, which comes from the semi-finished product processing and manufacturing unit.

[0092] Warehouse inventory balance constraint

[0093] Involves the warehouse balance in the recycling link:

[0094]

[0095] wherein, the inventory balance of material i in period t is the inventory level in period (t - 1) plus the quantity transported from the previous link plus the quantity purchased from the supplier minus the quantity to be transported to the next link.

[0096] Warehouse balance not involving the recycling link

[0097]

[0098]

[0099] wherein, the inventory balance of material i in period t is the inventory level in period (t - 1) plus the quantity transported from the previous link minus the quantity to be transported to the next link; is the inventory level of the final product fp in period t, specifically: the remaining products from the previous period plus the transported products minus the required products, as shown in the appendix Figure 4 shown.

[0100] Furthermore, the closed-loop manufacturing production decision optimization model is solved according to the product demand. According to different types of constraint conditions as decision variables, the closed-loop manufacturing production decision optimization model is decomposed into a master problem and a sub-problem by Benders, that is, the master problem and sub-problem in step S3 are respectively:

[0101]

[0102] Step S4 performs a dual operation on the sub-problem, simplifies the original problem, and forms

[0103] New original problem:

[0104]

[0105] New dual sub - problem:

[0106]

[0107] Among them, w is a decision variable, and (BB - SP i,s,t ) is the coefficient of the decision variable w; BB is a constant; G, H, K, L represent dual cost coefficients; B i , C i , D i , E i are constants.

[0108] Furthermore, the solutions of the new dual sub - problem have multiple types:

[0109] The new dual sub - problem g(w) has no solution:

[0110]

[0111] Among them, the corresponding new original problem also has no solution;

[0112] The new dual sub - problem g(w) has a bounded solution:

[0113]

[0114] Among them, is the initial solution;

[0115] The new dual sub - problem g(w) obtains a bounded solution g(w * ), and an optimality constraint is added to the new original problem to obtain the following formula (22):

[0116]

[0117] Among them, w * is the decision solution of the new dual sub - problem of the closed - loop manufacturing production decision. The solution of the new original problem obtained by the constraint is used as or updates the upper bound up of the range of the closed - loop manufacturing production decision optimization model: g(w * );

[0118] The new dual sub - problem g(w) has an unbounded solution. A feasibility constraint is added to the new original problem to obtain the following formula (23):

[0119] 0≥(BB - SP i,s,t )w * (23)

[0120] Furthermore, the lower bound of the range of the closed - loop manufacturing production decision optimization model is updated

[0121]

[0122] Where lb is the lower bound, w * The production decision made at this time.

[0123] Step S5: Compare the upper bound and the lower bound, and perform iterative processing:

[0124]

[0125] Where up,lb represents the upper and lower bounds of the range of the closed-loop manufacturing production decision optimization model.

[0126] In step S7, the upper bound is close to the lower bound, i.e.

[0127] up≈lb (25)

[0128] Among them, the closeness of the upper and lower bounds indicates that the optimal solution that satisfies the closed-loop manufacturing production decision optimization model is calculated through Benders decomposition;

[0129] In step S6, the upper bound is not close to the lower bound, i.e.

[0130] up>lb (26)

[0131] The fact that the upper and lower bounds are not close indicates that: the optimal solution that satisfies the closed-loop manufacturing production decision optimization model is not calculated through Benders decomposition; Bring it back to the new dual subproblem and start iterating, repeating steps S4 to S6, and continuously iterating and approaching the upper and lower bounds of the closed-loop manufacturing production decision optimization model to obtain the optimal production decision.

[0132] In general, by establishing a closed-loop manufacturing production decision optimization model with mixed integer variables, the forward process is used for production and manufacturing, and the reverse process is used for recycling and processing. The materials that have passed the recycling process are transported in the next cycle and transported to the forward process for reuse; according to the type of decision variables, the original problem of closed-loop manufacturing production decision is decomposed into a main problem and sub-problems by Benders; the main problem contains 0-1 decision variables for production decisions, and the sub-problems contain integer decision variables for production decisions; by performing dual operations on sub-problems, the closed-loop manufacturing production decision optimization model is simplified to form new dual sub-problems and new original problems; solve the new dual sub-problems, and according to the type of solution obtained, add different types of constraints to the new original problem to accelerate the convergence speed and update the lower bound; solve the new original problem and update the upper bound, and bring it back to the new dual sub-problem. If the upper and lower bounds are not close, repeat the solution of the new dual sub-problem, add constraints, and iterate repeatedly; otherwise, stop the iteration and obtain the optimal solution that satisfies the production decision.

[0133] In this embodiment, for the system shown in the appendix Figure 1 the method of the present invention and the direct solution method are respectively used for calculation. The comparison table of the procurement plans for solving production decisions by the two methods is as follows:

[0134] Table 1 Comparison of procurement plans for solving production decisions by two methods

[0135]

[0136] As can be seen from Table 1, in terms of the procurement scope, the procurement scope of the method of the present invention is wider, and two types of suppliers are selected. This is because compared with the direct solution method, the method of the present invention can better reflect the superiority of decision-making, select a more satisfactory and rich procurement plan, and provide more possibilities for the production plan; in addition, in terms of the procurement quantity, the procurement quantity of the method of the present invention is 2,155 less than that of the direct solution method, reducing the redundant procurement quantity. The reason is that the method of the present invention makes more use of recyclable materials, realizing resource conservation and the sustainable development of the enterprise.

[0137] Table 2 Comparison of procurement plan details of two methods

[0138]

[0139]

[0140] As can be seen from Table 2, no procurement is carried out in the third cycle by using the method of the present invention, while procurement is still carried out in Scenario 2. This is because, by using the method of the present invention, the associations between various links in production manufacturing and the cost expenditures of each link itself can be taken into account more comprehensively. Therefore, according to the cost ratios required by each link, recyclable materials are fully used in the third cycle, saving a large amount of expenditures on procurement in the third cycle through the small expenditures on warehouse storage and recycling, reducing the production cost. From this decision-making plan, it can be seen that the method of the present invention can effectively make a satisfactory decision.

[0141] Table 3 Comparison of production costs of two methods

[0142]

[0143] As can be seen from Table 3, the total cost of the method of the present invention in closed-loop production and manufacturing is 601,390 yuan less than the total cost of direct solution. Among them, the procurement cost and manufacturing cost of the method of the present invention are less than those of direct solution by 724,450 yuan and 79,600 yuan respectively. However, in terms of warehouse cost, recycling cost and transportation cost, they are 45,500 yuan, 101,740 yuan and 55,420 yuan more respectively. This is because, through the method of the present invention, a better choice of production decision is made. Although the cost expenditures in aspects of warehouse, recycling and transportation are high, the savings in cost compared to procurement and manufacturing are much lower. Such a decision-making scheme not only reduces the cost expenditure of production, but also reduces the harm caused by production and manufacturing to the environment. Through recycling, pollution is reduced, the recycling of resources is improved, and sustainable development is achieved; the superiority of the present invention is obvious in that the Benders iterative decomposition method is adopted in closed-loop production and manufacturing, the performance of the optimization scheme is outstanding, which can bring high benefits to enterprises and green development to the environment.

[0144] Generally speaking, the present invention establishes a closed-loop manufacturing system combining forward production and reverse recycling in the manufacturing industry, realizes the full reuse of resources, and indirectly realizes environmental protection and sustainable development while carrying out production; the present invention introduces the idea of Benders iterative decomposition into the production planning of closed-loop manufacturing, realizes the rapid customization of production decision-making schemes, and the decision-making planning schemes are more reasonable and effective; the present invention considers the closed-loop production and manufacturing processes of multiple cycles, multiple products and multiple links, and improves the benefits of enterprises at different production and manufacturing stages in large-scale production plans, and greatly reduces the production cost.

[0145] It should be noted that although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principle and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A closed-loop manufacturing production decision optimization method based on Benders iterative decomposition, characterized in that Including the following steps: Step S1: Construct a closed-loop manufacturing production decision optimization model, as well as the feasibility constraints and optimality constraints of the closed-loop manufacturing production decision optimization model; Step S2: Perform Benders iterative decomposition on the closed-loop manufacturing production decision optimization model to obtain a master problem and a subproblem; Step S3: Perform a dual operation on the subproblem to form a new dual subproblem and a new primal problem; Step S4: Solve the new dual subproblem, and add feasibility constraints or optimality constraints to the new primal problem according to the type of solution to solve for the upper and lower bounds of the closed-loop manufacturing production decision scope optimization model; Step S5: Compare the upper bound and the lower bound, and perform iterative processing; Step S6: If the upper bound and the lower bound are not close, repeat Steps S4 - S6; Step S7: If the upper bound and the lower bound are close, obtain the optimal solution of the closed-loop manufacturing production decision optimization model; The closed-loop manufacturing production decision optimization model is: In the formula: The startup capital required to purchase the nth material from the s supplier in the t period is: When the supplier is selected, SP i,s,t is 1, otherwise 0; ST i,s,t is the corresponding startup capital when the supplier is selected; j mc is the manufacturing unit of material i; v is the inventory unit of material i; y is the recycling unit of material i; A i,s,t ,B i ,C i ,D i ,E i are the coefficients of the cost function; P i,s,t , MM i,t ,IV i,t ,MY i,t ,QT i,t are the procurement cost, manufacturing cost, warehouse cost, recycling cost and transportation cost respectively; The transportation cost includes: where \(i\in N\) and \(t\in T\) are the costs of transporting the material from manufacturing unit \(j\) mc to warehouse \(j\), wh the cost of transporting from warehouse \(j\) wh to manufacturing unit \(j\), mc the cost of transporting from manufacturing unit \(j\) mc to recycling unit \(j\), rc the cost of transporting from recycling unit \(j\) rc to warehouse \(j\), wh and the cost of transportation between recycling unit \(j\) rc ; The feasibility constraints include: production constraints in the production and manufacturing process, recycling constraints in the recycling process, production demand constraints, and inventory balance constraints in the warehouse; The optimality constraints include supplier selection constraints; The master problem and the subproblem in Step S3 are respectively: In Step S4, perform a dual operation on the subproblem, simplify the closed-loop manufacturing production decision optimization model, and form New primal problem: New dual subproblem: Among them, w is a decision variable, and (BB-SP i,s,t ) is the coefficient of the decision variable w; BB is a constant; G, H, K, L represent dual cost coefficients; B i , C i , D i , E i are constants; The solution of the new dual subproblem has multiple types: The new dual subproblem g(w) has no solution: Among them, the corresponding new primal problem also has no solution; The new dual subproblem g(w) has a bounded solution: Among them, is the initial solution; The bounded solution \(g(w)\) of the new dual sub-problem \(g(w)\) is obtained, and an optimality constraint is added to the new primal problem to obtain the following formula (22): * ) where w * is the decision solution of the new dual sub-problem of the closed-loop manufacturing production decision. The solution of the new primal problem obtained by the constraint is used as or updates the upper bound up: g(w * ) of the scope of the closed-loop manufacturing production decision optimization model; The new dual subproblem g(w) has an unbounded solution. Add feasibility constraints to the new primal problem to obtain the following formula (23): 0≥(BB - SP i,s,t )w * (23) Furthermore, update the lower bound of the closed-loop manufacturing production decision optimization model range where lb is the lower bound, w * is the production decision obtained at this time; In Step S5: Compare the upper bound and the lower bound, and perform iterative processing: Among them, up and lb represent the upper bound and the lower bound of the closed-loop manufacturing production decision optimization model range; In Step S7, the upper bound and the lower bound are close, that is up≈lb(25) Among them, the fact that the upper bound and the lower bound are close indicates that the optimal solution satisfying the closed-loop manufacturing production decision optimization model is calculated through Benders decomposition; In Step S6, the upper bound and the lower bound are not close, that is up>lb (26) Among them, the fact that the upper bound and the lower bound do not approach indicates that the optimal solution that satisfies the closed-loop manufacturing production decision optimization model has not been calculated through Benders decomposition; by bringing the initial solution back into the new dual sub-problem, iteration begins, repeating step S4-step S6. Through the continuous iteration and approaching of the upper bound and the lower bound of the closed-loop manufacturing production decision optimization model, the optimal production decision is obtained.

Images