Sequential Function Decomposition Method and System for a Two-Node Inventory System under Capacity Constraints

Abstract

The present invention relates to a sequential function decomposition method and system for a two-node inventory system under capacity constraints. The method includes: S1: constructing a dynamic programming recurrence model according to the inventory level conditions of different levels in the logistics system under capacity constraints; S2: based on the dynamic programming recurrence model, using an alternative equivalent formula, dividing a period into two sub-periods for inventory decision-making, and constructing an optimal inventory strategy for the sub-period existing in the two-node inventory system with the upstream as the bottleneck condition under the condition of a sequential process; S3: constructing an optimal inventory strategy for the two-node inventory with the downstream as the bottleneck condition based on the sequential process. The present invention allows the bottleneck restriction to be at any level of the upstream and downstream. Through a sequential process of converting a high-dimensional optimization problem into a series of low-dimensional optimization problems, the decision variables are sequentially determined one by one, and the optimal replenishment strategy for the two-node inventory system under capacity constraints is realized.

Description

Technical Field

[0001] The present invention relates to the field of upstream and downstream inventory management, and particularly to a method and system for sequence function decomposition of a two-node inventory system under capacity constraints. Background Art

[0002] In the upstream and downstream, the supply process from the supplier to the retailer has always been the key to inventory research problems, and the coordination problems of production, storage, and delivery among different levels of the supply chain are particularly important. Inventory management is the management of the quantity of goods in the logistics process. Excessive inventory occupies a large amount of funds for enterprises and increases the interest burden. However, if the inventory is reduced too much, there will be a state of supply falling short of demand. According to the external requirements for inventory, the characteristics of enterprise orders, forecasts, plans, and execute a behavior of replenishing inventory, and control this behavior. The key lies in how to determine the order, how much to order, and when to order.

[0003] The inventory management system is the basis of production, planning, and control. Due to the continuous increase in labor costs and advanced manufacturing technologies, production capacity limitations are quite common in actual production. The choice of inventory in the upstream and downstream is particularly important. The inventory level at the upstream level will affect the supply to the downstream and also have a certain impact on the subsequent processes of the downstream. The inventory level at the downstream level will directly affect the process of the retailer connecting to further consumers, resulting in the problem of supply falling short of demand.

[0004] Different levels will adopt different replenishment strategies when facing bottleneck problems. For the replenishment strategy of a two-node serial system with capacity constraints where the upstream level is the bottleneck, the standard decomposition method introduced previously for such problems is no longer applicable in this case. Therefore, how to formulate the optimal replenishment strategy for a multi-level system with capacity constraints where the upstream is the bottleneck level has become an urgent problem to be solved. Summary of the Invention

[0005] In order to solve the above technical problems, the present invention provides a method and system for sequence function decomposition of a two-node inventory system under capacity constraints.

[0006] The technical solution of the present invention is: A method for sequence function decomposition of a two-node inventory system under capacity constraints, comprising:

[0007] Step S1: Construct a dynamic programming recurrence model according to the inventory level conditions at different levels in the logistics system under capacity constraints;

[0008] Step S2: Based on the dynamic programming recurrence model, adopt an alternative equivalent formula to divide a period into two sub-periods for inventory decision-making, and construct an optimal inventory strategy for the sub-periods in the two-node inventory system with the upstream as the bottleneck condition under the condition of a sequential process.

[0009] Step S3: Based on the sequential process, construct an optimal inventory policy for the two-node inventory with the downstream as the bottleneck condition.

[0010] Compared with the prior art, the present invention has the following advantages:

[0011] The method proposed by the present invention realizes the optimal replenishment strategy for the two-node inventory system under capacity constraints. Instead of restricting the bottleneck stage to the downstream stage, the present invention restricts the bottleneck to the upstream stage. Through a sequential process that transforms a high-dimensional optimization problem into a series of low-dimensional optimization problems, the decision variables are determined one by one in sequence. Based on the decomposability of the sequential process and the sequence function, the optimal replenishment strategy for each stage is described as depending on the stage benchmark inventory strategy based on the sequence function. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 It is a flowchart of a method for decomposing a sequence function for a two-node inventory system under capacity constraints in an embodiment of the present invention;

[0013] Figure 2 It is a structural block diagram of a system for decomposing a sequence function for a two-node inventory system under capacity constraints in an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0014] The present invention provides a method for decomposing a sequence function for a two-node inventory system under capacity constraints.

[0015] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below through specific embodiments in conjunction with the accompanying drawings.

[0016] The present invention provides a two-node inventory system. The system of the first node usually targets inventory systems such as retail centers and self-pickup points close to the market, and the second node targets inventory systems of warehousing centers far from the market. The two-node inventory system can effectively handle the production, storage and coordination of goods at various stages in the supply chain. The class inventory in the present invention mainly targets the inventory levels of the upstream and downstream. The class inventory system is an effective inventory management system, making the management of different class inventories more convenient and efficient.

[0017] Embodiment 1

[0018] As Figure 1 shown, a method for decomposing a sequence function for a two-node inventory system under capacity constraints provided by an embodiment of the present invention includes the following steps:

[0019] Step S1: According to the inventory level conditions of different classes under capacity constraints in the logistics system, construct a dynamic programming recurrence model;

[0020] Step S2: Based on the dynamic programming recurrence model, using an alternative equivalent formula, divide a period into two sub-periods for inventory decision-making. For the sub-periods in the two-node inventory system with the upstream as the bottleneck condition, construct its optimal inventory strategy under the condition of a sequential process;

[0021] Step S3: Based on the sequential process, construct the optimal inventory strategy for the two-node inventory with the downstream as the bottleneck condition.

[0022] In one embodiment, the above Step S1: According to the inventory level conditions of different levels under capacity constraints in the logistics system, construct a dynamic programming recurrence model as shown in Formulas (1) to (3), specifically including:

[0023]

[0024]

[0025]

[0026] Among them, in Formula (1), t is the period, and t = 1,..., T; X is the level inventory vector; f t (X) is the optimal discounted cost from period t to period t in state X; Y is the level inventory vector after ordering, and Y needs to satisfy Among them, Y j is the inventory level after ordering for the j-th level, X j represents the inventory level before ordering for the j-th level, and X1 = x1, X2 = x1 + x2, where x j represents the inventory level of the j-th node in the traditional sense, K j represents the capacity of the j-th level; β is the discount factor, and β ∈ (0, 1); D t is the demand for each period t, G t (Y) is the expected value of D t ; e is a unit vector with all elements being 1;

[0027] In Formula (2), H j (j = 1, 2) is the unit holding cost of each node in the traditional supply chain, h j (j = 1, 2) is the unit holding cost of each level. Among them, h1 = H1 - H2 > 0 and h2 = H2 respectively represent the unit holding costs of the first level and the second level, and b is the unit backlog cost generated in each period; for the (T + 1)-th period, for any X, let f T+1 (X) ≡ 0;

[0028] In Formula (3), when j = 1, 2, g j，t(Y j ) is a univariate convex function that only contains Y j ;

[0029] Substitute G t (Y) = g 1，t (Y1) + g 2，t (Y2) into formula (1), and the following formula (4) can be obtained:

[0030]

[0031] Among them, the constraint condition is Let K1 = nK, K2 = K; M is a sufficiently large integer, which can make Y2 <= Y1 + MK hold necessarily in the entire variable value space.

[0032] In one embodiment, the above step S2: Based on the dynamic programming recurrence model, an alternative equivalent formula is adopted to divide a period into two sub-periods for inventory decision-making. For the sub-periods in the two-node inventory system with the upstream as the bottleneck condition, under the condition of the sequential process, its optimal inventory strategy is constructed, specifically including:

[0033] Set each period t to include 2 sub-periods: sub-period 1 and sub-period 2, and the starting states of the sub-periods are respectively represented as: and Use to represent the system state at the end of the sub-period; Follow the backward method of the dynamic programming recurrence model and start from the preprocessing step:

[0034] Step S21: This is the preprocessing step. Let contain all inventory decisions in period t; Then the following formula (5) is used for preprocessing:

[0035]

[0036] Let F 1，T+1 (·) = 0. This preprocessing step does not contain any decision-making problems;

[0037] Step S22: In sub-period 2, let Among them, the Y1 in is the optimal post-order inventory level in the first stage during the sub-period 1. According to the following formulas (6) - (7), Y2 can be obtained as the optimal post-order inventory level in the second stage during sub-period 2:

[0038]

[0039]

[0040] Among them, Y1 is regarded as a given known quantity in sub-period 2; Y2 must satisfy the constraint conditions: X2 ≤ Y2 ≤ (X2 + K) ∧ (Y1 + MK);

[0041] Let Ω2 = {Y1 + mK | m = 0, …, M} ∪ {Y2}, where Ω2 is the base set of the sub-period 2 at time t, and the definition of Ω2 depends on (Y1, Y2), that is, the vector Among them, the sequence of variables {Y1 + mK | m = 0, 1, …, M} is unique and must satisfy the condition Y1 ≤ Y1 + K ≤ … ≤ Y1 + MK; for the variable Y2, it cannot be greater than Y1 + MK, but it may be higher or lower than Y1 + mK for each m ∈ {1, …, M - 1}; for m ∈ {1, …, M}, let π 2，m correspond to the sequence of Ω2;

[0042] In π 2，m Y2 is between Y1 + (m - 1)K and Y1 + mK, and Π2 represents the set of these M sequences, that is, Π2 = {π 2，1 …, π 2，M}; for a given sequence π ∈ Π2, if takes values that satisfy the sequence π, then it is said that is in the sequence cone caused by π;

[0043] Step S23: In sub-period 1, let Using the following formulas (8) - (9), obtain the optimal post-order inventory level Y1:

[0044]

[0045]

[0046] Among them, Y1 must satisfy the constraints X1 ≤ Y1 ≤ X2 ∧ (X1 + nK);

[0047] Let Ω1 = {X2 - mK | m = 0, …, M} ∪ {Y1}, where Ω1 is the base set of the sub-period 1 at time t;

[0048] There are M sequences of Ω1, and these sequence sets are represented by Π1. For m = {1, …, M}, π 1，m corresponds to the sequence X2 - MK ≤ … ≤ X2 - mK ≤ Y1 ≤ X2 - (m - 1)K ≤ … ≤ X2. In π 1，m Y1 takes values between X2 - mK and X2 - (m - 1)K, and each sequence induces a sequence cone in the space;

[0049] Step S24: During the periods \(t = 1,\cdots,T\), given the initial class inventory state at sub - period 1 as \(X\), when \(m>n\), the optimal post - order inventory level, i.e., \(Y1\), is as shown in the following formula (10):

[0050]

[0051] When \(m < n\), the optimal post - order inventory level \(Y1\) is as shown in the following formula (11):

[0052]

[0053] At sub - period 2, the optimal post - order inventory level is as shown in the following formula (12):

[0054]

[0055] Wherein, and And it satisfies and are a set of univariate functions;

[0056] Wherein, and the initial class inventory state in the said sub - period 2 is Assume The corresponding sequence cone is induced by \(\pi\) 1，m and satisfies the following formula (13):

[0057]

[0058] In one embodiment, the above - mentioned step S3: According to the sequential process, construct the optimal inventory strategy of the two - node inventory with the downstream as the bottleneck condition, specifically including:

[0059] For the downstream class decision, consider a two - node serial system with capacity limits \(K1 < K2\); where let \(K1 = K\), there is \(\Omega1=\{X2 - K,X2\}\cup\{Y1\}\), and it must satisfy \(X2 - K\leq Y1\leq X2\);

[0060] When \(M = n = 1\) and there is only one sequence at each level, there exists \(s1\) such that the optimal \(Y1\) satisfies the optimal replenishment strategy formula (12) - (13) in the said sub - period 1 in step S2 above when \(m\leq n\), and it can be simplified As the following formula (14):

[0061]

[0062] There exists \(s2\) such that the optimal \(Y2\) satisfies the optimal replenishment strategy formula (15) in sub - period 2 in step S2 above, and it can be simplified As the following (15):

[0063]

[0064] Embodiments of the present invention formulate an optimal replenishment policy for a system under the dual-node capacity limit of the upstream bottleneck stage. The bottleneck limit is allowed to be in either the upstream or downstream stage. Embodiments of the present invention propose a sequential process that transforms a high-dimensional optimization problem into a series of low-dimensional optimization problems, and determines decision variables one by one in sequence. Based on the sequential process and the decomposability of the sequence function, the optimal replenishment policy for each stage is described as a class-based reference inventory policy that depends on the sequence function.

[0065] Embodiment 2

[0066] As Figure 2 shown, embodiments of the present invention provide a sequence function decomposition system for a dual-node inventory system under capacity constraints, including the following modules:

[0067] A module for constructing a dynamic programming recurrence model, which is used to construct a dynamic programming recurrence model according to the inventory level conditions at different stages in the logistics system under capacity constraints;

[0068] An optimal inventory policy module under the condition that the upstream is the bottleneck, which is used to make inventory decisions by dividing a period into two sub-periods based on the dynamic programming recurrence model and using an alternative equivalent formula, and construct its optimal inventory policy for the sub-period in the dual-node inventory system with the upstream as the bottleneck under the condition of the sequential process;

[0069] An optimal inventory policy module under the condition that the downstream is the bottleneck, which is used to construct the optimal inventory policy for the dual-node inventory with the downstream as the bottleneck based on the sequential process.

[0070] The above embodiments are provided only for the purpose of describing the present invention, and are not intended to limit the scope of the present invention. The scope of the present invention is defined by the appended claims. All equivalent substitutions and modifications made without departing from the spirit and principles of the present invention shall be covered within the scope of the present invention.

Claims

1. A sequential function decomposition method for a two-node inventory system under capacity constraints, characterized in that, Including: Step S1: Construct a dynamic programming recurrence model according to the inventory level conditions of different classes under capacity constraints in the logistics system, specifically including: (1) (2) (3) Among them, in formula (1), t is the period, and t = 1, …, T; X is the class inventory vector; is the optimal discounted cost from period t to period t in state X; Y is the class inventory vector after ordering, and Y needs to satisfy , where is the inventory level after ordering for the j-th class, represents the inventory level before ordering for the j-th class, and where represents the inventory level of the j-th node in the traditional sense, represents the capacity of the j-th class; is the discount factor, and ; is the demand in each period t, is the expected value of; e is a unit vector with all elements being 1; In formula (2), (j = 1, 2) is the unit holding cost of each node in the supply chain in the traditional sense, (j = 1, 2) is the unit holding cost of each class, where, respectively represent the unit holding costs of the first class and the second class, and b is the unit backlog cost generated in each period; for the (T + 1)-th period, for any X, let ; In formula (3), when j = 1, 2, is a univariate convex function that only contains ; Substitute into Equation (1), and the following Equation (4) can be obtained: (4) Among them, is the optimal discounted cost from period t to period t in state X; denotes that the set is , where M is a sufficiently large integer such that <= + MK must hold throughout the variable value space; Step S2: Based on the dynamic programming recurrence model, use an alternative equivalent formula to divide a period into two sub-periods for inventory decision-making. For the sub-periods in the two-node inventory system with the upstream as the bottleneck condition, construct its optimal inventory strategy under the condition of a sequential process, specifically including: Set each of the said periods to include 2 sub-periods: sub-period 1 and sub-period 2, and the starting states of the sub-periods are respectively represented as: and , using to represent the system state at the end of the sub-period; following the backward method of the said dynamic programming recurrence model, starting from the preprocessing step: Step S21: This is a preprocessing step. Let , include all inventory decisions in period t; perform preprocessing using the following formula (5): (5) Let , this preprocessing step does not involve any decision-making problems; Step S22: In sub-period 2, let , ; where in is the optimal post-order inventory level of the first stage in the sub-period 1. According to the following formulas (6) - (7), the optimal post-order inventory level of the second stage in the sub-period 2 can be obtained as : (6) (7) Among them, is regarded as a given known quantity in sub-period 2; The constraint condition must be satisfied: ； Let , where is the base set of the sub-period 2 at time t, is defined depending on , that is, the vector ; where the sequence of the variable is unique and must satisfy the condition The variable , which cannot be greater than , but for each it may be higher or lower than ; for , let correspond to the sequence of ; In in In and between, denotes the set of these sequences, i.e., ; for a given sequence , if takes a value that satisfies the sequence , then is said to be in the sequence cone caused by ; Step S23: In sub-period 1, let , , and use the following formulas (8) to (9) to obtain the optimal post-order inventory : (8) (9) Among them, the constraints must be satisfied ; Let ; There are M sequences, which are represented by to denote these sequence sets. For , corresponds to the sequence . In , the value of ranges between and . Each sequence induces a sequence cone in the space of . Step S24: During the periods \(t = 1,\ldots,T\), given that the initial class inventory status in sub - period 1 is \(X\), when \(m>n\), the optimal post - order inventory level, i.e., , is as shown in the following formula (10): (10) When m < n, the optimal post-order inventory level is as shown in the following formula (11): (11) In sub-period 2, the optimal inventory level after ordering is as shown in the following formula (12): (12) Among them, , and satisfy , is a set of univariate functions; Among them, , and the initial class inventory status in the sub-period 2 is ; Assume The corresponding sequence cone is induced by and satisfies the following formula (13): (13)； Step S3: Based on the sequential process, construct the optimal inventory strategy for the two-node inventory with the downstream as the bottleneck condition, specifically including: For downstream class decisions, consider a two-node serial system with a capacity limit of ; where let , there is , and it must satisfy ; When M = n = 1 and there is only one sequence at each level, there exists such that the optimal satisfies the optimal replenishment strategy formulas (12) - (13) in the described sub - period 1 in the above - mentioned step S2 when and can be simplified to the following formula (14): (14) There exists such that the optimal satisfies the optimal replenishment strategy formula (15) of sub-period 2 in step S2 above, which can be simplified as follows (15): (15)。 2. A sequential function decomposition system for a two-node inventory system under capacity constraints, characterized in that, Including the following modules: A module for constructing a dynamic programming recurrence model, which is used to construct a dynamic programming recurrence model according to the inventory level conditions of different classes under capacity constraints in the logistics system, specifically including: (1) (2) (3) Among them, in formula (1), t is the period, and t = 1, …, T; X is the class inventory vector; is the optimal discounted cost from period t to period t in state X; Y is the class inventory vector after ordering, and Y needs to satisfy , where is the inventory level after ordering for the j-th class, represents the inventory level before ordering for the j-th class, and where represents the inventory level of the j-th node in the traditional sense, represents the capacity of the j-th class; is the discount factor, and ; is the demand in each period t, is the expected value of; e is a unit vector with all elements being 1; In formula (2), (j = 1, 2) is the unit holding cost of each node in the supply chain in the traditional sense, (j = 1, 2) is the unit holding cost of each class, where, respectively represent the unit holding costs of the first class and the second class, and b is the unit backlog cost generated in each period; for the (T + 1)-th period, for any X, let ; In formula (3), when j = 1, 2, is a univariate convex function that only contains ; Substitute into Equation (1), and the following Equation (4) can be obtained: (4) where, is the optimal discounted cost from period t to period t in state X; denotes the set as , M is a sufficiently large integer such that <= + MK must hold throughout the variable value space; An optimal inventory strategy module under the upstream bottleneck condition, which uses the dynamic programming recurrence model and an alternative equivalent formula to divide a period into two sub-periods for inventory decision-making. For the sub-periods in the two-node inventory system with the upstream as the bottleneck condition, construct its optimal inventory strategy under the condition of a sequential process, specifically including: Each of the said periods is set to include 2 sub-periods: sub-period 1 and sub-period 2, and the starting states of the sub-periods are respectively expressed as: and , using to represent the system state at the end of the sub-period; following the backward method of the said dynamic programming recurrence model, starting from the preprocessing step: Step S21: This is a preprocessing step. Let , include all inventory decisions in period t; perform preprocessing using the following formula (5): (5) Let , this preprocessing step does not contain any decision problems; Step S22: In sub-period 2, let , ; where, in is the optimal post-order inventory of the first class in the sub-period 1. According to the following formulas (6) - (7), the optimal post-order inventory of the second class in sub-period 2 can be obtained as : (6) (7) Among them, is regarded as a known quantity given in sub-period 2; The constraint condition must be satisfied: ； Let , where is the base set of the sub-period 2 at time t, is defined depending on , i.e., the vector ; where the sequence of the variable is unique and must satisfy the condition The variable , which cannot be greater than , but for each it may be higher or lower than ; for , let correspond to the sequence of ; In in In and between, indicating the set of these sequences, i.e., ; for a given sequence , if takes a value that satisfies the sequence , then is said to be in the sequence cone caused by ; Step S23: In sub-period 1, let , , and use the following formulas (8) to (9) to obtain the optimal post-order inventory : (8) (9) Among them, the constraints must be satisfied ; Let ; There are M sequences, denoted by this set of sequences. For , corresponds to the sequence , and the value of in ranges between and . Each sequence induces a sequence cone in the space of ; Step S24: During the periods t = 1, …, T, given that the initial class inventory status in sub-period 1 is X, when m > n, the optimal post-order inventory level, i.e., , is as shown in the following formula (10): (10) When m < n, the optimal post-order inventory level is as shown in the following formula (11): (11) In sub-period 2, the optimal inventory level after ordering is as shown in the following formula (12): (12) Among them, , and satisfy , are a set of univariate functions; Among them, , and the initial class inventory status in the sub-period 2 is ; Suppose The corresponding sequence cone is induced by and satisfies the following formula (13): (13)； An optimal inventory strategy module under the downstream bottleneck condition, which is used to construct the optimal inventory strategy for the two-node inventory with the downstream as the bottleneck condition based on the sequential process, specifically including: For downstream class decisions, consider a two-node serial system with a capacity limit of ; where let , there is , and it must satisfy ; When M = n = 1 and there is only one sequence at each level, there exists such that the optimal satisfies the optimal replenishment strategy formulas (12)-(13) in the described sub-period 1 in the above step S2 when and can be simplified to the following formula (14): (14) There exists such that the optimal satisfies the optimal replenishment strategy formula (15) in sub-period 2 of the above step S2 and can be simplified as follows (15): (15)。

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