A robust optimization model for container shipping booking decision considering order uncertainty

Abstract

The application discloses a robust optimization model for container booking decision of sea shipping considering order uncertainty and an optimization solving method thereof, wherein the modeling process of the optimization model comprises the following steps: 1) determining a target function of a two-stage robust optimization model; and 2) determining constraint conditions contained in the model, including one-stage constraint conditions and two-stage constraint conditions. The container shipping booking problem considering customer order demand uncertainty is established as a two-stage robust optimization model. Since the uncertain set is a polyhedron, the model cannot be solved by simply enumerating all uncertain scenarios. The model is solved by using a row and column generation algorithm. The shipping container booking problem is abstracted as a mathematical model, and a row and column generation algorithm based on a main problem-subproblem framework is designed for the model. The algorithm can efficiently solve the problem and has high technical and economic value.

Description

TECHNICAL FIELD

[0001] The application relates to a robust optimization model for container booking decision of sea transportation considering order uncertainty and an optimization solving method thereof. TECHNICAL BACKGROUND

[0002] The sea container transportation has become an indispensable part of global trade due to its relatively low transportation price. In order to carry out international trade more efficiently, a general trade enterprise will seek help from a third-party logistics service provider in logistics transportation. For the third-party logistics service provider, due to the instability of the global market under various factors, they are difficult to allocate the transportation capacity, so that the waste of transportation capacity or the failure of customer service due to insufficient transportation capacity will have a great impact on their own cost and reputation. Therefore, the third-party logistics service provider will often provide more preferential prices for the pre-booking transportation capacity, so as to encourage customers to make pre-booking for the cabin, thereby optimizing their own transportation capacity scheduling.

[0003] On the other hand, from the perspective of international trade enterprises, they also face the fluctuating global market. Under the condition of uncertain customer demand, if the booking quantity is too small, it may lead to extremely high temporary booking price, or even the customer order cannot be sent in time; if the booking quantity is too large, it will cause the loss of transportation cost. Therefore, how to make container cabin reservation in the face of uncertain customer demand is of important technical and economic value to enterprises. SUMMARY

[0004] The application aims to provide a robust optimization model for container booking decision of sea transportation considering order uncertainty and an optimization solving method thereof, which can help enterprises reduce logistics cost and improve overall efficiency.

[0005] The application provides a two-stage robust optimization model for container shipping booking considering the uncertainty of customer order demand, characterized by the following modeling process:

[0006] 1) The symbols of the parameters and variables used in the specific mathematical model are defined:

[0007] Index and set

[0008] d: represents the customer index;

[0009] s: represents the route index;

[0010] j: represents the container type index;

[0011] n: represents the container index;

[0012] D: represents the customer set;

[0013] S: represents a set of routes;

[0014] I: represents a set of product types;

[0015] J: represents a set of container types;

[0016] S d : represents a set of routes containing the destination of customer d;

[0017] I d : represents a set of products demanded by customer d;

[0018] N sj : represents a set of j types of containers for route s;

[0019] Parameter

[0020] c dsj : represents the cost of j types of containers for route s to reach the destination of customer d;

[0021] w i : represents the inventory of product i;

[0022] represents the basic demand of customer d for product i;

[0023] represents the maximum fluctuation of the demand of customer d for product i;

[0024] o di : represents the actual demand of customer d for product i;

[0025] v j : represents the capacity of j types of containers;

[0026] C: represents the unit penalty of unfulfilled customer orders;

[0027] Decision variable

[0028] z dsj : one-stage decision variable, represents the number of j types of containers for route s scheduled for customer d;

[0029] y dsjn : one-stage decision variable, is 1 if the nth allocation of j types of containers for route s is assigned to customer d, otherwise 0;

[0030] x dsjni : two-stage decision variable, represents the amount of product i loaded in the nth allocation of j types of containers for route s assigned to customer d;

[0031] g di: Two-stage decision variable, denoted as the amount of demand fluctuation uncertainty allocated to customer d for product i;

[0032] 2) Determine the objective function of the two-stage robust optimization model:

[0033] In this invention, the objective function is divided into two parts, the first-stage objective function is to minimize the total booking cost, and the second-stage objective function is to find the maximum minimum order unfulfillment penalty cost given the first-stage booking decision:

[0034]

[0035] 3) The constraints contained in the model:

[0036] The first-stage constraints include:

[0037] The calculation method of the number of j type containers for customer d scheduled on route s:

[0038]

[0039] Each container can be allocated to only one customer at most:

[0040]

[0041] Decision variable y dsjn Can only take 0 or 1:

[0042]

[0043] Decision variable z dsj Can only take non-negative integers:

[0044]

[0045] The second-stage constraints include:

[0046] The total shipment quantity of each product does not exceed its inventory quantity:

[0047]

[0048] The total quantity of products sent to each customer does not exceed its demand:

[0049]

[0050] The total quantity of products loaded in each container does not exceed its maximum capacity:

[0051]

[0052] Finally, the decision variable x dsjni Is greater than or equal to 0:

[0053]

[0054] where the customer order demand uncertainty is described by a budget-uncertain set:

[0055]

[0056] The uncertain set indicates that the actual demand of customer d for product i fluctuates around its basic demand with a maximum fluctuation of where Γ is an integer parameter defined in advance to control the conservative level of uncertainty, when Γ is 0, it means that all demands have no fluctuation, and when Γ takes ∑ d∈D |I d |, all customer order demands can fluctuate.

[0057] The application provides an optimization solving method of a two-stage robust optimization model of container shipping booking considering customer order demand uncertainty, characterized in that a row and column generation algorithm is used to solve the model, including the following processes:

[0058] The application establishes a two-stage robust optimization model of the container shipping booking problem considering customer order demand uncertainty, since the uncertain set is a polyhedron, it cannot be solved simply by enumerating all uncertain scenarios. The application uses a row and column generation algorithm to solve the model.

[0059] The row and column generation algorithm is based on the main problem-subproblem framework for iterative solving: the main problem is the relaxation of the original problem, and the lower bound of the original problem can be obtained by solving the main problem; by solving the subproblem, important scenarios can be identified and added to the main problem, and better lower bounds can be obtained by generating rows and columns. At the same time, in each iteration, the upper bound of the original problem can be obtained according to the objective function value of the subproblem and the one-stage objective function value in the main problem. Through iterative solving, the upper and lower bounds can be continuously tightened, and finally the optimal solution is obtained when the upper and lower bounds are equal.

[0060] The main problem is modeled as:

[0061]

[0062] where is the important scenario identified in the lth iteration. The subproblem can be modeled as:

[0063]

[0064] The model is a double-layer optimization model and cannot be directly solved. Through the duality theory, the inner layer is dual processed, and the model is re-modeled as a single-layer optimization problem F(y):

[0065]

[0066] The basic idea is as follows:

[0067] 1) Initialization LB=-infinity, UB=+infinity, k=0

[0068] 2) Solve the main problem to obtain the optimal solution Update the lower bound at the same time

[0069] 3) Fix Solve the sub-problem and update the upper bound

[0070] 4) If UB-LB<=epsilon, terminate the algorithm and output the optimal solution; otherwise, add the identified important scenario to the main problem, and generate new variables and corresponding constraints, k=k+1.

[0071] The row and column generation algorithm process adopted by the application is as shown in Figure 1 .

[0072] The application abstracts the shipping container booking problem into a mathematical model, and designs a row and column generation algorithm based on the main problem-sub-problem framework for the model. The algorithm can efficiently solve the problem, and has high technical and economic value. BRIEF DESCRIPTION OF DRAWINGS

[0073] Figure 1 The row and column generation algorithm process. DETAILED DESCRIPTION

[0074] In order to facilitate the understanding of those skilled in the art, the application will be further described below in conjunction with examples and drawings. The content mentioned in the embodiments is not a limitation of the application.

[0075] A two-stage robust optimization model and optimization solving method for container shipping booking considering customer order demand uncertainty, the modeling process is as follows:

[0076] 1) Symbol definition

[0077] d: represents the customer index;

[0078] s: represents the route index;

[0079] j: represents the container type index;

[0080] n: represents the container index;

[0081] D: represents the customer set;

[0082] S: represents a set of routes;

[0083] I: represents a set of product types;

[0084] J: represents a set of container types;

[0085] S d : represents a set of routes containing the destination of customer d;

[0086] I d : represents a set of products demanded by customer d;

[0087] N sj : represents a set of j types of containers for route s;

[0088] c dsj : represents the cost of j types of containers for route s to reach the destination of customer d;

[0089] w i : represents the inventory of product i;

[0090] : represents the basic demand of customer d for product i;

[0091] : represents the maximum fluctuation of demand of customer d for product i;

[0092] o di : represents the actual demand of customer d for product i;

[0093] v j : represents the capacity of j types of containers;

[0094] C: represents the unit penalty of unfulfilled customer orders;

[0095] 2) Model establishment

[0096] Through the analysis of the characteristics of the problem, the mathematical model is as follows:

[0097] Find the maximum minimum order unfulfilled penalty cost under the condition of one-stage booking decision:

[0098]

[0099] Expression (28) represents that the objective function is to minimize the total cost;

[0100] Expression (29) represents the calculation method of the number of j types of containers for route s reserved for customer d;

[0101] Expression (30) represents that each container can be allocated to at most one customer;

[0102] Expression (31) represents the decision variable y dsjn Can only take 0 or 1;

[0103] Expression (32) represents the decision variable z dsj Can only take non-negative integers;

[0104] Expression (33) represents that the total delivery amount of each product does not exceed its inventory amount;

[0105] Expression (34) represents that the total amount of products sent to each customer does not exceed its demand;

[0106] Expression (35) represents that the total amount of products loaded in each container does not exceed its maximum capacity;

[0107] Expression (36) represents the decision variable x dsjni Greater than or equal to 0;

[0108] Expression (37) represents the budget uncertainty set, which is used to describe the uncertainty of customer order demand.

[0109] Embodiment:

[0110] In the example, the number of customers is 3, the number of routes is 5, the number of containers is 2 (the capacities are 1 and 0.5 respectively), the number of products is 10, and the customer order demand is randomly generated. For the budget uncertainty set, the maximum fluctuation is set to 60% of the basic demand, and Γ takes round(0.6∑ d∈D |I d |).

[0111] On a personal notebook computer with a CPU of Core i5-1135G7 (2.40GHz) and a memory of 16GB, the row and column generation algorithm of the application converges to the optimal solution after only 4 iterations, and the solving time is 88 seconds. As a comparison, the iteration number of the Benders-dual algorithm is 7, and the solving time is 195 seconds. Therefore, the algorithm of the application has obvious advantages.

Claims

1.A robust optimization model for container shipping booking decision considering order uncertainty, characterized in that The modeling process is as follows: 1) Define the symbols of the parameters and variables used in the specific mathematical model: Indices and sets d: represents the customer index; s: represents the route index; j: represents the container type index; n: represents the container index; D: represents the customer set; S: represents the route set; I: represents the product type set; J: represents the container type set; S d : denotes a set of routes containing the customer d destination; I d : represents the set of products demanded by the customer d; N sj : represents a j-type container set of the route s; Parameters c dsj : represents the cost required for a j type container of route s to arrive at the destination of customer d; w i : represents the inventory level of product i; denotes the basic demand of customer d for product i; the maximum fluctuation in the demand of the customer d for the product i; o di : represents the actual demand of the customer d for the product i; v j : represents the capacity of a j-type container; C: represents the unit penalty of the customer order not completed; Decision variables z dsj : one-stage decision variable, denoted as the number of j type containers that customer d reserves for route s; y dsjn : one-stage decision variable, 1 if the nth assignment of j type containers of predetermined route s is given to customer d, otherwise 0; x dsjni : two-stage decision variable representing the quantity of product i loaded in the nth container of j type assigned to route s for customer d; g di : Two-stage decision variable, denoted as the demand fluctuation uncertainty allocated to customer d for product i; 2) Determine the objective function of the two-stage robust optimization model: The objective function has two parts. The first-stage objective function is to minimize the total booking cost, and the second-stage objective function is to find the maximum minimum order not completed penalty cost given the first-stage booking decision: 3) Determine the constraints included in the model The constraints included in the model include first-stage constraints and second-stage constraints; First-stage constraints include: The calculation method of the number of j type containers for the route s reserved for the customer d: Each container can be assigned to at most one customer: Decision variable y dsjn Can only take 0 or 1: Decision variable z dsj Integer only non-negative: Second-stage constraints include: The total shipment quantity of each product does not exceed its inventory quantity: The total quantity of products sent to each customer does not exceed its demand: The total quantity of products loaded in each container does not exceed its maximum capacity: Finally the decision variable x dsjni greater than or equal to 0: Where the customer order demand uncertainty is described by a budget uncertainty set: This uncertain set represents the actual demand of customer d for product i fluctuating around its basic demand with a maximum fluctuation of where Γ is an integer parameter defined in advance to control the conservative level of uncertainty. When Γ is 0, it means that all demands have no fluctuation, while when Γ takes ∑ d∈D |I d |, all customer order demands can fluctuate.

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