To solve the charge planning problem involving charges and the orders in each charge, a traveling salesman problem based charge planning model and the improved cross entropy algorithm are proposed. Firstly, the charge planning problem with unknown charge number is modeled as a traveling salesman problem. The objective of the model is to minimize the dissimilarity costs between each order and its charge center order, the open order costs, and the unselected order costs. Secondly, the improved cross entropy algorithm is proposed with the improved initial state transition probability matrix which is constructed according to the differences of steel grades and order widths between orders. Finally, an actual numerical example shows the effectiveness of the model and the algorithm.
Primary steel making is a main stage of steel production process [
Charge planning problem is a NPhard combinatorial optimization problem, so how to improve the speed and accuracy of the algorithm for this problem is a key point. As mentioned above, the charge plan is made under some production constraints, which can be used to improve the speed and accuracy of the charge planning algorithm. The existing algorithms only consider these constraints as variables in the charge planning models while the proposed improved cross entropy (ICE) algorithm in this study could use the constraints efficiently. Cross entropy (CE) method [
In this research, the improved cross entropy algorithm is proposed to solve the charge planning problem with unknown charge numbers and its objective is to minimize the dissimilarity costs between each order and its charge center order. The rest of the paper is organized as follows. Section
Charge is the basic unit for the steel making process, and the problem in this study is to establish the optimum charge plan including the charge center orders and the orders merged into them. Each order has its own requirement on steel grade, specification, and due date. The requirement differences may cause some costs when an order merged into a charge center order. Therefore, minimize the requirement differences between charge center orders and the orders merged into them are one of three objectives of charge planning problem. The other two objectives are to minimize the costs of unselected orders and the amount of open orders which are used to fill in the furnace but do not belong to any current orders. The requirements for establishing a charge plan are listed as follows.
The steel grades of the orders in a charge should be in the same steel grade class.
The widths differences of the orders in a charge should not be larger than the allowed maximum adjustment width
The thicknesses of orders in a charge should be the same. In this study, the thicknesses of orders are supposed all same according to the actual production process.
The total weight of the contract products in a charge should not surpass the maximum furnace capacity.
The due dates of the contract products in a charge should be similar.
Given
Minimize
subject to
In the charge planning model mentioned above, objective function (
Regarding the orders as cities, the sum of dissimilarity costs caused by orders merged into charge centers, the costs resulted from open order and the penalties for unselected orders as distances, furnace capacity as the maximum distance a salesman travels, the charge planning problem can be regarded as a traveling salesman problem. The objective of the problem is to find an optimum order sequence minimum the penalty costs that calculated by formula
The piecewise function (
In the last section, the traveling salesman problem based model of charge planning problem is constructed. We can use cross entropy method, which has been proven as an efficient method for solving the traveling salesman problem, to solve the charge planning problem as the method. The result did not meet performance expectation when we use the method directly, so we need to improve it. The key point in using cross entropy method to solve a combinatorial optimization problem is the state transition probability matrix, which can be improved according to the characteristics of the problem. In the charge planning problem, two orders with different steel grades or widths have different dissimilarity costs when merged into a same charge center, so the matrix elements can be settled according to the dissimilarity costs as follows.
Assign values to matrix elements according to the process constraints of the problem:
Normalize
Based on the state transition probability matrix above, we proceed the improved cross entropy algorithm as follows.
Generate
Calculate
Order
Update
If
To avoid local optimum, instead of updating the transition matrix
We test our algorithm on the practical production data shown in [
To verify the efficiency of the proposed algorithm, we compare the results obtained by ICE and CE. The algorithms are run on Matlab 7.0 and a personal computer of Pentium R, 2 GB RAM. There are three parameters in the proposed improved cross entropy algorithm and cross entropy method: sample size
Comparison of results by ICE and CE.
ICE  CE  

Optimum value  2352  2354 
Mean running time  26.06  28.98 
Mean deviation  8.6  55.3 
The optimum values are the best values obtained by the two algorithms in 10 times, the mean times are the average times that the two algorithms run 10 times, and the mean deviations are the average distances between the 10 values and the optimum values of the two algorithms in 10 times. The details of the optimum charge plan obtained by ICE and CE are shown in Table
Optimum charge plans obtained by ICE and CE.
Charges  ICE  CE  

Slabs  Weights  Slabs  Weights  
1  12, 13, 17, 15  98  4, 8, 15, 21  99 
2  14, 21, 20, 19  95  2, 1, 5, 3  100 
3  26, 23, 25, 24  96  26, 24, 23, 25  96 
4  4, 9, 16, 18  99  14, 18, 12, 20  98 
5  5, 3, 1, 2  100  13, 16, 19, 17  95 
6  6, 8, 11, 10, 7  100  11, 10, 7, 9, 6  100 
7  29, 30  20  29, 30  20 


Unselected  22, 27, 28  63  22, 27, 28  63 
From Tables
In this paper, we described the charge planning problem that focuses on the unknown charge number and the dissimilarity costs between orders and those charge center orders in iron and steel production process. The charge planning model, which is based on traveling salesman problem, is formulated and the improved cross entropy algorithm is proposed. Through the improvement of state transition probability matrix, the proposed ICE algorithm could take advantage of the problem’s characteristics. As a result, the speed and accuracy of the algorithm are improved, which are proved by an actual production example. Future work will focus on improving the stability and accuracy of ICE and extending the model and algorithm to the actual charge planning systems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge financial support from the National High Technology Research and Development Program (863 Program) of China under Grant 2007AA04Z157, the Shandong Provincial Natural Science Foundation under Grant ZR2010FZ001, and the Graduate Independent Innovation Foundation of Shandong University (GIIFSDU) under Grant 2082012yzc12136.