1 A Classification Problem in the Credit Card Industry A Comparison of Three Solutions
A Classi?cation Problem in the Credit Card Industry: A Comparison of Three Solutions
Christopher H. Jolly
GE Corporate Research and Development, Schenectady, NY 12301, USA
Mukkai S. Krishnamoorthy
Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Abstract Classifying accounts is a critical part of ef?ciently servicing credit cards. A problem we call the Rule Coverage Problem (RCP) arrises when using rules for specifying a classi?cation strategy, and we develop and compare three approaches for solving this problem.
1 Introduction and RCP
GE Capital Services (GECS) is the world’s largest supplier and manager of private label credit cards, serving 60 million accounts for 300 retailers. With such a large number of accounts it is critical to perform all servicing functions effectively and ef?ciently since small improvements on a peraccount basis have a signi?cant impact on pro?ts. Two major servicing functions are collections and marketing. Collecting from delinquent accounts consists of creating and implementing a collection strategy that allocates resources (e.g., collector phone calls and computergenerated letters) to delinquent accounts. For example, “good” customers should be separated from “bad” customers because we collect ef?ciently from them in different ways. An effective collection strategy maximizes payments less expenses over the long term while maximizing customer good will. A collection strategy is created either by intuition and experience or more recently by optimization programs such as PAYMENT [Makuch90]. Marketing active accounts consists of creating and implementing a marketing strategy that augment billing statements with promotional messages and inserts. For example, credit card insurance only needs to be marketed to those card holders who do not already have it, and a new store opening only needs to be marketed to those living (or working or shopping) within a reasonable driving distance. An effective marketing strategy increases credit card usage, lowers advertising costs, and increases (credit card and retailer) brand loyalty.
Large, Dynamic Account Base
Classi?ed Account Base
Classi?cation Needs Rules
Intelligent System
Figure 1: Classi?cation Problem
Thus, GECS needs to be able to classify accounts into homogeneous groups as shown in Figure 1.
1
Objects Rules Classes
Figure 2: Using Rules For Classi?cation An ef?cient method for specifying a classi?cation strategy is with rules as shown in Figure 2. Each rule tests attributes of an object in the rule’s antecedent and concludes a class in its consequent (rules have the form “if antecedent then consequent”). A collection strategy typically consists of hundreds of rules, and a marketing strategy typically consists of tens of rules. if x < 50 then class AA if y < 10 then class BB Figure 3: Ambiguous for x = 40 and y = 4 Rules are ambiguous when more than one applies. For example, if x = 40 and y = 4 for the rules in Figure 3, the rules could assign either class AA or BB since both rules apply. To resolve the ambiguity, we assign priorities to the rules. The rules are written (from top to bottom) from highest priority to lowest priority. Therefore, in this example the rules would assign class AA.
1.1
Rule Coverage
Higher priority rules can cover (i.e., render useless) lower priority rules. In Figure 4, for example, whenever the antecedent of the second rule is true, the antecedent of the first rule is true. Since the first rule has higher priority, the second rule never will assign class BB to any object. if x < 500 then class AA if x < 400 then class BB Figure 4: Simple Example of Rule Coverage Under the assumption that a person is specifying classi?cation rules, then a rule coverage is a logical error similar to an unreachable statement in a programming language (see Figure 5). An intelligent rule editor should point out each rule covered and the rules covering it. if 1 = 1 then goto label AA x = 400 AA: continue Figure 5: Example of an Unreachable Program Statement Informally, the Rule Coverage Problem (RCP) is determining if any rules are covered by higherpriority rules. It can be shown that RCP is CoNPcomplete (see [Garey79] for an introduction to NPcomplete problems).
1.2
Solving RCP
We look for potential algorithms for solving RCP by reducing RCP to a better known problems and examining the algorithms developed for them. The reduction, however, should not be too complicated; otherwise, the algorithms that work well for the better known problem may not work well for RCP after the reduction. In this work we reduce
2
instances of RCP to instances of CoSAT, which can be solved by many methods [Hooker88]. We solved them using two methods: integer programming and resolution. Since RCP is CoNPcomplete and some practical classi?cation strategies have over 1000 rules, for our third approach we de?ne a class of subproblems of RCP and develop ef?cient algorithms for some of them. All three approaches were tested on sample cases, and the results are shown in Section 4.
2 Reducing RCP to a CoSAT
It can be shown that RCP reduces to the unsatis?ability problem, CoSAT, as shown in Figure 6. We reduce RCP to the Single Rule Coverage Problem (SRCP), SRCP to what we de?ne as a theorem proving problem over multiplevalued variables (MVTP), MVTP to what we de?ne as MultiSAT, and MultiSAT to SAT. Only the problem de?nitions are presented here; the reductions and proofs of correctness have been omitted to save space. RCP SRCP MVTP CoMultiSAT CoSAT Figure 6: Reductions to a Known Problem
2.1
The Rule Coverage Problem (RCP)
In this section we give the formal de?nition of the Rule Coverage Problem (RCP). We start with some de?nitions. Throughout, we assume n rules and m variables. Let V = { v 1, v 2, …, v m } be a set of variables. Let T = { t v , t v , …, t v } be a set of test sets. Each t v is a set of simple 1 2 m i boolean tests on variable v i . Simple boolean tests are not formally defined here, but some examples are x < 500 , x = 7 , and 3 < x < 20 . if x < 75 and y < 5 and z = 0 if x < 50 and z = 1 if x > 35 if y < 2 if y > 6 and z = 1 then class AA then class BB then class CC then class DD then class EE
Figure 7: Example Classi?cation Rules For example, for the rules in Figure 7, the variables V = { x, y, z } , and the set of test sets T = { t x, t y, t z } , where test set t x = { x < 50, x < 75, x > 35 } , test set t y = { y < 2, y < 5, y > 6 } , and test set t z = { z = 0, z = 1 } . Let R = { r 1, r 2, …, r n } be a sequence (order is important) of rules. A rule typically has two parts: an antecedent, the
3
condition under which a rule is true, and a consequent, the actions of a rule. Since only the antecedent is relevant here, only the antecedent is shown. A rule is defined here as a conjunction of simple boolean tests. In addition, we often write rules in a more compact form as is done with boolean expressions. We put each test inside parentheses and have implied and’s between tests. Figure 8 shows the rules in Figure 7 in shorthand notation. R1: R2: R3: R4: R5:
( x < 75 ) ( y < 5 ) ( z = 0 ) ( x < 50 ) ( z = 1 ) ( x > 35 ) ( y < 2) ( y > 6) ( z = 1)
Figure 8: Example Rules The Rule Coverage Problem RCP(V,T,R) is given variables V, tests T, and rules R, is there a rule r i that is not satisfiable whenever rules r j for j < i are false (or, equivalently, is there a rule r i such that whenever r i is true, rule r j , for some j < i , is true.) For example, consider the rules shown in Figure 8. Variable z takes on the values 0 and 1 only. In this case rule R4 is covered. When R4 is true, either x ≤ 35 or x > 35 . If x > 35 , then R3 is true. If x ≤ 35 , then either z = 1 or z = 0 . If z = 1 , then R2 is true. If z = 0 , then R1 is true ( y < 5 since R4 is true). Thus, rule R4 is covered.
2.2
The Single Rule Coverage Problem (SRCP)
A natural subproblem of RCP is the Single Rule Coverage Problem (SRCP). The Single Rule Coverage Problem SRCP(V,T,R) is given variable V, tests T, and rules R, is the last rule, r n , not satis?able whenever rules r j , for j < R , are false (or, equivalently, whenever r n is true, is there a rule r j , for some j < R , that is true). In ?rst order logic SRCP(V,T,R) is
? ? ? ?A ? ? ? ? j < n, r j ( A ) ? ? r n ( A ) ? ? A ( r n ( A ) ? ? j < n, r j ( A ) ) ,
or, equivalently,
where A represents the variable assignments, and r i ( A ) is the boolean value of the rule under assignment A.
2.3
Multiplevalued variables
Whereas boolean variables take on the values true and false (or 1 and 0), we write multiplevalued variables to indicate variables that take on integer values from 0 to some ?nite number. We also refer to such variables as multivalued variables or multivariables for short. We also de?ne some notation to simplify the use of multivalued variables. For boolean literals, a variable with a bar over it (such as x ) is true if the variable has the value 0, and a variable without a bar over it (such as x ) is true if the variable has the value 1. For multivalued literals, we list as subscripts the values that make the literal true. For example, if x is a multivalued variable, the literal x o is true when x has the value 0; x 1 is true when x has the value 1; and x 0, 1 is true when x has the value 0 or 1. When not ambiguous, we omit the subscripted commas and write x 0, 1 as x 01 . We often write multiliteral to mean a literal of a multivariable. If a multiliteral has all possible subscripts, then we say it reduces to true, since the literal is always satis?ed.
4
2.4
MultiVariable Theorem Proving Problem (MVTP)
A theorem proving problem using propositional logic is determining if P ? Q is a tautology, where P and Q are expressions over boolean variables. We define the MultiVariable Theorem Proving Problem (MVTP) as a theorem proving problem over multivalued variables. Formally, MVTP(V,L,P,Q) is given a set of multivariables V = { v 1, v 2, …, v m } and lengths L = { l v , l v , …, l v } , where v i can take on integer values between 0 and l v – 1 , and 1 2 m i two expressions, P and Q, over V, is P ? Q a tautology. For example, consider V = { w, x, y, z } , P = x234 ( x 04 + y 0 ) w 0 , and Q = x 34 + y 01 + z 0 , where l w = 2 , l x = 5 , l y = 3 , and l z = 2 . P ? Q is
x 234 ( x 04 + y 0 ) w 0 ? x 34 + y 01 + z 0 ,
which is a tautology that can be proved using algebra or a truth table on all possible values for the variables.
2.5
MultiSAT
MultiSAT, de?ned here, is similar to SAT except that the variables are multivalued variables, not boolean variables. Formally, MultiSAT(V,L,C) is given a set of multivariables V = { v 1, v 2, …, v m } and lengths L = { l v , l v , …, l v } , 1 2 m where v i can take on integer values between 0 and l v – 1 , and given a set of clauses C over the multiliterals from the i variables in V, is the conjunction of the clauses in C satisfiable. For example, consider V = { w, x, y, z } and C = x 234, x 04 + y 0, w 0, x 012, y 2, z 1 , where l w = 2 , l x = 5 , l y = 3 , and l z = 2 . The clauses in C are not all satisfiable, which can be shown by algebra or a truth table.
3 RCP and incomplete subproblems
Since RCP is CoNPcomplete, we may want to de?ne and solve a tractable subproblem. One natural class of subproblems is Nrule coverage. An algorithm performs Nrule coverage if it detects all rules that are covered by N or fewer rules, and it may or may not detect rules that are covered by more than N rules. An example of 1rule coverage is shown in Figure 4, and an example of 3rule coverage is shown in Figure 8.
3.1
Strategy for detecting Nrule coverage Rule 2 Rule 1 Rule 4
Rule 3
Figure 9: Graphical Representation of ThreeRule Coverage Rule coverage can be viewed graphically as shown in Figure 9, where rules 1, 2, and 3 cover rule 4. One way of deciding RCP is to determine if each rule is occupying space not covered by higher priority rules. If for the moment we restrict the variables in RCP to having two values, then we can view RCP as a boolean sum. Consider, for example, the rules in Figure 10. We can represent ( x < 50 ) by x , ( x ≥ 50 ) by x, ( y < 2 ) by y , and so on
5
(z only takes on the values 0 and 1), and we obtain the boolean expressions shown in Figure 11. We can record the R1: R2: R3: R4:
( x < 50 ) ( y < 2 ) ( z = 0 ) ( x < 50 ) ( z = 1 ) ( x ≥ 50 ) ( y < 2)
Figure 10: Rules for Boolean Sum R1: xyz R2: xz R3: x R4: y Figure 11: Expressions for Mathematical Sum space occupied by keeping track of the sum. Beginning with the highest priority rule, the space occupied by R1 is xyz . We sequentially add the next highest priority rule and check if the total space occupied by the sum increases. Adding R2, the sum, xyz + xz , simplifies to xy + xz , which occupies more space than xyz because the former includes xyz . Adding R3, the sum, xy + xz + x , simplifies to y + z + x , which occupies more space than the preceding sum because it includes xyz . Adding R4, the sum, y + z + x + x , simplifies to y + z + x . Since this sum is identical to the previous sum, no unoccupied space was added to the sum, and thus rule R4 is covered. One ?aw in this procedure is that minimal boolean sums are not unique, and thus determining if adding a term to a sum increases the total occupied space is a dif?cult problem (as dif?cult as SRCP). To overcome this ?aw we can represent the sum in a canonical form such as all prime implicants, which can be computed by the consensus procedure of Quine [Schneeweiss89]. Another ?aw of this summing procedure is that we restricted it to boolean variables. We overcome this ?aw by extending Quine’s consensus procedure for use with multivariables.
3.2
Multivariable consensus procedure
The consensus procedure of Quine can be extended for use with multivariables to form a multivariable consensus procedure. We also add a continuation rule, which like the simpli?cation rule speeds up the procedure when implemented on a computer. The multivariable consensus procedure is useful in solving RCP because we can compute all prime implicants in a running “sum”, and if a rule is absorbed by any multivariable prime implicant, it is covered. Before presenting the multivariable consensus rules, we introduce some notation. The superscripted minus on a variable or a term T ( T ) indicates that the range of each multivariable in T is a (nonstrict) nonempty subset of the range of the corresponding variable in T. For example, if T = x 12 y 04 , then T might be x 12 y 04 , x 1 y 04 , x 2 y 0 , etc. Simi+ + larly the superscripted plus on a variable or a term T ( T ) indicates that the range of each multivariable in T is a (nonstrict) superset of the range of the corresponding variable in T.
3.2.1 The multivariable absorption rule
The multivariable absorption rule is
T + T U = T,
where T and U are multivariable terms. (The absorption rule is T + TU = T .) For example, w 024 x 12 + w 04 x 12 y 23 z 01 becomes w 024 x 12 .

3.2.2 The multivariable continuation rule
The multivariable continuation rule is
6
xr T + xs T = xr ∪ s T ,
where x is a multivariable and T is a multivariable term. For example, x 024 y 12 + x 14 y 12 becomes x 0124 y 12 . Multivariable x r ∪ s is replaced by true if r ∪ s includes all ranges. The multivariable continuation rule is not applied if r ∪ s = r or r ∪ s = s . Note that the multivariable continuation rule is derived from the multivariable simpli?cation rule when T = T and U is empty, followed by the multivariable absorption rule.
3.2.3 The multivariable simpli?cation rule
The multivariable simpli?cation rule is
xr T + xs T U = xr T + xr ∪ s T U ,
where x is a multivariable and T and U are multivariable terms. (The simpli?cation rule is x + x T = x + T .) For ? ? ? example, x 024 y 12 + x 14 y 2 z 01 becomes x 024 y 12 + x 0124 y 2 z 01 . Multivariable x r ∪ s is replaced by true if r ∪ s includes all ranges. The multivariable simpli?cation rule is not applied if r ∪ s = s . (Note that Quine’s simpli?cation rule could be written more generally as x T + x TU = x T + TU , which is derived from one application of the consensus rule, fol? ? ? lowed by one application of the absorption rule.)


3.2.4 The multivariable consensus rule
The multivariable consensus rule is to add a multivariable consensus of two multivariable terms. The multivariable consensus of x r T and x s U is
Q( x r T , x s U ;x) = x r ∪ s TU .
(The consensus of xT and xU is Q( xT , xU ;x) = TU .) For example, x 024 y 12 w 1 + x 14 y 2 z 01 becomes x 024 y 12 w 1 + x 14 y 2 z 01 + x 0124 y 2 z 01 w 1 . Multivariable x r ∪ s is replaced by true if r ∪ s includes all ranges. The consensus rule is not applied if r ∪ s = r , r ∪ s = s , or TU = φ .
3.3
Rule coverage

The multivariable consensus rules can be used to create 1rule, 2rule, and 3rule coverage algorithms using the following additional rule, which we call the partial absorption rule.
xr T + xs T U = xr T + xs – r T U ,
where x s – r T U is omitted if s ? r . Note that the partial absorption rule is a more general form of the multivariable absorption rule. When s ? r , x r T absorbs x s T U . Only the part of x s T U that is absorbed is removed. Theorem 1: When a rule, R, is covered under 2rule coverage, either (1) applying partial absorption on R, once for each of the two rules, absorbs R, or (2) R is only absorbed by a result of applying a multivariable rule on the two covering rules where a variable is reduced to true.

We illustrate the two cases with simple examples. Assume terms T 1 and T 2 cover T a . If T 1 = x 024 y 12 w 1 and T 2 = x 14 y 2 z 01 , then T a = x 01 y 2 z 0 w 1 is absorbed after applying the partial absorption rule twice, ?rst with T 1 to reduce T a to x 1 y 2 z 0 w 1 , and then with T 2 , which absorbs it. If T 1 = x 024 y 12 w 1 and T 2 = x 134 y 2 z 01 , where x 01234 = true , then T a = y 2 z 0 w 1 is absorbed by the consensus of T 1 and T 2 , which is y 2 z 01 . Theorem 2: When a rule, R, is covered under 3rule coverage, either (1) applying partial absorption on R twice, once for each of the three rules and then again on the three rules, absorbs R, or (2) R is only absorbed by a result of applying the multivariable rules on the three covering rules where a variable is reduced to true.
We illustrate with an example. Assume terms T 1 , T 2 , and T 3 cover T a . If T 1 = x 0 y 1 , T 2 = x 1 y 1 , and T 3 = x 01 y 0 , then T a = x 01 y 01 is absorbed after applying the partial absorption rule in two passes. On the ?rst pass T a is not altered by T 1 or T 2 but is by T 3 to produce x 01 y 1 , which is partially absorbed on the second pass by T 1 and ?nally absorbed by
7
T2 .
Here is an algorithm for 1rule coverage for RCP. Let P be the set of incomplete prime implicants in a running sum. (In a complete algorithm, P would consist only of prime implicants, but in an incomplete algorithm the terms in P may not all be prime implicants, and we refer to these terms as incomplete prime implicants.) The first step is to convert the n rules (e.g., ( x < 5 ) ( y < 100 ) ) into n multivariable terms (e.g., x 01 y 012 ), and we label the terms T i , for 1 ≤ i ≤ n . We put the first term (corresponding to the first rule) in P. The following steps are repeated for the subsequent terms, where the next term is labeled T i . If T i is absorbed by a term in P, then the rule corresponding to T i is covered. Otherwise, add T i to P. Since at each step, P contains all previous terms, this algorithm performs 1rule coverage. The algorithm for using partial absorption for 2rule coverage is the same as the algorithm for 1rule coverage except for two differences. First, the partial absorption rule is used instead of the absorption rule. This ensures 2rule coverage for all coverages that do not reduce a variable to true. Second, only the results of applying a multivariable rule that reduce a variable to true are added to P (versus all results of applying a multivariable rule). This ensures 2rule coverage for all coverages that reduce a variable to true. The algorithm for using partial absorption for 3rule coverage is the same as the algorithm for 2rule coverage except for two differences. First, the partial absorption rule is applied for two passes on the terms in P (if the term is not partially absorbed on the ?rst pass, then the second pass is unnecessary). This ensures 3rule coverage for all coverages that do not reduce a variable to true. Second, the results of applying a multivariable rule that reduce a variable to true, in addition to being added to P, are used with the multivariable rules and the other terms in P (that correspond to the original rules) and those results that reduce a variable to true are also added to P. This ensures 3rule coverage when one or two variables are reduced to true. The running time of the 3rule coverage algorithm is O ( n 4 m 3 ) for n terms of m variables. Note that the size of P is n plus the number of results having one or two variables reduced to true. The number of results having one variable reduced to true is O ( n 2 m ) , and if all of these combine with the n original terms to reduce two variables to true, then P grows to O ( n 3 m 2 ) . If, however, the number of terms added to P is O ( n ) , which is likely for our applications, then the running time is O ( n 2 m ) .
4 Experimental results
In this section we present the results of testing the three methods for solving RCP. We developed two groups of test cases: ? actual classi?cation rules used at GE Capital, and ? random MultiSAT instances (satis?ability problems where each variable can take on more than two values). In the following graphs “CPLEX” represents the approach where instances of RCP are reduced to instances of CoSAT, which are solved using integer programming in the CPLEX package; “OTTER” represents the approach where instances of RCP are reduced to instances of CoSAT, which are solved using resolution in the OTTER package; and “RULE” represents the approach where instances of RCP are solved with the multivariable consensus procedure. The execution times in the graphs include only the time when searching for coverage (i.e., no overhead processing time from reading/interpreting the problem, etc.). We have fewer examples from the OTTER approach because its overhead processing uses days to weeks of CPU time.
8
4.1
Actual rules used at GE Capital
This group of test cases consists of the actual classi?cation rule sets used at GE Capital. However, we only have 10 test cases.
Figure 12: Rule Coverages for CPLEX, OTTER, and Multivariable Consensus Rule
Figure 12 shows the rule coverages detected by the three approaches, and Figure 13 shows the execution times. The CPLEX and OTTER solutions, being complete, found all of the rule coverages, but they are much slower than the multivariable consensus procedure. The multivariable consensus procedure missed only 2 of the 468 covered rules, and it is much faster.
Figure 13: Execution Times for CPLEX, OTTER, and Multivariable Consensus Rule
4.2
Random MultiSAT problems
This group of test cases were created to be similar to actual classi?cation rules in order to provide additional insight on the execution times of the three methods. However, no randomly generated instance of MultiSAT had any covered rules. This indicates that the instances of RCP that occur in practice may not be the harder instances, if the hard instances have a probability of being satis?ed equal to 0.5, as in the case of many hard instances of the satis?ability problem. Figure 14 shows the execution times for the CPLEX and multivariable consensus procedure solutions. The multivariable consensus procedure is much faster for these test cases. Due to the extremely long running times of the OTTER solution, the results for this approach are not available yet.
9
Figure 14: Execution Times for Instances of Random MultiSAT
5 Bibliography
[Makuch90] W. Makuch. Optimizing the Collection of Delinquent Consumer Credit. Ph.D. Thesis, Rensselaer Polytechnic Institute, May, 1990. W. Schneeweiss. Boolean Functions with Engineering Applications and Computer Programs. SpringerVerlag, New York, N.Y., 1989. J. Hooker. A Quantitative Approach to Logical Inference. Decision Support Systems, 4:4569, 1988. M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman and Company, New York, N.Y., 1979.
[Schneeweiss89]
[Hooker88]
[Garey79]
10
相关推荐
 Changes in the Credit Card Industry
 Remedying the Dismal State of the Credit Card System A Look at Solutions Past, Present, and
 ECONOMICS REPORT  Changes in the Credit Card Industry
 Coexistence of positive solutions of nonlinear threepoint boundary value and its conjugate problem
 Capacity credit of Wind power Generation problems and solutions_HE SIYUAN_G1101476E
 ECONOMICS REPORT  An Unsettled Time for the Credit Card Industry
 Comparison of threeclass classification performance metrics a case study in breast cancer
 Closed Form Approximation Solutions for the Restricted Circular Three Body Problem
 Linear stabilities of elliptic triangle solutions of the planar charged threebody problems
 Complete Solution Classification for the perspectiveThreePoint Problem
最新更新
 1 A Classification Problem in the Credit Card Industry A Comparison of Three Solutions
 计算机仿真技术在铝电解槽中的应用实践_论文
 【测控指导】20152016学年高一语文人教版必修3课时演练：2.7 李商隐诗两首 Word版含解析.doc
 第8章半导体物理基础第2讲
 湖北省孝感市孝南区2017_2018学年八年级语文上学期期中试题扫描版新人教版20180109267
 全域旅游背景下海口会展旅游发展对策建议
 社会学视野下的丁克家庭_孙懿俊
 海尔空调说明书
 《中国梦 我的梦》说话发言材料稿件
 I‘vegot教案Word972003文档(2)
 Gmail的IMAP协议要把我逼疯了！
 日本一中的重要节日及活动
 珠海市蒙鑫贸易有限公司企业信用报告天眼查
 鲁教版一年级上册语文课件2.5爷爷和小树2
 高三政治课更应重视情感态度和价值观教育
 大学化学实验心得体会
猜你喜欢
 三星s轻奢和s8拍照
 人教版小学数学二年级上册课件：第1课时 认识时间
 2019年InDesign基本操作中文简单易学
 药店处方笺怎么写
 入党思想汇报19
 阜阳市颖东区袁寨镇永生生猪养殖场(企业信用报告) 天眼查
 脓胸的症状怎么回事
 北京厚华广告传媒有限公司(企业信用报告) 天眼查
 彩色多普勒能量图检测子宫内膜血流评价IVFET结局研究进展
 临时道路施工合同范本
 php实现可扩展模块,php编译可扩展模块
 学生打架检讨书范文模板下载五篇
 六年级写景作文 ： 下雪了_200字
 控制箱出厂检验报告
 3第二章初识鲁班算量k
 *湖市广陈镇毅营逃?畔⒆裳??癫科笠敌畔⒈ǜ姝天眼查
 临清市博良轴承有限公司企业信息报告天眼查
 JasperJPEG2000的实现(1)
 大自然的语言教学设计1
 A Time Domain Simulation Approach for Micro Milling Processes
 20182019优秀幼师师德演讲稿范本word范文 (1页)
 参观电厂心得体会范文
 湖北省2016年下半年房地产估价师《理论与方法》：求土地价值的公式考试试题
 最新部编版小学语文三年级上册富饶的西沙群岛10课后反思公开优质课教学课件
 最新静脉留置针详细操作流程资料
 外研版20192020学年七年级上学期英语第一次月考试卷A卷
 高考英语写作技巧2.ppt
 *三年生物高考试题图表题分类汇编
 综述护理干预对甲状腺手术患者焦虑及疼痛不适的影响分析
 现代市场营销学课件第五章：市场营销调研与预测 尚可精品文档
 大红袍制作流程是什么
 赞美母亲文章
 014 服务酒精类饮品
 高中物理教师教育工作述职总结范文
 唯美个性名言句子
 四年级数学上册《商的变化规律的应用》教案分析
 建立以KPI为核心的业绩管理体系
 2010年社团文化展示月—社团招才纳新工作安排(最终)
 群文阅读：《反复结构的童话故事阅读》学习单
 多核处理器的关键技术及其发展趋势
 【新教材】粤教版科学小学一年级上册和下册【全册】教案教学设计(上下两册)
 季度销售个人总结