Research ArticleAn Incentive Dynamic Programming Method forthe Optimization of Scholarship Assignment

Di Huang1 Yu Gu 1 HansWang2 Zhiyuan Liu 1 and Jun Chen 1

1 Jiangsu Key Laboratory of Urban ITS Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic TechnologiesSchool of Transportation Southeast University China2Department of Logistics amp Maritime Studies The Hong Kong Polytechnic University Kowloon Hong Kong

Correspondence should be addressed to Zhiyuan Liu zhiyuanlseueducn

Received 8 June 2018 Accepted 31 July 2018 Published 12 August 2018

Academic Editor Xinchang Wang

Copyright copy 2018 DiHuang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Scholarship assignment is an operations management problem confronting university administrators which is traditionally solvedbased on administratorsrsquo personal experiences This paper proposes an incentive method inspired by dynamic programming toreplace the traditional decision-making process in the scholarship assignment The objective is to find the optimal scholarshipassignment scheme with the highest equity while accounting for both the practical constraints and the equity requirementMoreover with the proposed method the scholarship assignment avoids time- and energy-consuming application processesconducted by students A solution algorithm is used to find feasible assignment schemes by iteratively solving a series of knapsacksubproblems based on dynamic programming and adjusting the monetary value of a unit score The optimal assignment schemecan then be screened out by applying the Gini coefficient for quantifying the equity of each feasible scheme A numerical case isinvestigated to illustrate the applicability of the proposed method and solution algorithm The results indicate that the proposedmethod is an efficient tool to assign scholarships to students with consideration of the equity

1 Introduction

Scholarships act as an important incentive and plays a crucialrole in encouraging undergraduate and graduate studentsin universities to work hard excel and aim for highereducation in their academic careers It is also considered as animportant component from the teaching perspective Manyresearches have proved that scholarships have positive effectson studentsrsquo performances in college and encourage studentsto seek for further education [1ndash3] Different scholarships arefunded by different sponsors such as entrepreneurs insti-tutions individuals etc who are in charge of determiningthe number of awardees and amount of money awarded toeach recipient Departments of universities are responsiblefor assigning scholarships to candidates by ranking thembased on their performances in departments However theuniversity administrators have no right to further dividethe existing scholarships into new scholarships with smalleramounts of monetary awards The performances of students

are evaluated by the score they gained each year The finalscore of a student is the summation of the following twoparts (1) basic score the grade of the studentrsquos courseworkand (2) research score the equivalent score converted by thestudentrsquos academic achievements (eg winning competitionspublishing papers and patents)

In practice there are many factors to be considered inscholarship assignment First a scholarship from the sameprovider should not be assigned to the same student morethan once as sponsors require to cover as many deservingstudents as they can Second a scholarship cannot be furtherdivided as the amount of each scholarship is determinedby its sponsor Third all scholarships must be awarded tostudents Fourth scholarships should be assigned to studentsbased on their performances (scores gained from courseworkand research) stating that the student with a higher scoreshould receive an amount of scholarship no less than thatreceived by students with lower scores Fifth the amountof scholarship assigned to each student should be strictly

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 5206131 7 pageshttpsdoiorg10115520185206131

2 Discrete Dynamics in Nature and Society

in line with hisher score in order to guarantee equity ofscholarship assignment For instance when two students getsimilar scores but the difference between the amounts of thescholarships awarded to these students is too large to ignoresuch discrimination may lead to discontent as the studentsmay feel unfairly treated

Our personal experiences show that almost all schol-arships are not directly assigned to students but awardedto the best applicants who apply for the scholarships Stu-dents can separate their research scores into smaller partsand use different parts to apply for different scholarshipsThen the applicants for each scholarship will be ranked bythe summations of their basic scores and chosen researchscores The applicants with the highest rankings have higherchances of winning the scholarships As we all know theassignment mechanism is not only dependent on applicantsrsquoperformances but also strongly influenced by applicantsrsquostrategies One can receive a higher amount of scholarshipsthan students with higher scores by adopting a better strat-egy for dividing hisher research scores and applying forappropriate scholarships with fewer competitors Moreoversuch mechanism is both time- and energy-consuming forstudents as they tend to focus on collecting informationof their competitors and developing appropriate applicationstrategies at the expense of their studies and research thusexerts an overall negative impact on their academic perfor-mance In this regard it is imperative and of vital significancefor university administrators to develop a systematic tool toassign scholarships in an efficient and equitable way

11 Literature Review It is reported in a diversity ofresearches that in spite of providing financial assistancescholarships will help students in various aspects includingmotivating studying improving confidence and even chang-ing their future plans [4ndash6] The assignment of scholarshipsthus becomes an important issue that might highly affectliving and learning of graduate students To the best of ourknowledge few researches have investigated the scholarshipassignment problem except the research conducted in [7]Amoros et al admitted that the best student should receivethemost prestigious scholarship and proposed an assignmentmechanism inNash Equilibrium to achieve a socially optimaloutcome However they fail to consider the equity norm intheir assignment mechanism

The equity norm has gained increasing attention inhigher education [8ndash11] and is a pressing concern in thescholarship assignment as much as in other policy making[12ndash14] and resource or reward allocation problems [15ndash19] Equity unlike equality which implies that all membersshould be assigned to equal amount of resource requiresthat the allocated resource should be proportionate to themagnitude of eachmemberrsquos contribution [20 21] In a groupthe members who perform better often require equity whenassigning resources It is found that people who outperformtheir peers intend to ask for equity in resource allocation [22]Such mechanism implies that in the scholarship assignmentstudents with higher scores might possess greater demandsfor equity Therefore it is important to consider the equitynorm in scholarship assignment

As few efforts have focused on the equity concernof scholarship assignment problem this research aims todevelop a scholarship assignment model to optimize thequality of scholarship assignment scheme by consideringthe equity norm The model can guarantee a basic require-ment for equity ie students receive a higher amount ofscholarships only when they have higher scores Furtherthis proposed model can quantify the equity of all feasibleassignment schemes satisfying the above requirement andhelp select the scheme with the highest equity

12 Objectives and Contributions The objective of thisresearch is to develop an efficient tool to generate theassignment scheme of scholarships with the highest equityThe contribution of the paper is threefold First we developa new approach that allows the university administrators toassign scholarships to students according to the ranking ofcoursework and research performances This is because ourmethod prioritizes students with higher scores in scholarshipassignment As a result the equity of scholarship assignmentscheme can be guaranteed by meeting the requirement thata student with a higher score must receive an amount ofscholarships no less than students with lower scores Suchequity requirement however is hard to satisfy when studentstake the leading role in scholarship assignment process as ifscholarships should be applied by students On the contrarythe proposed method is led by university administratorswhich contributes to helping motivate and encourage thestudents to work hard in their study and research as betterperformances are strictly associated with higher monetaryrewards

Second as this assignment is done by university admin-istrators instead of students themselves who are naturallybusywith studying researching and other commitments ourmethod can save the valuable time for students

Third the method can assess the equity of differentfeasible assignment schemes Various schemes can meet therequirement that better students receive more scholarshipswhile the qualities of schemes differ from each other Onemay find that another student who performs little betterthan she can receive much more scholarships which isanother form of inequity This concern can be addressed byintroducing the Gini coefficient to measure the inequity ofeach feasible scholarship assignment scheme

The remainder of this paper is organized as follows Thedescription of considerations in the scholarship assignmentproblem is first introduced The method and solution algo-rithm to solve the proposed scholarship assignment problemare presented in detail followed by a small example toillustrate the application of the proposed method Then theresults and findings of the numerical example are reportedFinally the conclusion is drawn

2 The Scholarship Assignment Problem

This study investigates the assignment mechanism of schol-arships at a department In a department different kindsof scholarships are offered to graduate students who havedone excellent jobs in academic research Different types

Discrete Dynamics in Nature and Society 3

of scholarships may differ from each other in two aspects(1) the number of awardees and (2) the amount of moneyan awardee can receive Students are free to apply fordifferent scholarships The results of scholarship appraisalare dependent on the studentsrsquo scores they have obtainedin the previous year As mentioned above the total scoreof a student is composed of the basic score obtained fromcoursework and additional research scores based on hisacademic achievements such as published journal papersIn order to simplify the evaluation of studentsrsquo works thebasic score is measured in the hundred-mark system whiledifferent kinds of academic achievements are measured inthe same weight Additionally a standardization process isapplied to convert a specific work into standard scores Forinstance the score of a published paper can be calculated interms of two components journalrsquos impact factor (IF) rankingand authorrsquos order A student who has published a paper inthe journal which occupies in Q1 [23] and is also the firstauthor of that journal will achieve additional 10 points Thescore decreases linearly with the slip of IF or authorrsquos order

In this study we assume that the department is in chargeof the assignment of scholarships and the scholarship isallocated according to the ranking of studentsrsquo total scoresTo avoid the unfairness in assigning scholarships we includethe constraint that the student with a higher score should beoffered at least no less amount of scholarship than studentswith lower scores Meanwhile other practical constraints areconsidered in the study For instance a single scholarshipcannot be shared by different students a student can onlybe awarded by a certain type of scholarship no more thanonce a student can either fail to win any scholarship or winmultiple scholarships all the scholarships should be assignedto students The optimal assignment should be the schemethat best guarantee the amount of scholarship received bya student is proportionate to the magnitude of hisher totalscore

The notations used in the method are listed below

Sets

119869 Set of scholarships119868 Set of students119869119894 Set of remaining scholarships that can be assignedto student 119894

Indices

119895 An index 119895 isin 119869 that refers to a particular scholarship119894 119896 Indices 119894 119896 isin 119868 that refer to particular students

Parameters and Variables

119898119895 The amount of scholarship 119895 ($)119888119895 The number of scholarship 119895119890119895 Equivalent score of scholarship 119895119904119894 The score of student 119894V Monetary value of a unit score

119866 Gini coefficient119899 Number of students119882119894 Welfare of student 119894119882 Average welfare of individual students119882min Minimum welfare of students120581 A constant used to adjust individual welfare119882119904119894 Adjusted welfare of student 119894

119882119904 Adjusted average welfare of individual students120575119894119895 A binary decision variable which equals 1 ifscholarship 119895 is assigned to student 119894 and 0 otherwise

3 The Scholarship Assignment Mechanism

The scholarship assignment problem discussed above canbe deemed as a combination of two subproblems The firstsubproblem is to search for all feasible assignment schemeswhich meet the practical constraints and the basic equityrequirement simultaneously The second subproblem is toquantify the equity of each feasible scheme and select theoptimal scheme with the highest equity

To address the first subproblem a method composed ofknapsack subproblems is developed to assign scholarships tostudents in sequence In the knapsack subproblem the totalscore gained by each student is deemed as the knapsack andthe equivalent score of each scholarship is regarded as theitem to be put in the knapsack Given a certain value of aunit score the amounts of all scholarships can be convertedto equivalent scores Assigning scholarships to each studentby solving knapsack problems can guarantee that the amountof scholarships (equivalent score) awarding to a studentis the maximum one that can be allocated to himheramong remaining scholarshipsHence the students who havepriorities in such scholarship assignment process can receivemore scholarships Through ranking students according totheir scores and assigning scholarships in sequence theproposed method thus certainly satisfies the basic equityrequirement that students with higher scores should receivehigher amounts of scholarships

To address the second subproblem the Gini coefficient isintroduced to measure the inequity of each feasible solutionThe Gini coefficient can quantify the equity of distributionof welfare (difference between the actual and ideal amountof scholarship gained by each individual student) The Ginicoefficient equals 0 when students receive the same welfareie there exists no difference between the actual and idealamounts of scholarship gained by each student A higherGini coefficient is associated with a higher inequity ofthe investigated assignment scheme The proposed methodmodifies the original calculation of Gini coefficient used inthe field of economics making it applicable to the scholarshipassignment problem The feasible assignment scheme withthe lowest value of Gini coefficient is selected as the optimalassignment scheme

31 Searching for Feasible Solutions To satisfy the require-ment that students with higher scores can obtain an amount

4 Discrete Dynamics in Nature and Society

of scholarships no less than students with lower scores thescholarship assignment problem is converted to a series ofone-dimensional 0-1 knapsack problems [24 25]

Firstly the initial monetary value V of a unit score can becalculated by

V =sum119895isin119869119898119895 sdot 119888119895sum119894isin119868 119904119894

(1)

According to the monetary value of a unit score theamount of scholarships 119898119895 can then be converted to equiv-alent scores 119890119895 The process can be expressed as

119890119895 =119898119895V forall119895 isin 119869 (2)

The sequence of assigning scholarships to students isin accordance with the ranking of studentsrsquo scores In eachiteration the scholarships are allocated to the studentwith thehighest score among unsigned students and the amount ofscholarships to be allocated to the student is derived throughsolving a knapsack subproblem Let the score of a student119894 119904119894 be the capacity of the knapsack and the equivalentscore of scholarship 119895 119890119895 be the items to be put into theknapsack Thus the scholarship assignment scheme of thechosen student 119894 can be derived from

max sum119895isin119869119894

119890119895 sdot 120575119894119895 (3)

st sum119895isin119869119894

119890119895 sdot 120575119894119895 le 119904119894 (4)

By iteratively solving knapsack subproblems for eachchosen student and assigning scholarships in accordance tothe solutions the students with higher scores who possesspriority in the scholarship assignment can certainly obtaingreater amounts of scholarships Hence given a certainmonetary value of the score V we can derive a correspondingfeasible solution set (scholarship assignment scheme) usingthe abovemethod By iteratively adjusting themonetary valueof a unit score we can enumerate all the feasible solution sets

However the qualities of feasible solution sets derivedby the above method cannot be judged As this paper aimsto assign scholarships to students considering the equitynorm the finally selected assignment schememust be the onewith the highest equity The following section introduces themethod to quantify equity of each feasible solution

32 Measuring Equity of Scholarship Assignment SchemeNote that perfect equity means that every student can receivea proportion of scholarship same as the proportion of hisherscore in the total score To quantify the equity of scholarshipassignment schemeswhich aim to assigning different scholar-ships to a group of students with different scores a suggestionis associated with modifying an existing metric of incomeinequity ie Gini coefficient In the transportation fieldthe Gini coefficient is widely utilized to analyze inequity inaccessibility [26ndash28] and effect of transport policies [29 30]Herein the Gini coefficient is adopted to measure the social

welfare (for instance the accessibility to certain opportunitiesor the welfare of toll pricing scheme) achieved by transportusers In scholarship assignment the Gini coefficient 119866 canexpress an inequalitymetric that evaluates the departure fromperfect equity which can be expressed as

119866 = 121198992119882sum119894 =119896isin119868

1003816100381610038161003816119882119894 minus1198821198961003816100381610038161003816 (5)

where119882119894 denotes the benefit that student 119894 receives from theactual assignment scheme119882 denotes the average individualwelfare and 119899 is the number of students Considering thepractical constraints the ideal assignment result with perfectequity is hard to reach An individual student might gaineither positive benefit (receive more scholarship than shedeserves) or negative benefit (receive less scholarship thanshe deserves) in certain scholarship assignment schemeTo measure the benefit each student receives from theassignment scheme we define the welfare of student 119894 119882119894as the difference between the studentrsquos actual scholarship andideal scholarship which can be derived by

119882119894 = 119904119886119894 minus 119904119901119894 (6)

where 119904119886119894 and 119904119901119894 denote the actual and ideal results ofscholarship assignment with regards to student 119894 respectivelyThe average individual welfare119882 can be expressed as

119882 =sum119894isin119868119882119894119899 (7)

It is evident that the average individual welfare119882 equals0 and the individual welfare can be negativeThese propertiesprohibit the usage of original Gini coefficient in that thevalue of average and individual income used in the originalGini coefficient evaluation can only be positive Therefore arescaling process is necessary to implement (5)The re-scaledindividual welfare119882119904119894 can be derived by

119882119904119894 = 119882119894 +1003816100381610038161003816119882min1003816100381610038161003816 + 120581 (8)

where119882min denotes the minimum individual welfare and 120581is a given constant

By using 119882119904119894 to replace 119882119894 in (5) and (7) the Ginicoefficient 119866 of each solution can be obtained

119866 = 121198992119882119904sum119894 =119896isin119868

1003816100381610038161003816119882119904119894 minus119882

1199041198961003816100381610038161003816 (9)

4 Solution Algorithm

As discussed in Section 3 the task of searching for feasibleassignment scheme can be converted to a series of one-dimensional 01 knapsack problem which can be exactlysolved by dynamic programming algorithms [31ndash36] In thissection a solution algorithm for the knapsack subproblembased on dynamic programming is used to find the optimalscholarship assignment for each student in sequence On thebasis of it a solution algorithm for finding feasible assignmentschemes and selecting the optimal scheme is proposed

Discrete Dynamics in Nature and Society 5

41 Algorithm 1 Algorithm for One-Dimensional 0-1 KnapsackSubproblem Assuming there are totally 119901 types of schol-arships let stage ℎ denote the first ℎ types of scholarshipsthat are considered in scholarship assignment 119904 denote theequivalent score that is currently assigned to the selectedstudent and 119891ℎ(119904) denote the maximum equivalent score theselected student can obtain at stage ℎ

Step 1 (initialization) Set 1198910(119904) = 0 forall119904 le 119904119894 ℎ = 1 119904 = 0119895 = 1

Step 2 Update 119891ℎ(119904)

Step 21 119891ℎ(119904) = max119891ℎminus1(119904) 119891ℎminus1(119904 minus119898119894) +119898119895 where119898119894 isthe equivalent score of the scholarship previously assigned tothe selected student

Step 22 119895 = 119895 + 1 If 119895 gt ℎ go to Step 23 otherwise go toStep 21

Step 23 119904 = 119904 + 1 If 119904 gt 119904119894 go to Step 3 otherwise go to Step21

Step 3 ℎ = ℎ + 1 If ℎ le 119901 go to Step 2 otherwise return119891ℎminus1(119904119894) and stop the algorithm

42 Algorithm 2 Derive Feasible Schemes andSelect Optimal Scheme

Step 1 Initialize the monetary value of a unit score V using(1) set the value of step size 119889 and upper bound of monetaryscore value Vmax

Step 2 Obtain the equivalent score of each scholarship 119898119895using (2)

Step 3 Derive feasible scholarship assignment

Step 31 Rank students according to their scores and putstudents into set R1 2 119894

Step 32 Assign scholarships to the first student i in set R usingAlgorithm 1

Step 33 Delete student 119894 from set R If set R is not empty119894 = 119894 + 1 go to Step 32 otherwise go to Step 4

Step 4 Store assignment result of Step 3 Increase monetaryscore value V = V + 119889 if V lt Vmax go to Step 2 otherwise goto Step 5

Step 5 Using (9) to calculate the Gini coefficient of eachassignment scheme stored in Step 4 return the optimalassignment scheme with lowest Gini coefficient value

5 An Illustrative Example

To illustrate the proposed method we propose a small casewith 3 types of scholarships and 5 candidates as an exampleThe unit of scholarship is US dollar ($) in Tables 1 3 and 4

Table 1 Details of scholarships

ID Number of awardees Amount of scholarship1 2 80002 3 50003 1 10000

Table 2 Data of candidates

ID Score1 252 203 304 355 15

Table 3 Ideal assignment results

ID Amount of scholarship1 82002 65603 98404 114805 4920

Tables 1 and 2 show the details of scholarships and scholarshipcandidates

It can be seen that the number of awardees of eachscholarship is separately 2 3 and 1 the monetary awardsprovided by each scholarship are respectively $8000 5000and 10000 The total scores of each student are 25 20 3035 and 15 Following the equity norm that the ideal amountof scholarship received by each student is proportionateto hisher score the ideal assignment result neglecting thepractical constraints is shown in Table 3

Using the method discussed in Section 31 four feasibleschemes can be obtained by iteratively solving the knapsacksubproblem for each student and then adjusting themonetaryvalue of the score Note that the derived schemes only satisfythe basic equity requirement that a student with higher scoremust receive an amount of scholarship no less than thatreceived by students with comparatively lower scores Thedetails regarding the 4 schemes are shown in Table 4

Through the method described in Section 32 we canthen compute the Gini coefficient of each scheme which isshown in Table 5

As per the Gini coefficients of each scheme it can befound that among 4 feasible schemes scheme 1 is the onewith the highest equity as the Gini coefficient of scheme1 is the lowest Therefore scheme 1 is chosen as the finalscholarship assignment scheme in this numerical example

From this example it can be concluded that the proposedmethod can enumerate several feasible scholarship assign-ment schemes and then choose the optimal one with theminimum value of Gini coefficient This demonstrates thepractical relevance of the proposed method

6 Discrete Dynamics in Nature and Society

Table 4 Derived feasible schemes of scholarship assignment

Scheme 1 Scheme 2

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 8000 1 1 25 8000 12 20 5000 2 2 20 5000 23 30 10000 3 3 30 13000 124 35 13000 12 4 35 15000 135 15 5000 2 5 15 0

Scheme 3 Scheme 4

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 5000 2 1 25 5000 22 20 5000 2 2 20 0 3 30 13000 12 3 30 13000 124 35 18000 13 4 35 23000 1235 15 0 5 15 0

Table 5 Gini coefficients of each scheme

Scheme Gini coefficient1 03342 03513 04754 0540

6 Conclusion

This paper develops a method to derive the optimal scholar-ship assignment schemewith the highest equity for universityadministrators The method is applicable as (1) it meets theequity requirement that students who perform better oughtto receive scholarships equal to or more than those receivedby less-achieving students (2) the scholarship assignmenteliminates the need of students to manually apply for specificscholarships which is a time- and energy-consuming pro-cess and (3) the equity of derived assignment schemes can bequantified through the Gini coefficient and the scheme withthe maximum equity (minimum Gini coefficient value) canbe identified and selected An illustrative example is adoptedto show the applicability of the proposed methodThe resultsindicate that the proposedmethod can obtain a set of feasibleassignment schemes efficiently and then derive the optimalone with the highest equity

Data Availability

The hypothetical data used to support the findings of thisstudy are included within the article No external data wereused to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the Top-Notch Academic Pro-grams Project of Jiangsu Higher Education Institutions(PPZY2015B148) and the Project of Educational Reform andPractice in Southeast University (2017-071)

References

[1] P E Peterson D Myers and W G Howell An Evaluation ofthe New York City School Choice Scholarships ProgramThe FirstYear Mathematica Policy Research Washington DC USA1998

[2] S L Desjardins ldquoThe impact of Washington State AchieversScholarship on student outcomesrdquo in Annual meeting of Associ-ation for the Study of Higher Education Vancouver BC Canada2009

[3] S L DesJardins and B P McCall ldquoThe impact of the GatesMillennium Scholars Program on college and post-collegerelated choices of high ability low-income minority studentsrdquoEconomics of Education Review vol 38 no 1 pp 124ndash138 2014

[4] M R Anderson-Rowland ldquoEvaluation of a ten year life plan-ning assignment for an academic scholarship success classrdquo inProceedings of the 41st Annual Frontiers in EducationConferenceCelebrating 41 Years of Monumental Innovations from Aroundthe World FIE 2011 pp 1ndash7 2012

[5] A Porter R Yang J Hwang J McMaken and J Rorison ldquoTheEffects of ScholarshipAmount onYield and Success forMasterrsquosStudents in Educationrdquo Journal of Research on EducationalEffectiveness vol 7 no 2 pp 166ndash182 2014

[6] H Forsyth and A Cairnduff ldquoA scholarship of social inclusionin higher education why we need it and what it should looklikerdquo Higher Education Research amp Development vol 34 no 1pp 219ndash222 2015

[7] P Amoros L C Corchon and B Moreno ldquoThe scholarshipassignment problemrdquo Games amp Economic Behavior vol 38 no1 pp 1ndash18 2002

[8] G Goastellec ldquoGlobalization and implementation of an equitynorm in higher education Admission processes and funding

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

Discrete Dynamics in Nature and Society 3

of scholarships may differ from each other in two aspects(1) the number of awardees and (2) the amount of moneyan awardee can receive Students are free to apply fordifferent scholarships The results of scholarship appraisalare dependent on the studentsrsquo scores they have obtainedin the previous year As mentioned above the total scoreof a student is composed of the basic score obtained fromcoursework and additional research scores based on hisacademic achievements such as published journal papersIn order to simplify the evaluation of studentsrsquo works thebasic score is measured in the hundred-mark system whiledifferent kinds of academic achievements are measured inthe same weight Additionally a standardization process isapplied to convert a specific work into standard scores Forinstance the score of a published paper can be calculated interms of two components journalrsquos impact factor (IF) rankingand authorrsquos order A student who has published a paper inthe journal which occupies in Q1 [23] and is also the firstauthor of that journal will achieve additional 10 points Thescore decreases linearly with the slip of IF or authorrsquos order

In this study we assume that the department is in chargeof the assignment of scholarships and the scholarship isallocated according to the ranking of studentsrsquo total scoresTo avoid the unfairness in assigning scholarships we includethe constraint that the student with a higher score should beoffered at least no less amount of scholarship than studentswith lower scores Meanwhile other practical constraints areconsidered in the study For instance a single scholarshipcannot be shared by different students a student can onlybe awarded by a certain type of scholarship no more thanonce a student can either fail to win any scholarship or winmultiple scholarships all the scholarships should be assignedto students The optimal assignment should be the schemethat best guarantee the amount of scholarship received bya student is proportionate to the magnitude of hisher totalscore

The notations used in the method are listed below

Sets

119869 Set of scholarships119868 Set of students119869119894 Set of remaining scholarships that can be assignedto student 119894

Indices

119895 An index 119895 isin 119869 that refers to a particular scholarship119894 119896 Indices 119894 119896 isin 119868 that refer to particular students

Parameters and Variables

119898119895 The amount of scholarship 119895 ($)119888119895 The number of scholarship 119895119890119895 Equivalent score of scholarship 119895119904119894 The score of student 119894V Monetary value of a unit score

119866 Gini coefficient119899 Number of students119882119894 Welfare of student 119894119882 Average welfare of individual students119882min Minimum welfare of students120581 A constant used to adjust individual welfare119882119904119894 Adjusted welfare of student 119894

119882119904 Adjusted average welfare of individual students120575119894119895 A binary decision variable which equals 1 ifscholarship 119895 is assigned to student 119894 and 0 otherwise

3 The Scholarship Assignment Mechanism

The scholarship assignment problem discussed above canbe deemed as a combination of two subproblems The firstsubproblem is to search for all feasible assignment schemeswhich meet the practical constraints and the basic equityrequirement simultaneously The second subproblem is toquantify the equity of each feasible scheme and select theoptimal scheme with the highest equity

To address the first subproblem a method composed ofknapsack subproblems is developed to assign scholarships tostudents in sequence In the knapsack subproblem the totalscore gained by each student is deemed as the knapsack andthe equivalent score of each scholarship is regarded as theitem to be put in the knapsack Given a certain value of aunit score the amounts of all scholarships can be convertedto equivalent scores Assigning scholarships to each studentby solving knapsack problems can guarantee that the amountof scholarships (equivalent score) awarding to a studentis the maximum one that can be allocated to himheramong remaining scholarshipsHence the students who havepriorities in such scholarship assignment process can receivemore scholarships Through ranking students according totheir scores and assigning scholarships in sequence theproposed method thus certainly satisfies the basic equityrequirement that students with higher scores should receivehigher amounts of scholarships

To address the second subproblem the Gini coefficient isintroduced to measure the inequity of each feasible solutionThe Gini coefficient can quantify the equity of distributionof welfare (difference between the actual and ideal amountof scholarship gained by each individual student) The Ginicoefficient equals 0 when students receive the same welfareie there exists no difference between the actual and idealamounts of scholarship gained by each student A higherGini coefficient is associated with a higher inequity ofthe investigated assignment scheme The proposed methodmodifies the original calculation of Gini coefficient used inthe field of economics making it applicable to the scholarshipassignment problem The feasible assignment scheme withthe lowest value of Gini coefficient is selected as the optimalassignment scheme

31 Searching for Feasible Solutions To satisfy the require-ment that students with higher scores can obtain an amount

4 Discrete Dynamics in Nature and Society

of scholarships no less than students with lower scores thescholarship assignment problem is converted to a series ofone-dimensional 0-1 knapsack problems [24 25]

Firstly the initial monetary value V of a unit score can becalculated by

V =sum119895isin119869119898119895 sdot 119888119895sum119894isin119868 119904119894

(1)

According to the monetary value of a unit score theamount of scholarships 119898119895 can then be converted to equiv-alent scores 119890119895 The process can be expressed as

119890119895 =119898119895V forall119895 isin 119869 (2)

The sequence of assigning scholarships to students isin accordance with the ranking of studentsrsquo scores In eachiteration the scholarships are allocated to the studentwith thehighest score among unsigned students and the amount ofscholarships to be allocated to the student is derived throughsolving a knapsack subproblem Let the score of a student119894 119904119894 be the capacity of the knapsack and the equivalentscore of scholarship 119895 119890119895 be the items to be put into theknapsack Thus the scholarship assignment scheme of thechosen student 119894 can be derived from

max sum119895isin119869119894

119890119895 sdot 120575119894119895 (3)

st sum119895isin119869119894

119890119895 sdot 120575119894119895 le 119904119894 (4)

By iteratively solving knapsack subproblems for eachchosen student and assigning scholarships in accordance tothe solutions the students with higher scores who possesspriority in the scholarship assignment can certainly obtaingreater amounts of scholarships Hence given a certainmonetary value of the score V we can derive a correspondingfeasible solution set (scholarship assignment scheme) usingthe abovemethod By iteratively adjusting themonetary valueof a unit score we can enumerate all the feasible solution sets

However the qualities of feasible solution sets derivedby the above method cannot be judged As this paper aimsto assign scholarships to students considering the equitynorm the finally selected assignment schememust be the onewith the highest equity The following section introduces themethod to quantify equity of each feasible solution

32 Measuring Equity of Scholarship Assignment SchemeNote that perfect equity means that every student can receivea proportion of scholarship same as the proportion of hisherscore in the total score To quantify the equity of scholarshipassignment schemeswhich aim to assigning different scholar-ships to a group of students with different scores a suggestionis associated with modifying an existing metric of incomeinequity ie Gini coefficient In the transportation fieldthe Gini coefficient is widely utilized to analyze inequity inaccessibility [26ndash28] and effect of transport policies [29 30]Herein the Gini coefficient is adopted to measure the social

welfare (for instance the accessibility to certain opportunitiesor the welfare of toll pricing scheme) achieved by transportusers In scholarship assignment the Gini coefficient 119866 canexpress an inequalitymetric that evaluates the departure fromperfect equity which can be expressed as

119866 = 121198992119882sum119894 =119896isin119868

1003816100381610038161003816119882119894 minus1198821198961003816100381610038161003816 (5)

where119882119894 denotes the benefit that student 119894 receives from theactual assignment scheme119882 denotes the average individualwelfare and 119899 is the number of students Considering thepractical constraints the ideal assignment result with perfectequity is hard to reach An individual student might gaineither positive benefit (receive more scholarship than shedeserves) or negative benefit (receive less scholarship thanshe deserves) in certain scholarship assignment schemeTo measure the benefit each student receives from theassignment scheme we define the welfare of student 119894 119882119894as the difference between the studentrsquos actual scholarship andideal scholarship which can be derived by

119882119894 = 119904119886119894 minus 119904119901119894 (6)

where 119904119886119894 and 119904119901119894 denote the actual and ideal results ofscholarship assignment with regards to student 119894 respectivelyThe average individual welfare119882 can be expressed as

119882 =sum119894isin119868119882119894119899 (7)

It is evident that the average individual welfare119882 equals0 and the individual welfare can be negativeThese propertiesprohibit the usage of original Gini coefficient in that thevalue of average and individual income used in the originalGini coefficient evaluation can only be positive Therefore arescaling process is necessary to implement (5)The re-scaledindividual welfare119882119904119894 can be derived by

119882119904119894 = 119882119894 +1003816100381610038161003816119882min1003816100381610038161003816 + 120581 (8)

where119882min denotes the minimum individual welfare and 120581is a given constant

By using 119882119904119894 to replace 119882119894 in (5) and (7) the Ginicoefficient 119866 of each solution can be obtained

119866 = 121198992119882119904sum119894 =119896isin119868

1003816100381610038161003816119882119904119894 minus119882

1199041198961003816100381610038161003816 (9)

4 Solution Algorithm

As discussed in Section 3 the task of searching for feasibleassignment scheme can be converted to a series of one-dimensional 01 knapsack problem which can be exactlysolved by dynamic programming algorithms [31ndash36] In thissection a solution algorithm for the knapsack subproblembased on dynamic programming is used to find the optimalscholarship assignment for each student in sequence On thebasis of it a solution algorithm for finding feasible assignmentschemes and selecting the optimal scheme is proposed

Discrete Dynamics in Nature and Society 5

41 Algorithm 1 Algorithm for One-Dimensional 0-1 KnapsackSubproblem Assuming there are totally 119901 types of schol-arships let stage ℎ denote the first ℎ types of scholarshipsthat are considered in scholarship assignment 119904 denote theequivalent score that is currently assigned to the selectedstudent and 119891ℎ(119904) denote the maximum equivalent score theselected student can obtain at stage ℎ

Step 1 (initialization) Set 1198910(119904) = 0 forall119904 le 119904119894 ℎ = 1 119904 = 0119895 = 1

Step 2 Update 119891ℎ(119904)

Step 21 119891ℎ(119904) = max119891ℎminus1(119904) 119891ℎminus1(119904 minus119898119894) +119898119895 where119898119894 isthe equivalent score of the scholarship previously assigned tothe selected student

Step 22 119895 = 119895 + 1 If 119895 gt ℎ go to Step 23 otherwise go toStep 21

Step 23 119904 = 119904 + 1 If 119904 gt 119904119894 go to Step 3 otherwise go to Step21

Step 3 ℎ = ℎ + 1 If ℎ le 119901 go to Step 2 otherwise return119891ℎminus1(119904119894) and stop the algorithm

42 Algorithm 2 Derive Feasible Schemes andSelect Optimal Scheme

Step 1 Initialize the monetary value of a unit score V using(1) set the value of step size 119889 and upper bound of monetaryscore value Vmax

Step 2 Obtain the equivalent score of each scholarship 119898119895using (2)

Step 3 Derive feasible scholarship assignment

Step 31 Rank students according to their scores and putstudents into set R1 2 119894

Step 32 Assign scholarships to the first student i in set R usingAlgorithm 1

Step 33 Delete student 119894 from set R If set R is not empty119894 = 119894 + 1 go to Step 32 otherwise go to Step 4

Step 4 Store assignment result of Step 3 Increase monetaryscore value V = V + 119889 if V lt Vmax go to Step 2 otherwise goto Step 5

Step 5 Using (9) to calculate the Gini coefficient of eachassignment scheme stored in Step 4 return the optimalassignment scheme with lowest Gini coefficient value

5 An Illustrative Example

To illustrate the proposed method we propose a small casewith 3 types of scholarships and 5 candidates as an exampleThe unit of scholarship is US dollar ($) in Tables 1 3 and 4

Table 1 Details of scholarships

ID Number of awardees Amount of scholarship1 2 80002 3 50003 1 10000

Table 2 Data of candidates

ID Score1 252 203 304 355 15

Table 3 Ideal assignment results

ID Amount of scholarship1 82002 65603 98404 114805 4920

Tables 1 and 2 show the details of scholarships and scholarshipcandidates

It can be seen that the number of awardees of eachscholarship is separately 2 3 and 1 the monetary awardsprovided by each scholarship are respectively $8000 5000and 10000 The total scores of each student are 25 20 3035 and 15 Following the equity norm that the ideal amountof scholarship received by each student is proportionateto hisher score the ideal assignment result neglecting thepractical constraints is shown in Table 3

Using the method discussed in Section 31 four feasibleschemes can be obtained by iteratively solving the knapsacksubproblem for each student and then adjusting themonetaryvalue of the score Note that the derived schemes only satisfythe basic equity requirement that a student with higher scoremust receive an amount of scholarship no less than thatreceived by students with comparatively lower scores Thedetails regarding the 4 schemes are shown in Table 4

Through the method described in Section 32 we canthen compute the Gini coefficient of each scheme which isshown in Table 5

As per the Gini coefficients of each scheme it can befound that among 4 feasible schemes scheme 1 is the onewith the highest equity as the Gini coefficient of scheme1 is the lowest Therefore scheme 1 is chosen as the finalscholarship assignment scheme in this numerical example

From this example it can be concluded that the proposedmethod can enumerate several feasible scholarship assign-ment schemes and then choose the optimal one with theminimum value of Gini coefficient This demonstrates thepractical relevance of the proposed method

6 Discrete Dynamics in Nature and Society

Table 4 Derived feasible schemes of scholarship assignment

Scheme 1 Scheme 2

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 8000 1 1 25 8000 12 20 5000 2 2 20 5000 23 30 10000 3 3 30 13000 124 35 13000 12 4 35 15000 135 15 5000 2 5 15 0

Scheme 3 Scheme 4

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 5000 2 1 25 5000 22 20 5000 2 2 20 0 3 30 13000 12 3 30 13000 124 35 18000 13 4 35 23000 1235 15 0 5 15 0

Table 5 Gini coefficients of each scheme

Scheme Gini coefficient1 03342 03513 04754 0540

6 Conclusion

This paper develops a method to derive the optimal scholar-ship assignment schemewith the highest equity for universityadministrators The method is applicable as (1) it meets theequity requirement that students who perform better oughtto receive scholarships equal to or more than those receivedby less-achieving students (2) the scholarship assignmenteliminates the need of students to manually apply for specificscholarships which is a time- and energy-consuming pro-cess and (3) the equity of derived assignment schemes can bequantified through the Gini coefficient and the scheme withthe maximum equity (minimum Gini coefficient value) canbe identified and selected An illustrative example is adoptedto show the applicability of the proposed methodThe resultsindicate that the proposedmethod can obtain a set of feasibleassignment schemes efficiently and then derive the optimalone with the highest equity

Data Availability

The hypothetical data used to support the findings of thisstudy are included within the article No external data wereused to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the Top-Notch Academic Pro-grams Project of Jiangsu Higher Education Institutions(PPZY2015B148) and the Project of Educational Reform andPractice in Southeast University (2017-071)

References

[1] P E Peterson D Myers and W G Howell An Evaluation ofthe New York City School Choice Scholarships ProgramThe FirstYear Mathematica Policy Research Washington DC USA1998

[2] S L Desjardins ldquoThe impact of Washington State AchieversScholarship on student outcomesrdquo in Annual meeting of Associ-ation for the Study of Higher Education Vancouver BC Canada2009

[3] S L DesJardins and B P McCall ldquoThe impact of the GatesMillennium Scholars Program on college and post-collegerelated choices of high ability low-income minority studentsrdquoEconomics of Education Review vol 38 no 1 pp 124ndash138 2014

[4] M R Anderson-Rowland ldquoEvaluation of a ten year life plan-ning assignment for an academic scholarship success classrdquo inProceedings of the 41st Annual Frontiers in EducationConferenceCelebrating 41 Years of Monumental Innovations from Aroundthe World FIE 2011 pp 1ndash7 2012

[5] A Porter R Yang J Hwang J McMaken and J Rorison ldquoTheEffects of ScholarshipAmount onYield and Success forMasterrsquosStudents in Educationrdquo Journal of Research on EducationalEffectiveness vol 7 no 2 pp 166ndash182 2014

[6] H Forsyth and A Cairnduff ldquoA scholarship of social inclusionin higher education why we need it and what it should looklikerdquo Higher Education Research amp Development vol 34 no 1pp 219ndash222 2015

[7] P Amoros L C Corchon and B Moreno ldquoThe scholarshipassignment problemrdquo Games amp Economic Behavior vol 38 no1 pp 1ndash18 2002

[8] G Goastellec ldquoGlobalization and implementation of an equitynorm in higher education Admission processes and funding

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

4 Discrete Dynamics in Nature and Society

of scholarships no less than students with lower scores thescholarship assignment problem is converted to a series ofone-dimensional 0-1 knapsack problems [24 25]

Firstly the initial monetary value V of a unit score can becalculated by

V =sum119895isin119869119898119895 sdot 119888119895sum119894isin119868 119904119894

(1)

According to the monetary value of a unit score theamount of scholarships 119898119895 can then be converted to equiv-alent scores 119890119895 The process can be expressed as

119890119895 =119898119895V forall119895 isin 119869 (2)

The sequence of assigning scholarships to students isin accordance with the ranking of studentsrsquo scores In eachiteration the scholarships are allocated to the studentwith thehighest score among unsigned students and the amount ofscholarships to be allocated to the student is derived throughsolving a knapsack subproblem Let the score of a student119894 119904119894 be the capacity of the knapsack and the equivalentscore of scholarship 119895 119890119895 be the items to be put into theknapsack Thus the scholarship assignment scheme of thechosen student 119894 can be derived from

max sum119895isin119869119894

119890119895 sdot 120575119894119895 (3)

st sum119895isin119869119894

119890119895 sdot 120575119894119895 le 119904119894 (4)

By iteratively solving knapsack subproblems for eachchosen student and assigning scholarships in accordance tothe solutions the students with higher scores who possesspriority in the scholarship assignment can certainly obtaingreater amounts of scholarships Hence given a certainmonetary value of the score V we can derive a correspondingfeasible solution set (scholarship assignment scheme) usingthe abovemethod By iteratively adjusting themonetary valueof a unit score we can enumerate all the feasible solution sets

However the qualities of feasible solution sets derivedby the above method cannot be judged As this paper aimsto assign scholarships to students considering the equitynorm the finally selected assignment schememust be the onewith the highest equity The following section introduces themethod to quantify equity of each feasible solution

32 Measuring Equity of Scholarship Assignment SchemeNote that perfect equity means that every student can receivea proportion of scholarship same as the proportion of hisherscore in the total score To quantify the equity of scholarshipassignment schemeswhich aim to assigning different scholar-ships to a group of students with different scores a suggestionis associated with modifying an existing metric of incomeinequity ie Gini coefficient In the transportation fieldthe Gini coefficient is widely utilized to analyze inequity inaccessibility [26ndash28] and effect of transport policies [29 30]Herein the Gini coefficient is adopted to measure the social

welfare (for instance the accessibility to certain opportunitiesor the welfare of toll pricing scheme) achieved by transportusers In scholarship assignment the Gini coefficient 119866 canexpress an inequalitymetric that evaluates the departure fromperfect equity which can be expressed as

119866 = 121198992119882sum119894 =119896isin119868

1003816100381610038161003816119882119894 minus1198821198961003816100381610038161003816 (5)

where119882119894 denotes the benefit that student 119894 receives from theactual assignment scheme119882 denotes the average individualwelfare and 119899 is the number of students Considering thepractical constraints the ideal assignment result with perfectequity is hard to reach An individual student might gaineither positive benefit (receive more scholarship than shedeserves) or negative benefit (receive less scholarship thanshe deserves) in certain scholarship assignment schemeTo measure the benefit each student receives from theassignment scheme we define the welfare of student 119894 119882119894as the difference between the studentrsquos actual scholarship andideal scholarship which can be derived by

119882119894 = 119904119886119894 minus 119904119901119894 (6)

where 119904119886119894 and 119904119901119894 denote the actual and ideal results ofscholarship assignment with regards to student 119894 respectivelyThe average individual welfare119882 can be expressed as

119882 =sum119894isin119868119882119894119899 (7)

It is evident that the average individual welfare119882 equals0 and the individual welfare can be negativeThese propertiesprohibit the usage of original Gini coefficient in that thevalue of average and individual income used in the originalGini coefficient evaluation can only be positive Therefore arescaling process is necessary to implement (5)The re-scaledindividual welfare119882119904119894 can be derived by

119882119904119894 = 119882119894 +1003816100381610038161003816119882min1003816100381610038161003816 + 120581 (8)

where119882min denotes the minimum individual welfare and 120581is a given constant

By using 119882119904119894 to replace 119882119894 in (5) and (7) the Ginicoefficient 119866 of each solution can be obtained

119866 = 121198992119882119904sum119894 =119896isin119868

1003816100381610038161003816119882119904119894 minus119882

1199041198961003816100381610038161003816 (9)

4 Solution Algorithm

As discussed in Section 3 the task of searching for feasibleassignment scheme can be converted to a series of one-dimensional 01 knapsack problem which can be exactlysolved by dynamic programming algorithms [31ndash36] In thissection a solution algorithm for the knapsack subproblembased on dynamic programming is used to find the optimalscholarship assignment for each student in sequence On thebasis of it a solution algorithm for finding feasible assignmentschemes and selecting the optimal scheme is proposed

Discrete Dynamics in Nature and Society 5

41 Algorithm 1 Algorithm for One-Dimensional 0-1 KnapsackSubproblem Assuming there are totally 119901 types of schol-arships let stage ℎ denote the first ℎ types of scholarshipsthat are considered in scholarship assignment 119904 denote theequivalent score that is currently assigned to the selectedstudent and 119891ℎ(119904) denote the maximum equivalent score theselected student can obtain at stage ℎ

Step 1 (initialization) Set 1198910(119904) = 0 forall119904 le 119904119894 ℎ = 1 119904 = 0119895 = 1

Step 2 Update 119891ℎ(119904)

Step 21 119891ℎ(119904) = max119891ℎminus1(119904) 119891ℎminus1(119904 minus119898119894) +119898119895 where119898119894 isthe equivalent score of the scholarship previously assigned tothe selected student

Step 22 119895 = 119895 + 1 If 119895 gt ℎ go to Step 23 otherwise go toStep 21

Step 23 119904 = 119904 + 1 If 119904 gt 119904119894 go to Step 3 otherwise go to Step21

Step 3 ℎ = ℎ + 1 If ℎ le 119901 go to Step 2 otherwise return119891ℎminus1(119904119894) and stop the algorithm

42 Algorithm 2 Derive Feasible Schemes andSelect Optimal Scheme

Step 1 Initialize the monetary value of a unit score V using(1) set the value of step size 119889 and upper bound of monetaryscore value Vmax

Step 2 Obtain the equivalent score of each scholarship 119898119895using (2)

Step 3 Derive feasible scholarship assignment

Step 31 Rank students according to their scores and putstudents into set R1 2 119894

Step 32 Assign scholarships to the first student i in set R usingAlgorithm 1

Step 33 Delete student 119894 from set R If set R is not empty119894 = 119894 + 1 go to Step 32 otherwise go to Step 4

Step 4 Store assignment result of Step 3 Increase monetaryscore value V = V + 119889 if V lt Vmax go to Step 2 otherwise goto Step 5

Step 5 Using (9) to calculate the Gini coefficient of eachassignment scheme stored in Step 4 return the optimalassignment scheme with lowest Gini coefficient value

5 An Illustrative Example

To illustrate the proposed method we propose a small casewith 3 types of scholarships and 5 candidates as an exampleThe unit of scholarship is US dollar ($) in Tables 1 3 and 4

Table 1 Details of scholarships

ID Number of awardees Amount of scholarship1 2 80002 3 50003 1 10000

Table 2 Data of candidates

ID Score1 252 203 304 355 15

Table 3 Ideal assignment results

ID Amount of scholarship1 82002 65603 98404 114805 4920

Tables 1 and 2 show the details of scholarships and scholarshipcandidates

It can be seen that the number of awardees of eachscholarship is separately 2 3 and 1 the monetary awardsprovided by each scholarship are respectively $8000 5000and 10000 The total scores of each student are 25 20 3035 and 15 Following the equity norm that the ideal amountof scholarship received by each student is proportionateto hisher score the ideal assignment result neglecting thepractical constraints is shown in Table 3

Using the method discussed in Section 31 four feasibleschemes can be obtained by iteratively solving the knapsacksubproblem for each student and then adjusting themonetaryvalue of the score Note that the derived schemes only satisfythe basic equity requirement that a student with higher scoremust receive an amount of scholarship no less than thatreceived by students with comparatively lower scores Thedetails regarding the 4 schemes are shown in Table 4

Through the method described in Section 32 we canthen compute the Gini coefficient of each scheme which isshown in Table 5

As per the Gini coefficients of each scheme it can befound that among 4 feasible schemes scheme 1 is the onewith the highest equity as the Gini coefficient of scheme1 is the lowest Therefore scheme 1 is chosen as the finalscholarship assignment scheme in this numerical example

From this example it can be concluded that the proposedmethod can enumerate several feasible scholarship assign-ment schemes and then choose the optimal one with theminimum value of Gini coefficient This demonstrates thepractical relevance of the proposed method

6 Discrete Dynamics in Nature and Society

Table 4 Derived feasible schemes of scholarship assignment

Scheme 1 Scheme 2

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 8000 1 1 25 8000 12 20 5000 2 2 20 5000 23 30 10000 3 3 30 13000 124 35 13000 12 4 35 15000 135 15 5000 2 5 15 0

Scheme 3 Scheme 4

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 5000 2 1 25 5000 22 20 5000 2 2 20 0 3 30 13000 12 3 30 13000 124 35 18000 13 4 35 23000 1235 15 0 5 15 0

Table 5 Gini coefficients of each scheme

Scheme Gini coefficient1 03342 03513 04754 0540

6 Conclusion

This paper develops a method to derive the optimal scholar-ship assignment schemewith the highest equity for universityadministrators The method is applicable as (1) it meets theequity requirement that students who perform better oughtto receive scholarships equal to or more than those receivedby less-achieving students (2) the scholarship assignmenteliminates the need of students to manually apply for specificscholarships which is a time- and energy-consuming pro-cess and (3) the equity of derived assignment schemes can bequantified through the Gini coefficient and the scheme withthe maximum equity (minimum Gini coefficient value) canbe identified and selected An illustrative example is adoptedto show the applicability of the proposed methodThe resultsindicate that the proposedmethod can obtain a set of feasibleassignment schemes efficiently and then derive the optimalone with the highest equity

Data Availability

The hypothetical data used to support the findings of thisstudy are included within the article No external data wereused to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the Top-Notch Academic Pro-grams Project of Jiangsu Higher Education Institutions(PPZY2015B148) and the Project of Educational Reform andPractice in Southeast University (2017-071)

References

[1] P E Peterson D Myers and W G Howell An Evaluation ofthe New York City School Choice Scholarships ProgramThe FirstYear Mathematica Policy Research Washington DC USA1998

[2] S L Desjardins ldquoThe impact of Washington State AchieversScholarship on student outcomesrdquo in Annual meeting of Associ-ation for the Study of Higher Education Vancouver BC Canada2009

[3] S L DesJardins and B P McCall ldquoThe impact of the GatesMillennium Scholars Program on college and post-collegerelated choices of high ability low-income minority studentsrdquoEconomics of Education Review vol 38 no 1 pp 124ndash138 2014

[4] M R Anderson-Rowland ldquoEvaluation of a ten year life plan-ning assignment for an academic scholarship success classrdquo inProceedings of the 41st Annual Frontiers in EducationConferenceCelebrating 41 Years of Monumental Innovations from Aroundthe World FIE 2011 pp 1ndash7 2012

[5] A Porter R Yang J Hwang J McMaken and J Rorison ldquoTheEffects of ScholarshipAmount onYield and Success forMasterrsquosStudents in Educationrdquo Journal of Research on EducationalEffectiveness vol 7 no 2 pp 166ndash182 2014

[6] H Forsyth and A Cairnduff ldquoA scholarship of social inclusionin higher education why we need it and what it should looklikerdquo Higher Education Research amp Development vol 34 no 1pp 219ndash222 2015

[7] P Amoros L C Corchon and B Moreno ldquoThe scholarshipassignment problemrdquo Games amp Economic Behavior vol 38 no1 pp 1ndash18 2002

[8] G Goastellec ldquoGlobalization and implementation of an equitynorm in higher education Admission processes and funding

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

Discrete Dynamics in Nature and Society 5

41 Algorithm 1 Algorithm for One-Dimensional 0-1 KnapsackSubproblem Assuming there are totally 119901 types of schol-arships let stage ℎ denote the first ℎ types of scholarshipsthat are considered in scholarship assignment 119904 denote theequivalent score that is currently assigned to the selectedstudent and 119891ℎ(119904) denote the maximum equivalent score theselected student can obtain at stage ℎ

Step 1 (initialization) Set 1198910(119904) = 0 forall119904 le 119904119894 ℎ = 1 119904 = 0119895 = 1

Step 2 Update 119891ℎ(119904)

Step 21 119891ℎ(119904) = max119891ℎminus1(119904) 119891ℎminus1(119904 minus119898119894) +119898119895 where119898119894 isthe equivalent score of the scholarship previously assigned tothe selected student

Step 22 119895 = 119895 + 1 If 119895 gt ℎ go to Step 23 otherwise go toStep 21

Step 23 119904 = 119904 + 1 If 119904 gt 119904119894 go to Step 3 otherwise go to Step21

Step 3 ℎ = ℎ + 1 If ℎ le 119901 go to Step 2 otherwise return119891ℎminus1(119904119894) and stop the algorithm

42 Algorithm 2 Derive Feasible Schemes andSelect Optimal Scheme

Step 1 Initialize the monetary value of a unit score V using(1) set the value of step size 119889 and upper bound of monetaryscore value Vmax

Step 2 Obtain the equivalent score of each scholarship 119898119895using (2)

Step 3 Derive feasible scholarship assignment

Step 31 Rank students according to their scores and putstudents into set R1 2 119894

Step 32 Assign scholarships to the first student i in set R usingAlgorithm 1

Step 33 Delete student 119894 from set R If set R is not empty119894 = 119894 + 1 go to Step 32 otherwise go to Step 4

Step 4 Store assignment result of Step 3 Increase monetaryscore value V = V + 119889 if V lt Vmax go to Step 2 otherwise goto Step 5

Step 5 Using (9) to calculate the Gini coefficient of eachassignment scheme stored in Step 4 return the optimalassignment scheme with lowest Gini coefficient value

5 An Illustrative Example

To illustrate the proposed method we propose a small casewith 3 types of scholarships and 5 candidates as an exampleThe unit of scholarship is US dollar ($) in Tables 1 3 and 4

Table 1 Details of scholarships

ID Number of awardees Amount of scholarship1 2 80002 3 50003 1 10000

Table 2 Data of candidates

ID Score1 252 203 304 355 15

Table 3 Ideal assignment results

ID Amount of scholarship1 82002 65603 98404 114805 4920

Tables 1 and 2 show the details of scholarships and scholarshipcandidates

It can be seen that the number of awardees of eachscholarship is separately 2 3 and 1 the monetary awardsprovided by each scholarship are respectively $8000 5000and 10000 The total scores of each student are 25 20 3035 and 15 Following the equity norm that the ideal amountof scholarship received by each student is proportionateto hisher score the ideal assignment result neglecting thepractical constraints is shown in Table 3

Using the method discussed in Section 31 four feasibleschemes can be obtained by iteratively solving the knapsacksubproblem for each student and then adjusting themonetaryvalue of the score Note that the derived schemes only satisfythe basic equity requirement that a student with higher scoremust receive an amount of scholarship no less than thatreceived by students with comparatively lower scores Thedetails regarding the 4 schemes are shown in Table 4

Through the method described in Section 32 we canthen compute the Gini coefficient of each scheme which isshown in Table 5

As per the Gini coefficients of each scheme it can befound that among 4 feasible schemes scheme 1 is the onewith the highest equity as the Gini coefficient of scheme1 is the lowest Therefore scheme 1 is chosen as the finalscholarship assignment scheme in this numerical example

From this example it can be concluded that the proposedmethod can enumerate several feasible scholarship assign-ment schemes and then choose the optimal one with theminimum value of Gini coefficient This demonstrates thepractical relevance of the proposed method

6 Discrete Dynamics in Nature and Society

Table 4 Derived feasible schemes of scholarship assignment

Scheme 1 Scheme 2

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 8000 1 1 25 8000 12 20 5000 2 2 20 5000 23 30 10000 3 3 30 13000 124 35 13000 12 4 35 15000 135 15 5000 2 5 15 0

Scheme 3 Scheme 4

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 5000 2 1 25 5000 22 20 5000 2 2 20 0 3 30 13000 12 3 30 13000 124 35 18000 13 4 35 23000 1235 15 0 5 15 0

Table 5 Gini coefficients of each scheme

Scheme Gini coefficient1 03342 03513 04754 0540

6 Conclusion

This paper develops a method to derive the optimal scholar-ship assignment schemewith the highest equity for universityadministrators The method is applicable as (1) it meets theequity requirement that students who perform better oughtto receive scholarships equal to or more than those receivedby less-achieving students (2) the scholarship assignmenteliminates the need of students to manually apply for specificscholarships which is a time- and energy-consuming pro-cess and (3) the equity of derived assignment schemes can bequantified through the Gini coefficient and the scheme withthe maximum equity (minimum Gini coefficient value) canbe identified and selected An illustrative example is adoptedto show the applicability of the proposed methodThe resultsindicate that the proposedmethod can obtain a set of feasibleassignment schemes efficiently and then derive the optimalone with the highest equity

Data Availability

The hypothetical data used to support the findings of thisstudy are included within the article No external data wereused to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the Top-Notch Academic Pro-grams Project of Jiangsu Higher Education Institutions(PPZY2015B148) and the Project of Educational Reform andPractice in Southeast University (2017-071)

References

[1] P E Peterson D Myers and W G Howell An Evaluation ofthe New York City School Choice Scholarships ProgramThe FirstYear Mathematica Policy Research Washington DC USA1998

[2] S L Desjardins ldquoThe impact of Washington State AchieversScholarship on student outcomesrdquo in Annual meeting of Associ-ation for the Study of Higher Education Vancouver BC Canada2009

[3] S L DesJardins and B P McCall ldquoThe impact of the GatesMillennium Scholars Program on college and post-collegerelated choices of high ability low-income minority studentsrdquoEconomics of Education Review vol 38 no 1 pp 124ndash138 2014

[4] M R Anderson-Rowland ldquoEvaluation of a ten year life plan-ning assignment for an academic scholarship success classrdquo inProceedings of the 41st Annual Frontiers in EducationConferenceCelebrating 41 Years of Monumental Innovations from Aroundthe World FIE 2011 pp 1ndash7 2012

[5] A Porter R Yang J Hwang J McMaken and J Rorison ldquoTheEffects of ScholarshipAmount onYield and Success forMasterrsquosStudents in Educationrdquo Journal of Research on EducationalEffectiveness vol 7 no 2 pp 166ndash182 2014

[6] H Forsyth and A Cairnduff ldquoA scholarship of social inclusionin higher education why we need it and what it should looklikerdquo Higher Education Research amp Development vol 34 no 1pp 219ndash222 2015

[7] P Amoros L C Corchon and B Moreno ldquoThe scholarshipassignment problemrdquo Games amp Economic Behavior vol 38 no1 pp 1ndash18 2002

[8] G Goastellec ldquoGlobalization and implementation of an equitynorm in higher education Admission processes and funding

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

6 Discrete Dynamics in Nature and Society

Table 4 Derived feasible schemes of scholarship assignment

Scheme 1 Scheme 2

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 8000 1 1 25 8000 12 20 5000 2 2 20 5000 23 30 10000 3 3 30 13000 124 35 13000 12 4 35 15000 135 15 5000 2 5 15 0

Scheme 3 Scheme 4

ID Score Amount ofscholarship

Type ofscholarship ID Score Amount of

scholarshipType of

scholarship1 25 5000 2 1 25 5000 22 20 5000 2 2 20 0 3 30 13000 12 3 30 13000 124 35 18000 13 4 35 23000 1235 15 0 5 15 0

Table 5 Gini coefficients of each scheme

Scheme Gini coefficient1 03342 03513 04754 0540

6 Conclusion

This paper develops a method to derive the optimal scholar-ship assignment schemewith the highest equity for universityadministrators The method is applicable as (1) it meets theequity requirement that students who perform better oughtto receive scholarships equal to or more than those receivedby less-achieving students (2) the scholarship assignmenteliminates the need of students to manually apply for specificscholarships which is a time- and energy-consuming pro-cess and (3) the equity of derived assignment schemes can bequantified through the Gini coefficient and the scheme withthe maximum equity (minimum Gini coefficient value) canbe identified and selected An illustrative example is adoptedto show the applicability of the proposed methodThe resultsindicate that the proposedmethod can obtain a set of feasibleassignment schemes efficiently and then derive the optimalone with the highest equity

Data Availability

The hypothetical data used to support the findings of thisstudy are included within the article No external data wereused to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the Top-Notch Academic Pro-grams Project of Jiangsu Higher Education Institutions(PPZY2015B148) and the Project of Educational Reform andPractice in Southeast University (2017-071)

References

[1] P E Peterson D Myers and W G Howell An Evaluation ofthe New York City School Choice Scholarships ProgramThe FirstYear Mathematica Policy Research Washington DC USA1998

[2] S L Desjardins ldquoThe impact of Washington State AchieversScholarship on student outcomesrdquo in Annual meeting of Associ-ation for the Study of Higher Education Vancouver BC Canada2009

[3] S L DesJardins and B P McCall ldquoThe impact of the GatesMillennium Scholars Program on college and post-collegerelated choices of high ability low-income minority studentsrdquoEconomics of Education Review vol 38 no 1 pp 124ndash138 2014

[4] M R Anderson-Rowland ldquoEvaluation of a ten year life plan-ning assignment for an academic scholarship success classrdquo inProceedings of the 41st Annual Frontiers in EducationConferenceCelebrating 41 Years of Monumental Innovations from Aroundthe World FIE 2011 pp 1ndash7 2012

[5] A Porter R Yang J Hwang J McMaken and J Rorison ldquoTheEffects of ScholarshipAmount onYield and Success forMasterrsquosStudents in Educationrdquo Journal of Research on EducationalEffectiveness vol 7 no 2 pp 166ndash182 2014

[6] H Forsyth and A Cairnduff ldquoA scholarship of social inclusionin higher education why we need it and what it should looklikerdquo Higher Education Research amp Development vol 34 no 1pp 219ndash222 2015

[7] P Amoros L C Corchon and B Moreno ldquoThe scholarshipassignment problemrdquo Games amp Economic Behavior vol 38 no1 pp 1ndash18 2002

[8] G Goastellec ldquoGlobalization and implementation of an equitynorm in higher education Admission processes and funding

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

Discrete Dynamics in Nature and Society 7

framework under scrutinyrdquo Peabody Journal of Education vol83 no 1 pp 71ndash85 2008

[9] W O Lee and M Manzon ldquoThe issue of equity and qual-ity of education in Hong Kongrdquo The Asia-Pacific EducationResearcher vol 23 no 4 pp 823ndash833 2014

[10] T Pitman ldquoUnlocking the gates to the peasants are policiesof lsquofairnessrsquo or lsquoinclusionrsquo more important for equity in highereducationrdquo Cambridge Journal of Education vol 45 no 2 pp281ndash293 2015

[11] X Qu W Yi T Wang S Wang L Xiao and Z Liu ldquoMixed-integer linear programming models for teaching assistantassignment and extensionsrdquo Scientific Programming vol 2017Article ID 9057947 7 pages 2017

[12] Z Liu S Wang and Q Meng ldquoOptimal joint distance andtime toll for cordon-based congestion pricingrdquo TransportationResearch Part B Methodological vol 69 pp 81ndash97 2014

[13] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[14] Z Liu S Wang B Zhou and Q Cheng ldquoRobust optimizationof distance-based tolls in a network considering stochastic dayto day dynamicsrdquo Transportation Research Part C EmergingTechnologies vol 79 pp 58ndash72 2017

[15] E A Mannix M A Neale and G B Northcraft ldquoEquityequality or need The effects of organizational culture on theallocation of benefits and burdensrdquo Organizational Behaviorand Human Decision Processes vol 63 no 3 pp 276ndash286 1995

[16] SWang ldquoEfficiency and equity of speed limits in transportationnetworksrdquo Transportation Research Part C Emerging Technolo-gies vol 32 pp 61ndash75 2013

[17] S Wang Q Meng and Z Liu ldquoContainership schedulingwith transit-time-sensitive container shipment demandrdquoTrans-portation Research Part B Methodological vol 54 pp 68ndash832013

[18] L Anselmi M Lagarde and K Hanson ldquoGoing beyondhorizontal equity An analysis of health expenditure allocationacross geographic areas in Mozambiquerdquo Social Science ampMedicine vol 130 pp 216ndash224 2015

[19] S Caleo ldquoWhen distributive justice and gender stereotypescoincide Reactions to equity and equality violationsrdquo Journalof Applied Social Psychology vol 48 no 5 pp 257ndash268 2018

[20] J S Adams ldquoInequity in social exchangerdquo Advances in Experi-mental Social Psychology vol 2 no 4 pp 267ndash299 1965

[21] E Walster G Walster and E Berscheid Equity Theory andResearch Allyn amp Bacon Boston MA USA 1978

[22] D M Messick and K P Sentis ldquoFairness and preferencerdquoJournal of Experimental Social Psychology vol 15 no 4 pp 418ndash434 1979

[23] ISI (Institute for Scientific Information) ldquoData fromJournal Citation Reportsrdquo (Dataset) Clarivate AnalyticsAccessed July 20 2018 httpjcrincitesthomsonreuterscomJCRJournalHomeActionaction

[24] S Martello and P Toth ldquoAlgorithms for Knapsack problemsrdquoNorth-Holland Mathematics Studies vol 132 pp 213ndash257 1987

[25] HM Salkin andKCADe ldquoTheknapsack problemA surveyrdquoNaval Research Logistics vol 22 no 1 pp 127ndash144 2010

[26] T Neutens ldquoAccessibility equity and health care Review andresearch directions for transport geographersrdquo Journal of Trans-port Geography vol 43 pp 14ndash27 2015

[27] H R Waters ldquoMeasuring equity in access to health carerdquo SocialScience amp Medicine vol 51 no 4 pp 599ndash612 2000

[28] T FWelch and SMishra ldquoAmeasure of equity for public transitconnectivityrdquo Journal of Transport Geography vol 33 no 33 pp29ndash41 2013

[29] A Sumalee ldquoOptimal toll ring design with equity constraintAn evolutionary approachrdquo Journal of Eastern Asia Society forTransportation Studies vol 5 pp 1813ndash1828 2003

[30] X Sun Z Liu and S Chen ldquoTheEquity Issue for Cordon-BasedCongestionPricingwithDistanceTollrdquo inProceedings of the 2ndInternational Conference on Vulnerability and Risk Analysis andManagement ICVRAM 2014 pp 2310ndash2319 July 2014

[31] S Martello D Pisinger and P Toth ldquoDynamic programmingand strong bounds for the 0-1 Knapsack ProblemrdquoManagementScience vol 45 no 3 pp 414ndash424 1999

[32] P Toth ldquoDynamic programming algorithms for the Zero-OneKnapsack problemrdquo Computing vol 25 no 1 pp 29ndash45 1980

[33] D Pisinger ldquoA minimal algorithm for the 0-1 knapsack prob-lemrdquo Operations Research vol 45 no 5 pp 758ndash767 1997

[34] F Li Z L Chen and L Tang ldquoIntegrated Production Inven-tory and Delivery Problems Complexity and AlgorithmsrdquoINFORMS Journal on Computing vol 29 no 2 pp 232ndash2502017

[35] H Liu and D ZWWang ldquoLocating multiple types of chargingfacilities for battery electric vehiclesrdquo Transportation ResearchPart B Methodological vol 103 pp 30ndash55 2017

[36] J Chen SWang Z Liu and Y Guo ldquoNetwork-based optimiza-tion modeling of manhole setting for pipeline transportationrdquoTransportation Research Part E Logistics and TransportationReview vol 113 pp 38ndash55 2018

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