A Reinforced Hungarian Algorithm for Task Allocation in Global Software Development

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1 A Reiforced Hugaria Algorithm for Task Allocatio i Global Software Developmet Xiao Yu State Key Lab. of Software Egieerig, Computer School, Wuha Uiversity, Wuha, Chia xiaoyu_whu@yahoo.com Ma Wu School of Computer Sciece ad Iformatio Egieerig, HuBei Uiversity, Wuha, Chia wuma_1@qq.com Xiagyag Jia * State Key Lab. of Software Egieerig, Computer School, Wuha Uiversity, Wuha, Chia Correspodig author jxy@whu.edu.c Ye Liu School of Computer Sciece ad Iformatio Egieerig, HuBei Uiversity, Wuha, Chia liuye061@qq.com Abstract The allocatio of software developmet tasks is a critical maagemet activity i distributed developmet projects. Oe of the most importat problem is to fid the lowest-cost way to assig tasks i global software developmet, which ca be solved by Hugaria algorithm. However, the origial Hugaria algorithm oly assume that a task ca oly be solved by oe developmet site. The assumptio is ot agreed with the actual case where a software developmet task is usually be solved through a collaboratio amog several sites. To address such a issue, this paper proposes a reiforced Hugaria algorithm (RHA) for task assigmet i global software developmet. RHA cosists of three major stages. First, RHA trasforms a m cost matrix ito two cost matrix by addig (2-m) virtual developmet sites. Secod, RHA performs the origial Hugaria algorithm o the two cost matrix to get the optimal assigmet results. Fially, RHA removes the (2-m) virtual developmet sites ad gets the fial optimal assigmet result for m tasks. Simulatio results idicate that RHA is a viable approach for the task assigmet problem i global software developmet. 1 Keywords task allocatio; global software developmet; Hugaria algorithm I. INTRODUCTION As global ad distributed software developmet is becomig a orm i the software idustry [1-2], a tight budget ad a shortage of resources ad time have motivated may software eterprises to start lookig for outside parters. With the advet of big data era [3-4] ad the developmet of etwork techology [5-6], kow as global software developmet (GSD), the allocatio of software developmet tasks over several sites, which may be spread across differet coutries, has become a commo practice i idustrial software egieerig [7-8]. 1 DOI referece umber: /SEKE Software developmet task assigmet is oe of the core steps to reasoably allocate limited developmet resources ad effectively exploit the capabilities of the developmet sites i global software developmet. However, task allocatio is still oe of the major challeges i global software developmet. A large umber of possible combiatios for task assigmet makes the process more complicated. For example, if a software developmet project cosists of five differet tasks which eed to be assiged amog four developmet sites, theoretically there are 4 5 =1024 differet combiatios available for task assigmet [9]. It is evidet that evaluatig all possibilities is impossible ad it eeds a approach to assig the task to the available developmet sites. There are several possible strategies for allocatig tasks that use differet criteria for allocatio. Bass et.al [10] state that i practice, allocatig task to lowcost coutries has become a cost-savig strategy for may orgaizatios. Therefore, oe of the most importat problem is to fid the lowest-cost way to assig tasks i global software developmet, which usually be solved by Hugaria algorithm [11]. However, the origial Hugaria algorithm solves the problem that the project maager allocates tasks to sites with the least total cost by performig all operatios o a cost matrix. It oly assumes that the umber of tasks is equal to the umber of sites ad a task ca oly be solved by oe developmet site. The assumptio is ot agreed with the actual case where the umber of tasks ad the umber of site are ot equal i geeral, ad a software developmet task is usually be solved through a collaboratio amog several sites. To address such a issue, this paper proposes a reiforced Hugaria algorithm (RHA) for task assigmet i global software developmet. The algorithm solves the problem that the project maager allocates tasks to m sites at the least total cost by performig all operatios o a m matrix. RHA

2 cosists of three major stages. First, RHA trasforms a m matrix ito two matrix by addig (m-) virtual tasks. Secod, RHA performs the origial Hugaria algorithm o the two cost matrix to get the optimal assigmet for 2 tasks. Fially, RHA removes the (m-) virtual tasks ad gets the fial optimal assigmet for m tasks. The effectiveess of RHA is demostrated by comparig it with the expasio matrix method [12]. Simulatio results idicate that RHA ca get the optimal assigmet with the lowest cost i global software developmet. The remaider of this paper is orgaized as follows. Sectio II reviews the Hugaria algorithm. Sectio III describes the proposed reiforced Hugaria algorithm (RHA) for task assigmet i global software developmet ad Sectio IV shows the operatioal process of the algorithm by a example. Sectio V demostrates the experimetal results. Sectio VI presets the related work. Fially, Sectio VII addresses the coclusio ad poits out the future work. II. HUNGARIAN ALGORITHM I this sectio, we first give the problem defiitio. The, we briefly review the Hugaria algorithm. A. Problem defiitio The usual assigmet problem is defied as follows: assig tasks to developmet sites with the least total cost, if task i is assiged to site j with a o-egative iteger cost c ij. This problem is a special case of the liear programmig problem, defied as follows: i=1 j=1 (1) Mi S= c ij x ij Subject to i=1 x ij = 1 (j=1, 2,..., ), (2) j=1 x ij = 1 (i=1, 2,..., ), (3) x ij=1 or 0. (4) where x ij= 1 meas task i is assiged to site j, otherwise x ij=0, task i is ot assiged to site j. The costs c ij form a cost matrix, deoted C. Theorem 1: If a umber is added to or subtracted from all of the etries of ay oe row or colum of a cost matrix C, the the optimal assigmet for the resultig cost matrix C is also a optimal assigmet for the origial cost matrix C. Proof: Assume that u i is subtracted from all of the etries i each row ad v j is subtracted from all of the etries i each colum from the origial cost matrix C. For the resultig cost matrix C, the problem is defied as follows: Mi S = i=1 j=1(c ij u i v j ) x ij (5) = i=1 j=1 c ij x ij i=1 j=1 u ij x ij v ij x ij i=1 j=1 Because i=1 x ij = 1 (j=1,2,3,) ad j=1 x ij = 1 (i=1,2,3,), the Mi S = i=1 j=1 c ij x ij i=1 u i j=1 v j = Mi S i=1 u i j=1 v j Sice i=1 u i + j=1 v j is a costat umber, the optimal assigmet for the resultig cost matrix C is also a optimal assigmet for the origial cost matrix C B. The Hugaria algorithm The Hugaria algorithm applies the above mathematical model ad theorem 1 to a give cost matrix to fid a optimal assigmet. Step 1. Subtract the smallest etry i each row from all the etries of its row. Step 2. Subtract the smallest etry i each colum from all the etries of its colum. Step 3. Draw lies through appropriate rows ad colums so that all the zero etries of the cost matrix are covered ad the miimum umber of such lies is used. Step 4. Test for Optimality: (1) If the miimum umber of coverig lies is, a optimal assigmet of zeros is possible ad we are fiished. (2)If the miimum umber of coverig lies is less tha, a optimal assigmet of zeros is ot yet possible. I that case, proceed to Step 5. Step 5. Determie the smallest etry ot covered by ay lie. Subtract this etry from each ucovered row, ad the add it to each covered colum. Retur to Step 3. C. Example A software developmet project cosists of four differet tasks: Task 1, Task 2, Task 3 ad Task 4. There are four developmet sites: S1, S2, S3, ad S4. They each demad differet pay for various tasks. The problem is to fid the lowest-cost way to assig the tasks. The 4 4 cost matrix of this problem is show as follows Step 1. Subtract 4.8 from Row 1, 6.4 from Row 2, 6.6 from Row 3, ad 5 from Row 4. The resultig matrix is as follows Step 2. Subtract 1 from Colum 4. The resultig matrix is as follows.

3 Step 3. Cover all the zeros of the matrix with the miimum umber of horizotal or vertical lies Step 4. Sice the miimal umber of lies is less tha 4, we have to proceed to Step 5. Step 5. Note that 1.4 is the smallest etry ot covered by ay lie. Subtract 1.4 from each ucovered row. The resultig matrix is as follows Now add 1.4 to each covered colum. The resultig matrix is as follows Now retur to Step 3. Step 3. Cover all the zeros of the matrix with the miimum umber of horizotal or vertical lies Step 4. Sice the miimal umber of lies is 4, a optimal assigmet of zeros is possible ad we are fiished Sice the total cost for this assigmet is 0, it must be a optimal assigmet. Here is the same assigmet made to the origial cost matrix So we should assig Task 1 to Site 2, Task 2 to Site 4, Task 3 to Site 1, ad Task 4 to Site 3. III. RHA I this sectio, we first give the problem defiitio. The, we preset our RHA method for task allocatio i global software developmet. A. Problem defiitio The origial Hugaria algorithm oly assume that the umber of tasks is equal to the umber of sites ad a task ca oly be solved by oe developmet site ad a task ca oly be solved by oe developmet site. The assumptio is ot agreed with the actual case i global software developmet where the umber of tasks ad the umber of site are ot equal i geeral, ad a software developmet task is usually be solved through a collaboratio amog several sites. Therefore, the task assigmet problem i global software developmet is defied as follows: assig tasks to m developmet sites with the least total cost (m>), if task i is assiged to site j with a o-egative iteger cost c ij. This problem is a special case of the liear programmig problem, defied as follows: m i=1 j=1 (6) Mi S= c ij x ij m Subject to i=1 x ij = 1 (j=1, 2,..., ), (7) j=1 x ij = 1 (i=1, 2,..., m), (8) x ij=1 or 0. (9) For this case, we eed to fid a optimal assigmet give a m cost matrix. Theorem 2: If a m matrix is trasformed ito two matrix by addig (2-m) virtual sites with the zero etries, the a optimal assigmet for the resultig cost matrixes is also a optimal assigmet for the origial cost matrix. Proof: For the resultig cost matrix C, the problem is defied as follows: 2 Mi S = i=1 j=1 c ij x ij (10) Subject to i=1 x ij = 1 (j = 1,2,,2) (11) j=1 x ij = 1 (i = 1,2,, ) (12) 2 j=+1 (13) x ij = 1 (i = 1,2,, ) x ij = 1 or 0 (i = 1,2,, ; j = 1,2,,2) (14) i=1 2 j=1 Mi S = c ij m x ij 2 = i=1 j=1 c ij x ij + i=1 j=m+1 c ij x ij Because c ij = 0 (i = 1,2,, ; j = m + 1, m + 2,,2) 2 The Mi S = i=1 j=1 c ij x ij + 0 i=1 2 j=1 = c ij x ij

4 = Mi S B. The RHA method The followig algorithm applies the above mathematical model ad theorem 2 to a give m cost matrix to fid a optimal assigmet. Algorithm 1 presets the pseudo-code of our proposed RHA method. Algorithm 1. RHA approach Iput: m cost matrix Output: optimal assigmet result 1. Subtract the smallest etry i each row from all the etries of its row; 2. Subtract the smallest etry i each colum from all the etries of its colum; 3. Trasform m matrix ito two matrix by addig (2-m) virtual sites with the zero etries; 4. Performs the origial Hugaria algorithm o the two cost matrix; 5. Removes the (2-m) virtual developmet sites ; 6. retur optimal assigmet result; IV. EXAMPLE I this sectio, we illustrate the executio of RHA by a example. Example 1: A software developmet project cosists of four differet tasks: Task 1, Task 2, Task 3 ad Task 4. There are six developmet sites: S1, S2, S3, S4, S5, ad S6. They each demad differet pay for various tasks. The problem is to fid the lowest-cost way to assig the tasks. The 6 8 cost matrix of this problem is show as follows E ij = Step 1. Subtract 4.8 from Row 1, 6.4 from Row 2, 6.6 from Row 3, ad 5 from Row 4. The resultig matrix is as follows Step 2. Subtract 1 from Colum 4, ad 1.4 from Colum 5. The resultig matrix is as follows Step 3. Trasform 6 4 matrix ito two 4 4 matrix by addig 2 virtual sites with the zero etries. The resultig matrixes are as follows The first matrix: The secod matrix: Step 4. Performs the origial Hugaria algorithm o the two 4 4 cost matrix. The assigmet result of the first cost matrix is as follows: Task1--->S2 Task2--->S4 Task3--->S1 Task4--->S3 The assigmet result of the secod cost matrix is as follows: Task1--->S7 Task2--->S8 Task3--->S6 Task4--->S5 Step 5.Removes the 2 virtual developmet sites ad output the fial assigmet result: Task1--->S2 Task2--->S4 Task3--->S1,S6 Task4--->S3,S5 V. EXPERIMENTS I this sectio, we evaluate our proposed reiforced Hugaria algorithm (RHA) for task assigmet empirically. We first itroduce the performace measures. The, i order to ivestigate the performace of RHA, we perform some empirical experimets.

5 A. Performace measures I the experimet, we employ oe commoly used performace measures, i.e, successful allocatio rate. It is defied ad summarized as follows. Successful allocatio rate (SAR) is the measure of tasks that are successfully allocated to developmet sites. The SAR=M s/m, where M s is the umber of cost matrixes that are successfully allocated ad M is the umber of all cost matrixes. The higher the value of successful allocatio rate, the more effective the algorithm is. B. Experimetal results I this experimet, we aalyze the effectiveess of RHA with differet umber of developmet sites. Here we fix the umber of tasks to 4 ad evaluate the performace of the evaluated algorithms by icreasig the umber of developmet sites from 6 to 12 with a icremet of 2. Therefore, we geerated four sets of matrix data, which cotais matrixes, matrixes, matrixes ad matrixes, respectively. Result Aalysis: As we ca see i Fig.1, the successful allocatio rate of RHA is always 1. The successful allocatio rate of the expasio matrix method goes up whe the umber of developmet sites icreases. It shows that RHA ca effectively allocate tasks to developmet sites with the lowest cost whe the umber of developmet sites chages. Fig. 1. SAR performaces with differet umber of developmet sites Experimet 2: I this experimet, we aalyze the effectiveess of RHA with differet umber of tasks. Here we fix the umber of developmet sites to 4 ad evaluate the performace of the evaluated algorithms by icreasig the umber of tasks from 4 to 10 with a icremet of 2. Therefore, we geerated four sets of matrix data, which cotais matrixes, matrixes, matrixes ad matrixes, respectively. Result Aalysis: As we ca see i Fig.2, the successful allocatio rate of RHA is always 1. The successful allocatio rate of the expasio matrix method decreases whe the umber of tasks icreases It shows that RHA ca effectively allocate tasks to developmet sites with the lowest cost whe the umber of tasks chages. Fig. 2. SAR performaces with differet umber of tasks VI. RELATED WORK I this sectio, we briefly review the existig task assigmet methods [13-26] i global software distributio. Geographically distributed developmet creates ew questios about how to coordiate muhi-site work. Griter proposes four methods product developmet orgaizatios to coordiate muhi-site work. He fids that o matter what model is used, it is difficulty to acquire expertise ad the distributio of project mass may be uequal [13-14]. Lamersdorf et.al thik that distributed developmet is ofte drive by some factors which are ot distiguished, such as risks, workforce capabilities, the iovatio potetial of differet regios, ad cultural factors. They discuss about these factors ad fid a lack of empirically-based multi-criteria distributio models [15]. A qualitative study aimig to idetify ad uderstad the criteria used i practice was coducted by Lamersdorf, et al. The results show that the sourcig strategy ad the type of software to be developed have a sigificat effect o the applied criteria [16]. The criteria ad causal relatioships were idetified i a literature study ad refied i a qualitative iterview study. Lamersdorf et.al itroduce a model which aims at improvig maagemet processes i globally distributed projects by givig decisio support for task allocatio that systematically regards multiple criteria [18]. A plaig tool to idetify task assigmet based o multiple criteria ad weighted project goals was developed by Lamersdorf ad Much. The model is called TAMRI (Task Allocatio based o Multiple criteria) ad its implemetatio is based o Bayesia etworks. The applicatio of the tool requires a large amout of kowledge o casual relatioships i distributed developmet. [17]. Though these process models have helped compaies to achieve global stadards, some social aspects are ot cosidered. Amrit proposes a methodology to test a hypothesis based o how social etworks ca be used to improve coordiatio i software idustry [21]. Setamait describes GSD-a hybrid computer simulatio model of the software developmet process i order to idetify the practices of work distributio ad try to classify criteria [22]. Marques presets a

6 domai otology to represet cocepts related to task allocatio i distributed teams i his paper. This method deals with the lack of stadardized vocabulary ad achieve kowledge acquisitio ad sharig [23]. VII. CONCLUSION AND FUTURE WORK I this paper, we address the issue of how to fid the lowest-cost way to allocate tasks amog developmet sites. We propose a reiforced Hugaria algorithm (RHA) for task assigmet i global software developmet. RHA cosist of two stages. I the first stage, the m matrix are trasformed ito two matrix by addig virtual developmet sites. I the secod stage, the two matrix are solved by the origial Hugaria algorithm. Fially, the virtual tasks are removed to get the optimal assigmet results. We coduct experimets o the sythetic datasets to evaluate the performace of the proposed algorithm. The experimetal results idicate that the proposed algorithm ca effectively allocate tasks to developmet sites with the lowest cost. 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