A Two-Step Tabu Search Heuristic for Multi-Period Multi-Site Assignment Problem with Joint Requirement of Multiple Resource Types

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1 Aticle A Two-Step Tabu Seach Heuistic fo Multi-Peiod Multi-Site Assignment Poblem with Joint Requiement of Multiple Resouce Types Siavit Swangnop and Paveena Chaovalitwongse* Depatment of Industial Engineeing, Faculty of Engineeing, Chulalongkon Univesity, Bangkok 10330, Thailand * paveena.c@chula.ac.th (Coesponding autho) Abstact. An assignment poblem has been extensively studied and applied in many industies. Vaiations of assignment poblem have been poposed and appeaed in liteatues fo many yeas. This pape extends the vaiation of assignment poblem in the dimension of task and esouce by poposing the joint equiement of multiple esouce types in a multi-peiod multi-site assignment poblem. The specific chaacteistic is that thee ae many multi-skill esouce types and tasks equie joint equiement of moe than one esouce type to opeate. An application of this model can be found in healthcae industy, especially in clinic netwoks o hostal netwoks, which have many sevice locations, have many esouce types such as doctos, nuses o medical equipments and definitely equie moe than one esouce type fo opeations. This pape poposes a two step Tabu seach heuistic fo multi-peiod multi-site assignment poblem with joint equiement of multiple esouce types. The specified neighbohood stategy, shot-tem memoy and long-tem memoy ae designed fo the addessed poblem in ode to geneate an efficient move to impove solutions. Fom computational study, solutions fom Tabu seach algoithm ae compaed with optimal solutions fom, and the esult shows that, fo small size poblems, most solutions ae close to optimal solutions (aveage gap =.%), fo medium size poblems, the algoithm can povide good solutions in a shot time compaing with (aveage gap = 5.8%), and fo lage size poblem, fou out of five solutions fom the poposed algoithm ae bette than solutions fom in a limit of time. Keywods: Assignment poblem, Tabu seach, heuistic, health esouce, joint equiement. ENGINEERING JOURNAL Volume 18 Issue 3 Received 4 Novembe 013 Accepted 0 Januay 014 Published 10 July 014 Online at DOI: /ej

2 1. Intoduction An assignment poblem has been extensively studied and applied in many industies, namely daiy [1], clothing [], mining [3], ailines [4], automated manufactuing [5] and sevice industies [6]. Fist appeaing in 195 [7], the classic assignment poblem is to find a one-to-one matching between n tasks and m agents and the objective function is to minimize the total cost. Ove the past few decades, the classic assignment poblem has been extended and many vaiations of the assignment poblem ae poposed, fo example, vaiation in objective function such as maximizing pofit[8, 9] o minimizing the maximum numbe of tavelling time[10], vaiation in planning peiod such as thee-dimensional assignment poblem [11], multipeiod assignment poblems fo medical esidents [1], multi-peiod machine assignment [13] o vaiation in task and esouce such as multi-esouce genealized assignment poblem [14], esouce-constained assignment scheduling [15], assignment poblem with senioity and job oity constaints [16] and genealization of multi-esouce genealized assignment poblem [9]. In this study, we extend the vaiation of assignment poblem in the dimension of task and esouce by poposing the joint equiement of multiple esouce types in a multi-peiod multi-site assignment poblem. This specific chaacteistic is that thee ae multiple esouce types and tasks equie joint of moe than one esouce type fo opeations. This model can be found in healthcae industy, especially in clinic netwoks o hostal netwoks, which have many sevice locations, have many esouce types such as physicians, nuses o medical equipments and equie moe than one esouce type fo opeations. Only qualified esouces can do tasks o teatments and allocating each esouce to each site has diffeent opeation cost. The esouce plannes have to decide whee thei esouces should be assigned to maximize total pofit. The emaining pats of this pape ae oganized as follows. The elated wok is eviewed in section. In section 3, the statement of poblem and mathematical model ae descibed. Then, Tabu seach algoithm and computational expeiment ae pesented in section 4 and 5. Finally, the conclusion and futue woks ae summaized in section 6.. Related Wok In academic view, this kind of multi-peiod multi-site assignment poblems can be found in a poblem of emegency esouce allocation [17-1] and a poblem of health staff scheduling [-6]. An emegency esouce allocation is a poblem of allocating esouces such as health staff, equipments and medical supplies fom possible depots o esponds units to disaste sites in the disaste situations such as eathquakes, floods o huicanes while health staff scheduling is a poblem of assigning physician, nuses o aides to hostals, wads o shifts. Both poblems conside allocating o assigning esouces to suitable shifts o sites as in the poposed poblem and many of them conside multiple esouce types. Fo emegency esouce allocation, thei esouces can be one o moe than one type and time peiod fo planning can also be one o moe than one peiod. The pefomance of allocation pocesses and decisions in a few days afte disaste stikes is an impotant key to educe the numbe of fatalities [1]. Zhang et al. [17] poposed the model and algoithm fo allocating multiple esouces fom emegency depots to disaste sites to fulfill all demands. Ozdama et al. [18] poposed emegency logistics planning model in natual disaste, which consists of two decisions: allocating multiple commodities to the disaste sites and scheduling the vehicles. Tzeng et al. [0] designed a elief-distibution model fo distibuting elief items fom collection points to tansfe candidate depots and elief demand points. Moe details of this poblem can be seen in Caunhye et al. [7] eviewing the optimization models in emegency logistics, Altay and Geen Iii [8] suveying the existing OR/MS liteatues in disaste opeations managements and Fiedich et al. [1] poviding the definitions of coe components in Emegency esouce allocation poblem. Fo health staff scheduling, a scope of allocation and assignment can be limited in one wad/depatment [, 3, 9-3], in many wads/depatments in one hostal [4] o in many hostals [5, 6, 33]. Most eseaches have focused on nuse scheduling moe than physician scheduling [34] and both staff ae mostly consideed sepaately. Enst et al. [34] eviewed a poblem of staff scheduling and osteing in which health staff scheduling is included. Tivedi and Wane [4] poposed algoithm fo allocating available float pool of nusing pesonnel to vaious inpatient units in a hostal. The nusing pesonnel ae divided into thee types which ae egisteed nuses, licensed pactical nuses and aides and substituting of those staff is allowed with diffeent pefomance. Aickelin and Dowsland [, 9] and Dowsland [30] also consideed the poblem of nuse scheduling. In thei models, thee ae sets of shift patten and nuses ae 84 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

3 divided into many gades in which highe gaded nuses can substitute nuses in lowe gades. The decision is to assign nuses to shift pattens, which povides the diffeent penalty cost in each shift patten. Fo physician scheduling, Cate and Laee [3] studied the poblem of physician scheduling in emegency oom while Goyal and Yadav [33] developed mathematical model and heuistic fo allocating physicians to vaious medical institutions. Othe health staff scheduling models can be seen in [5, 6, 31, 3]. Most of these studies mainly focus on allocating o assigning esouces to woking sites o shifts. Howeve, they usually conside assigning only one type of esouce and if they conside multiple esouces types, they do not concen joint equiement of esouce types. Anothe kind of poblem elated to the consideed poblem is the genealized assignment poblem (GAP) in which thee ae many agents who has own capacity fo doing tasks and they can do many tasks as long as they have enough capacity. In the past decades, many extensions of GAP ae poposed and models concening multiple esouce types and joint of multiple esouces ae aisen. Gavish and Pikul [14] poposed multi-esouce genealized assignment poblem (MRGAP), which extends GAP by allowing agents to have multiple esouces and thei esouces ae consumed when accomplishing thei tasks. Mazzola and Neebe [15] defined the assignment poblem with side constaints (APSC) which extends GAP as MRGAP but the esouces in the system is not sepaated by individual. All agents can use the esouces in the system until esouces ae out. Toktas et al. [35] combined the chaacteistics of the GAP and MRGAP with the APSC and geneates two additional poblems: collectively capacitated genealized assignment poblem (CCGAP) and assignment poblem with individual capacities (AIPC). Alidaee et al. [9] pesented the assignment model that includes the model of MRGAP as special cases. This new model is called genealization of MRGAP (GMRGAP). An extension is that thei tasks consist of many opeations and if tasks ae done, all its opeations must be completed. Also, although models in this poblem conside multiple esouces assignment and thei esouces can be assigned to many tasks, most models have the condition that each task must to be assigned to only one esouce. Thee is only a model of Alidaee et al. [9] which studied deeply in assigning multiple esouces to tasks. Howeve, they have the diffeent objective and do not conside a dimension of multiple peiod and multiple sites as in ou models. Although this kind of poblem is in the focus of many eseaches and have been extended in many aeas, thee ae no any models consideing multi-peiod multi-site assignment poblems with joint equiement of multiple esouce types as in the consideed model. The objective of this eseach is to develop the mathematical model and the solution method based on Tabu seach algoithm. 3. Poblem Desciption A multi-peiod and multi-site assignment poblem detemines whee esouces ae and what esouces do in each peiod. In ou model, thee ae many esouce types and tasks equie multiple esouce types fo opeation. All esouces ae allocated to the site with diffeent opeation cost, and each esouce is limited to be in one site. Each task at sites povides diffeent benefit, and esouces ae assigned to do tasks fo total pofit maximization. A mathematical model can be witten as follows: Index i = index fo tasks; i {1,,3,..., I} j = index fo esouces; j {1,,3,..., J} p = index fo peiods; p {1,,3,..., P} = index fo esouce types; {1,,3,..., R} s = index fo sites; s {1,,3,..., S} Set I ps = set of task i occuing in site s in peiod p. Paametes g j =1 if esouce j in type is qualified to do task i in peiod p. = 00 othewise. [Big M value] ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 85

4 b = 1 if task i in peiod p equies esouce type. = 0 othewise. B = benefit when task i in peiod p is executed. C = opeation cost when esouce j in type is assigned to site s. I ps = set of task i occuing in site s in peiod p. Decision vaiables Y j = 1 if esouce j in type is assigned to task i in peiod p. = 0 othewise. Z = 1 if esouce j in type is assigned to site s. = 0 othewise. W = 1 if task i in peiod p is executed. = 0 othewise. Objective function Maximize total pofit = P I R J S BW CZ (1) p 1 i 1 1 j 1 s 1 Constaints Qualification constaint: Location constaint: Joint equiement constaint: Available task constaint: I g jy 1 ; {1,..., R}, j {1,..., J}, p {1,..., P} () j i 1 S Z 1 ; {1,..., R}, j {1,..., J} (3) s 1 J gjy b j W ; {1,..., R}, p {1,..., P}, i {1,..., I} j 1 Z (4) Yj ; {1,..., R}, j {1,..., J}, p {1,..., P}, s {1,..., S}, i I ps (5) The objective function, Eq. (1), maximizes the total pofit, which is calculated fom total benefits and total opeation cost of all esouces. Eq. () enfoces that only qualified esouces can do tasks and each esouce is assigned to only one task pe peiod. Eq. (3) enfoces that each esouce must be assigned to only one site. Eq. (4) states that only qualified esouces can do tasks and tasks can be done when joint equiements of esouces ae satisfied, fo example, if a task equies esouce type 1 and, this task can be done ( W =1) when thee ae two qualified esouces, selected fom esouce type 1 and, assigned to do this task. Each site has diffeent tasks and esouces can do only tasks in the site whee they ae assigned. Eq. (5) is used fo enfocing this estiction. 4. Tabu Seach Algoithm Tabu seach (TS) is a well-known meta-heuistic fo solving a combinatoial optimization initiated by Glove [36-38]. A basic pocess fo finding solutions by Tabu seach algoithm is oughly divided into thee steps: set an initial solution, find neighbohoods and select neighbohood to be new solution. The second and thid steps ae done iteatively until stopng citeia is met. Efficiency of Tabu seach algoithm mainly depends on the stuctue of neighbohood, Tabu list and long tem memoy. Good neighbohood lets the algoithm find the best solution in a shot time. Because moving to wose solutions is allowed, Tabu list is used to pevent algoithm fom cycling o being stuck in a local optimum. Long tem memoy is usually 86 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

5 used to identify the good o bad elements of the solutions o the unvisited egions and then povide the good diection of the next move. Tabu seach algoithm is widely applied in allocation, scheduling and assignment poblem [5, 31, 39-43]. Fo poblems whose decision can be divided into many steps as ou model, one appoach fo develong algoithm is to sepaate the decision into many steps depending on chaacteistics of the poblem and algoithms fo each step ae developed [3, 34, 43-45]. Fo ou poblem, as descibed in pevious section, the decision of the model is to find the suitable sites (allocation) and suitable tasks (assignment) fo esouces. To develop algoithm fo the consideed poblem, we sepaate the decision into two steps: allocating esouce to sites and then assigning esouces to tasks. We popose a two-step Tabu seach algoithm fo solving the consideed poblem: Main Tabu seah algoithm (MTS) fo esouce allocation in the fist step and Sub Tabu seach algoithm (STS) fo esouce assignment in the second step. A stuctue of two-step Tabu seach algoithm is illustated in Fig. 1. The algoithm stats fom geneating an initial solution in Main Tabu seah algoithm (MTS). Then, a pocess of finding all neighbohoods is done. Each neighbohood is a set of esouces which should be moved to some sites to povide a bette solution. Because getting tue objective function of all neighbohoods by using Sub Tabu seach algoithm (STS) takes a lot of computational time, we have a pocess of educing the numbe of neighbohood by selecting only some neighbohoods with some citeia to be candidates. Afte candidate list is geneated, Sub Tabu Seach Algoithm (STS) will be done to find a solution of esouce assignment of each candidate and the tue objective function will be calculated. In ou STS, we design a specific Tabu list, neighbohood and divesification technique fo getting bette solution. Afte all candidates ae calculated by STS, the best candidate is selected to be a new initial solution fo MTS and the pocess of updating Tabu list and best known solution in MTS ae done. The pocess is done iteatively until eaching the stopng citeia. The detail of MTS and STS is descibed as follows: Main Tabu Seach Algoithm (MTS) [Resouce allocation] 0. Geneate an initial solution 1. Find all neighbohoods. Geneate candidate list 3. Select best candidate and update Tabu list, new solution and best known solution Find solution and objective function of each candidate Sub Tabu Seach Algoithm (STS) [Resouce assignment] 1. Find all neighbohoods. Select best candidate and update tabu list, long-tem memoy, new solution and best known solution Fig. 1. Stuctue of two-step Tabu seach algoithm Main Tabu Seach Algoithm (MTS) Details of Main Tabu Seach Algoithm ae descibed as follows: Geneate an initial solution: All esouces ae assigned to the site that has the lowest opeation cost. Then, the poblem is split up into many sub-poblems. One sub-poblem is a poblem of one site and one peiod. Then, is used to find an optimal solution of each sub-poblem. Find all neighbohoods: The objective of this step (MTS) is to move esouce to the bette site. Neighbohoods ae geneated fom moving set of esouces fom sites to anothe o, in othe wods, changing the value of Z fom 1 to 0 and the value of Z ' fom 0 to 1. The destination site to which these esouces ae moved is the site whee thee ae unassigned tasks (tasks that nobody does). The moved esouces ae the esouce in each type that can do those unassigned tasks, which selects only one esouce ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 87

6 pe type. Fo example, in Fig., task i 1 in site s is an unassigned task equiing esouce type and 4. The esouce j and j of type and j and j of type can do this task. Suppose j and j ae selected to move and, in initial solution, j is in s and j is in s. Geneating neighbohood is to move esouce j fom site s and j fom site s to site s o, in othe wods, to change the value of Z and 4 Z fom 1 to 0 and the value of Z and Z 3 fom 0 to 1. Othe neighbohoods can be geneated with the same concept which is moving j and j to site s 1 4, moving j and j to site s and moving j and j 3 4 to site s. This pocess will be done with all unassigned tasks to geneate all neighbohoods. 1 3 Initial solution Conside all esouces that can do task i1 Resouce Site Neighbohood solution Geneate neighbohood by changing esouces to wok at the destination site Resouce Site Resouce type j 1 S 3 Resouce type j 1 S 3 j S 4 j S 4 Resouce type 4 j 3 j 4 S 5 S 6 Resouce type 4 j 3 j 4 S 5 S 6 Destination site S Destination site S Fig.. Example of geneating neighbohood in MTS. Geneate candidate list: Because thee ae a lot of neighbohoods in this step, we educe the numbe of neighbohoods by selecting only some neighbohoods. We calculate suogate objective function of each neighbohood, which consumes shot computational time, and select only the best M neighbohoods to be the candidate in the candidate list. If thee is moe than one candidate geneated fom one task, only the candidate which has the highest suogate objective function is consideed to be in candidate list. Suogate objective function = Tbg - Tbl + Toc o Tbg= Total possible benefit gain, which is the sum of highest benefits of unassigned tasks that each moved esouce can do in each peiod in new site. o Tbl = Total benefit lost fom moving all esouce to new site. o Toc = Total additional opeation cost fom moving all esouce to new site. Tabu list: Tabu list is a shot tem memoy used fo peventing cycling. In ou model, all moved esouces fom the best neighbohood ae added to Tabu list. The esouces in Tabu list ae not allowed to be moved to othe sites fo N iteations. Stopng ule: MTS will be un iteatively until eaching the maximum iteation W o the maximum computational time V. 4.. Sub Tabu Seach Algoithm (STS) Fo each candidate, one o moe esouces ae moved to the new site and then esouces in some sites and peiods ae changed. The algoithm in this step is to assign esouces in these sites and peiods to unassigned tasks to get as much benefit as possible. Those sites and peiods will be calculated by this algoithm one by one until all of them ae consideed. An initial solution in this step is the solution fom MTS. Details of Sub Tabu Seach Algoithm ae descibed as follows: Find all neighbohoods: Because the objective of this step is to assign esouce to unassigned tasks, the neighbohood is geneated fom selecting some esouces to do unassigned tasks ( W 0 ). Both available esouces (esouces which ae not assigned to any tasks) and unavailable esouces (esouces which ae assigned to some tasks) ae able to be eassigned to do unassigned tasks. A set of esouces which povides the minimum benefit lost fom eassigning them to do the unassigned task is selected to be the 88 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

7 neighbohood. The benefit lost of each neighbohood is calculated fom the sum of benefit of all tasks which ae cancelled because of changing esouces fom doing the tasks that they wee assigned to doing new unassigned tasks. Fo example, Fig. 3, an unassigned task i 5 in peiod p 1 equies esouce type, which esouce j 3 and j 4 can do, and type 3, which esouce j and j can do. Suppose, in initial solution, 5 6 j and j do task i and i whose benefit is 100 and 50 espectively while j and j do task i and i 4 whose benefit is 00 and 150 espectively. To geneate neighbohood, we select esouce j and j to do 4 6 this unassigned task because they povide the minimum benefit lost fom cancelling tasks ( B p i 3 B ). The neighbohood is to change the value of Y, W, Y and p i fom 1 to 0 and change the value of Y, Y and j p i j p i j p i j p i W 14 W fom 0 to 1. Qualified esouces in this step 15 must not only be able to do unassigned tasks but also be in the site which has those unassigned tasks. This step is done with all unassigned tasks to geneate all neighbohoods. Afte having all neighbohoods, the neighbohood which has the highest objective function is selected to be a new initial solution. This pocess is done iteatively until the objective function does not impove fo P iteations. Set of unassigned tasks(w =0) W W Resouce type Resouce type 3 Initial solution Conside qualified esouces Resouce j 3 j 4 j 5 j 6 Task (Benefit) 1 (100) i (50) i i (00) 3 i (150) 4 Geneate neighbohood by changing the best esouce in each type to do the unassigned task Resouce Task (Benefit) Resouce type Resouce type 3 Neighbohood solution j 3 j 4 j 5 j 6 1 (100) i (50) i i (00) 3 i (150) 4 Unassigned task i 5 Unassigned task i 5 Fig. 3. Example of geneating neighbohood in STS. Tabu list: All changed esouces and tasks fom the best neighbohood ae added to Tabu list. They ae not allowed to be changed fo Q iteations. Divesification technique: This is a technique in Tabu seach algoithm to enable the seaching pocess to move and find neighbohoods in diffeent aea of solution space. We collect a fequency of tasks done in each iteation to long tem memoy. If the objective function of solution does not impove fo D iteations, the benefit of each task will be divided by this fequency and the new benefit will be used to find the neighbohood. The pocess will be done fo E iteations and then the long tem memoy will be eset. 5. Computational Expeiments A Tabu seach algoithm is tested to evaluate the efficiency of the poposed algoithm to the consideed poblem. The algoithm was coded in C# 010 and un on the Windows 7 Ultimate with Intel Coe i5-410m, CPU.30GHz and RAM 4GB. We compae ou esults with solutions fom commecial optimization tool (ILOG 1.1.0). Test poblems ae geneated into thee diffeent sizes. The fist set of poblem is a small size poblem which takes shot computational time. That is, can find an optimal solution in a few second. The second set is a medium size poblem which takes less than 4,000 seconds to find an optimal solution while the thid set, a lage size poblem, takes moe than 4000 seconds. Fo the small size poblem, the numbe of esouce type is fixed to (atio of task that equies 1 type and types is set to 5%: 75%). The numbe of peiod is set to 3 and 6 while the numbe of site is set to 5 and 10. A atio of esouce and task is set to 1: [6 esouces: 1 tasks and 10 esouces: 0 tasks] and atio of esouce that can do each task is set to 0.4. Opeation cost and benefit ae andomized unifomly between 500 to,000 and 400 to 4,000 espectively. Fo each poblem set, 10 tests ae geneated. Fo the medium size poblem, the expeiment is sepaated into pats. A atio of esouce and task is vaied in the fist pat while the numbe of esouce type and the atio of task that equies each type ae vaied in the second pat. In the fist pat, the numbe of esouce is vaied fom 10 to 16 and the atio of ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 89

8 esouce and task is set to 1: and 1:4. The numbe of esouce type, peiod and site ae fixed to, 1 and 5 espectively. In the second pat, the numbe of esouce type is set to and 3. The atio of tasks that equies each esouce type is vaied fom 0% to 100%. The numbe of esouce, peiod, site and task ae fixed to 14, 9, 5 and 30 espectively and atio of esouce that can do each task is set to 0.4. Opeation cost and benefit ae andomized unifomly between,000 to 10,000 and 400 to 4,000 espectively. Fo each poblem set, 5 tests ae geneated. Fo the lage size poblem, the numbe of esouce, peiod, site and task ae fixed to 0, 1, 5 and 60 espectively. The numbe of esouce type is set to 4, 6, 8, 10 and 1 and atio of esouce that can do each task is set to 0.4. Opeation cost is unifomly andomized between,000 to 10,000 fo all poblems while a benefit is unifomly andomized between 400 to 4,000 fo 4, 6 and 8 esouce types, 800 to 8,000 fo 10 esouce types and 1,00 to 1,000 fo 1 esouce types. The anges of benefit in each poblem size ae diffeent to maintain the value of objective function to be positive. Fo each poblem set, 1 test is geneated. The details of all poblem sizes ae shown in Table1. The fist seven columns ae the desciption of tested poblems, which is the size of poblem, the set of poblem, the numbe of esouce, the numbe of task, the numbe of peiod, the numbe of site and the numbe of esouce type, and the est ae the atio of tasks that equies each esouce type fo opeations. Table 1. Poblem size Small poblem Medium poblem (Pat1) Medium poblem (Pat) Lage poblem Poblem set Details of all poblem sizes. Numbe of esouces Numbe of tasks Numbe of peiods Numbe of sites Numbe of esouce types Ratio of each esouce type S S S S MA MA MA MA MA MA MA MA MB MB MB MB MB MB MB MB MB MB L L L L L All paametes of Tabu seach algoithm ae set accoding to size of the poblem. Fom peliminay expeiments, the suitable paametes fo STS can be set as follows: P ROUNDUP(6*(numbe of task),0) (6) Q ROUNDUP MIN 0.7 * (numbe of esouce), 0.1* (numbe of task), 0 (7) D ROUNDUP(0.4*(numbe of task),0) (8) 90 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

9 E (9) Fo MTS, a suitable size of Tabu list (N) to pevent algoithm fom being stuck in a local optimum is 3 o 4 depending on the poblem size. The numbe of maximum iteation (V) and candidate (M) affect diectly to the quality of the solution. A lage numbe of V and M incease the oppotunity to find bette solutions; howeve, it takes moe computational time. V and M ae limited to the suitable value accoding to the poblem size and the computational time. All MTS and STS paametes ae shown in Table. The paametes ae divided into goups: MTS and STS. In MTS, the Max iteation (W)/time (V), Tabu list (N) and Candidate list (M) is the maximum iteation o maximum time fo unning MTS, the numbe of iteation fo keeng moved esouces in Tabu list and the numbe of neighbohood in candidate list espectively. In STS, the Max iteation (P), Tabu list (Q) Ticke fo Divet (D) and Duation fo Divet (E) is the maximum iteation fo unning STS, the numbe of iteation fo keeng changed esouces and tasks in Tabu list, the numbe of iteation fo using divesification technique and the duation fo using divesification technique espectively. Table. Poblem size Small poblem Medium poblem (Pat1) Medium poblem (Pat) Lage poblem Paametes fo all test poblems. Poblem set Paametes MTS STS Candidate Max Tabu Ticke Max iteation Tabu list list iteation list fo Divet (W) /time (V) (N) (M) (P) (Q) (D) S1 5 seconds S 5 seconds S3 5 seconds S4 5 seconds Duation fo Divet (E) MA iteations MA1. 00 iteations MA iteations MA iteations MA.1 00 iteations MA. 00 iteations MA.3 00 iteations MA.4 00 iteations MB iteations MB1. 00 iteations MB iteations MB iteations MB iteations MB.1 80 iteations MB. 80 iteations MB.3 80 iteations MB.4 80 iteations MB.5 80 iteations L1 100 iteations L 100 iteations L3 100 iteations L4 100 iteations L5 100 iteations *The value of all paametes in STS is calculated by equation (6) (9). The expeiment of small size poblems In the small size poblem, 4 poblems ae geneated: S1, S, S3 and S4. Fo S1 and S3, thee ae only few peiods but many sites wheeas, fo S and S4, thee ae only few sites but many peiods. The numbe of esouce and task in S3 and S4 is moe than in S1 and S. Time fo unning MTS fo all poblems is limited to 5 seconds. The esults of the expeiment ae illustated in Table 3. The second column (#OPT by Tabu) shows the numbe of optimal solution found by Tabu seach algoithm. The aveage optimal gap is shown in the thid column, which is calculated fom [(solution of ) (solution of Tabu seach)] *100 / (solution of ). The esult shows that, fo all the test poblems (40 tests), 13 optimal solutions ae found and the aveage optimal gap anges fom 0.6 to 4.0. ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 91

10 OPtimal gap OPtimal gap Computational time(second) Computational time(second) DOI: /ej Table 3. The esults fom expeiments of small size poblems. #OPT by Tabu Poblem set (10 tests) Aveage gap (%) S S S3 1.7 S The expeiment of medium size poblems [pat1] In the medium size poblem [pat1], the expeiment is sepaated into two goups, and fou poblems pe goup ae geneated: MA1.1 to MA1.4 fo the fist goup, and MA.1 to MA.4 fo the second goup. All paametes in both goups ae the same except the numbe of task which is set by the atio of esouce and task (1: fo the fist goup and 1:4 fo the second goup). The esult of the expeiment is shown in Table 4. The second column shows the computational time of while the thid column shows time to find the best solution of Tabu seach algoithm. The foth to sixth column show the minimum optimal gap, maximum optimal gap and aveage optimal gap. The aveage optimal gap and computational time ae plotted in Fig. 4. The esult shows that the computational time of consideably inceases when the numbe and atio of esouce and task incease. Tabu seach algoithm can find good solutions in a shot time compaing with and the quality of the solution emains good when the numbe and atio of the esouce and task incease. Table 4. The esults fom expeiments of medium size poblems [pat 1]. Poblem set time (sec) Tabu time(sec) Minimum gap (%) Maximum gap (%) Aveage gap (%) MA MA MA MA MA MA MA MA Computational time [Resouce:Task=1:] Computational time [Resouce:Task=1:4] Tabu seach 400 Tabu seach MA1.1 MA1. MA1.3 MA1.4 0 MA.1 MA. MA.3 MA.4 Poblem set Poblem set Optimal gap [Resouce:Task=1:4] Optimal gap [Resouce:Task=1:] MA.1 MA. MA.3 MA.4 Poblem set Fig. 4. Computational time and optimal gap of medium size poblem [pat 1]. The expeiment of medium size poblems [pat] In the medium size poblem [pat], the expeiment is sepaated into two goups by the numbe of esouce type, and five poblems pe goup ae geneated: MB1.1. to MB1.5 fo the fist goup and MB.1 to MB.5 fo the second goup. All paametes in both goups ae the same except the numbe of esouce 3 MA1.1 MA1. MA1.3 MA1.4 Poblem set 9 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

11 OPtimal gap OPtimal gap Computational time(second) Computational time(second) DOI: /ej type vaied fom to 3 and the atio of task that equies each type vaied fom 0% to 100%. The esult of the expeiment is shown in Table 5 and the computational time and optimal gap ae illustated in Fig. 5. The esult shows that the computational time of extemely inceases when the numbe of esouce type and the atio incease (fo this expeiment, the numbe of esouce type just inceases by 1). The optimal gap slightly inceases when the atio inceases. Howeve, the algoithm can still povide good solutions in a shot time compaing with the. Table 5. The esults fom expeiments of medium size poblems [pat ]. Poblem set time (sec) Tabu time(sec) Minimum gap (%) Maximum gap (%) Aveage gap (%) MB MB MB MB MB MB MB MB MB MB Computational time [ esouce types] Computational time [3 esouce types] Tabu seach Tabu seach MB1.1 MB1. MB1.3 MB1.4 MB1.5 Poblem set 0 MB.1 MB. MB.3 MB.4 MB.5 Poblem set Optimal gap [ esouce types] MB1.1 MB1. MB1.3 MB1.4 MB1.5 Poblem set Fig. 5. Computational time and optimal gap of medium size poblem [pat ]. The expeiment of lage size poblems In the lage size poblem, five poblems ae geneated: L1, L, L3, L4 and L5. Because finding an optimal solution fo this poblem set takes a lot of time, the computational time fo unning is limited to 4,000 seconds while time fo unning MTS is limited to 100 iteations and the numbe of candidate list is fixed to 3. We know fom the esult of expeiment in medium size poblem that the computational time o complexity of the poblem extemely inceases when the numbe of esouce type inceases. In this expeiment, the numbe of esouce type is vaied fom 4 to 1, which is a lage numbe compaing with the pevious expeiment ( and 3 esouce types), to evaluate the efficiency of the algoithm when being applied to high complexity poblems. The computational time and optimal gap ae shown in Table 6. The value of optimal gap in this poblem set can be less than zeo because unning time of is limited and solutions fom can be wose than the solutions fom Tabu seach algoithm. As shown in the Table5, the optimal gap of fou out of five poblems is less than zeo, which means that fou out of five solutions fom Tabu seach algoithm ae bette than solutions fom. Figue 6 illustates the computational time and the best solution found in each time by and Tabu seach algoithm. Pats of line that have no data means that cannot find any feasible solution. As can be seen, when the numbe of esouce type inceases, time to find a feasible solution by inceases. In Optimal gap [3 esouce types] MB.1 MB. MB.3 MB.4 MB.5 Poblem set ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 93

12 Objective function value Objective function value Objective function value Objective function value Objective function value DOI: /ej contast to, Tabu seach algoithm can povide good feasible solutions in a vey shot computational time. Table 6. The esults fom expeiments of lage size poblems. Poblem set Tabu time(sec) Gap L L L L L esouce types 6 esouce types Tabu seach Tabu seach Computational time (second) Computational time (second) 10 esouce types 8 esouce types Tabu seach Tabu seach Computational tme (second) Computational tme (second) 1 esouce types Tabu seach Fig. 6. Computational time and optimal gap of lage size poblem [pat ]. In summay, the complexity of poblems dastically inceases when the numbe of esouce type and the numbe of task that equies joint of multiple esouce types incease. Tabu seach algoithm pefoms well in all poblem sizes. Many solutions in small size poblem ae optimal and the aveage gap fo all poblems is.%. In the medium size poblem, good solutions can be found in a shot time compaing with (aveage gap = 5.8%). Fo the lage size poblem, the poposed algoithm clealy outpefoms. Most solutions fom Tabu seach algoithm ae bette than solutions fom and all best solutions can be found quickly compaing with. 6. Conclusion Computational tme (second) In most allocation and assignment models, only one esouce type is consideed. If multiple esouces types ae consideed, they fail to conside joint equiement of multiple esouce types. In this study, we have consideed this aspect in a multi-peiod multi-site assignment poblem (multi-peiod multi-site assignment poblem with joint equiement of multiple esouce types). This kind of poblem can be found in planning multiple health esouces in clinic o hostal netwok. The mathematical model and heuistic based on Tabu seach algoithm was developed. The poposed Tabu seach algoithm is comsed of two steps (two-step Tabu seach algoithm). The fist step aims to allocate esouces to site while the second step means to assign esouces to task. The computational expeiments ae done to evaluate the efficiency of the 94 ENGINEERING JOURNAL Volume 18 Issue 3, ISSN (

13 algoithm. Test poblems ae gouped into thee sizes: small, medium and lage size poblems. The esult shows that the developed algoithm povides good solutions in all poblem sizes. Although the algoithm pefoms well fo all test poblems, the quality of solution tends to be dopped when the atio of task that equies moe than one esouce type inceases. Futue wok should find ways to impove the quality of the solution when this atio is high. In addition, the developed model does not allow thei esouces to be otated, so anothe subject of futue study is to develop model that allow thei esouces to be otated to many sites. Acknowledgements The potion of this wok was pesented at 17 th Intenational Confeence of the Intenational Jounal of Industial Engineeing Theoy, Applications, and Pactice (IJIE) [46]. We wish to thank Pof. Rein Boondiskulchok and Sakol Suethanaponkul fo guidance and feedback on the ealie dafts of this pape and anonymous eviewes fo insightful comments. Refeences [1] L. R. Foulds and J. M. Wilson, A vaiation of the genealized assignment poblem aising in the New Zealand daiy industy, Annals of Opeations Reseach, vol. 69, no. 0, pp , [] S. Haji-Gabouj, A fuzzy genetic multiobjective optimization algoithm fo a multilevel genealized assignment poblem, Systems, Man, and Cybenetics, Pat C: Applications and Reviews, IEEE Tansactions on, vol. 33, no., pp. 14-4, 003. [3] G. H. Reynolds, A shuttle ca assignment poblem in the mining industy, Management Science, vol. 17, no. 9, pp , [4] H. D. Sheali, E. K. Bish, and X. Zhu, Ailine fleet assignment concepts, models, and algoithms, Euopean Jounal of Opeational Reseach, vol. 17, no. 1, pp. 1-30, 006. [5] S. Bilgin, and M. Azizoglu, Opeation assignment and capacity allocation poblem in automated manufactuing systems, Computes & Industial Engineeing, vol. 56, no., pp , 009. [6] A. Coominas, R. Pasto, and E. Rodiguez, Rotational allocation of tasks to multifunctional wokes in a sevice industy, Intenational Jounal of Poduction Economics, vol. 103, no. 1, pp. 3-9, 006. [7] D. W. Pentico, Assignment poblems: A golden annivesay suvey, Euopean Jounal of Opeational Reseach, vol. 176, no., pp , 007. [8] C. Rainwate, J. Geunes, and H. Edwin Romeijn, The genealized assignment poblem with flexible jobs, Discete Applied Mathematics, vol. 157, no. 1, pp , 009. [9] B. Alidaee, H. Gao, and H. Wang, A note on task assignment of seveal poblems, Computes & Industial Engineeing, vol. 59, no. 4, pp , 010. [10] A. Ravindan and V. Ramaswami, On the bottleneck assignment poblem, Jounal of Optimization Theoy and Applications, vol. 1, no. 4, pp , [11] K. C. Gilbet and R. B. Hofsta, Multidimensional Assignment Poblems. Blackwell Publishing Ltd, 1988, pp [1] L. S. Fanz and J. L. Mille, Scheduling medical esidents to otations: solving the lage-scale multipeiod staff assignment poblem, Opeations Reseach, vol. 41, no., pp , [13] X. Zhang and J. F. Bad, A multi-peiod machine assignment poblem, Euopean Jounal of Opeational Reseach, vol. 170, no., pp , 006. [14] B. Gavish and H. Pikul, Efficient algoithms fo solving multiconstaint zeo-one knapsack poblems to optimality, Mathematical Pogamming, vol. 31, no. 1, pp , [15] J. B. Mazzola and A. W. Neebe, Resouce-constained assignment scheduling, Opeations Reseach, vol. 34, no. 4, pp , [16] G. Caon, P. Hansen, and B. Jaumad, The assignment poblem with senioity and job oity constaints, Opeations Reseach, vol. 47, no. 3, pp , [17] J.-H. Zhang, J. Li, and Z.-P. Liu, Multiple-esouce and multiple-depot emegency esponse poblem consideing seconday disastes, Expet Systems with Applications, vol. 39, no. 1, pp , 01. [18] L. Ozdama, E. Ekinci, and B. Kucukyazici, Emegency Logistics Planning in Natual Disastes, Annals of Opeations Reseach, vol. 19, no. 1-4, pp , 004. ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 95

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15 [44] H. Li and K. Wome, Scheduling pojects with multi-skilled pesonnel by a hybid MILP/CP bendes decomposition algoithm, Jounal of Scheduling, vol. 1, no. 3, pp , 009. [45] L. Chen, A. Langevin, and D. Riopel, A tabu seach algoithm fo the elocation poblem in a waehousing system, Intenational Jounal of Poduction Economics, vol. 19, no. 1, pp , 011. [46] S. Swangnop and P. Chaovalitwongse, A Tabu seach heuistic fo multi-peiod multi-site assignment poblem with joint equiement of multiple esouce types, in Poceedings of 17th Intenational Confeence on Industial Engineeing Theoy, Applications and Pactice (IJIE 013), 013, pp ENGINEERING JOURNAL Volume 18 Issue 3, ISSN ( 97

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