Advanced Sensitivity Analysis of the Semi-Assignment Problem

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1 Proeedigs of the 202 Iteratioal Coferee o Idustrial Egieerig ad Operatios Maagemet Istabul, Turkey, July 3 6, 202 Advaed Sesitivity Aalysis of the Semi-Assigmet Problem Shih-Tig Zeg ad Ue-Pyg We Departmet of Idustrial Egieerig ad Egieerig Maagemet Natioal Tsig Hua Uiversity, 0, Setio 2, Kuag-Fu Rd., Hsihu, 300, Taiwa Chi-Je Li Departmet of Idustrial Egieerig ad Maagemet Ta Hwa Istitute of Tehology No., Dahua Rd., Qiogli Towship, Hsihu Couty 307, Taiwa Abstrat Sesitivity aalysis of the semi-assigmet problem (semi-ap) is oered with obtaiig the perturbed rage of the ost that a be perturbed without the urret positive variable set hagig. Due to the high degeeray of the semi-ap, the traditioal sesitivity aalysis, whih deides the perturbed rage where the urret optimal basis remais optimal, is impratial. By usig our proposed algorithm, the degeerate basi variable a be pivoted out, ad the hose basi variable a be pivoted i without hagig the assiged ell. Coerig the high degeeray of the semi-ap, we a broade the perturbed rage by allowig this basis hagig, as log as maitaiig the same positive variable set of the urret problem. I this paper, we propose a algorithm for determiig the perturbed rage of the assiged ad uassiged ell, ad omputatioal result is also provide to demostrate the effiiet of the proposed algorithm. Keywords Semi-assigmet problem, sesitivity aalysis, degeeray. Itrodutio The key assumptio of the lassi assigmet problem is that the tasks ad the agets to whih they are to be assiged are uique; however, Petio (2007) metioed that although all the agets are uique, it is a ommo oditio that some of the tasks are idetial i a ompay or fatory. Therefore, that kid of problem is more pratial tha the lassi assigmet problem. The problem for that situatio is alled the semi-ap, ad the model for the situatio where there are agets should be assiged to m task groups, with d i tasks i group i. Cosiderig m the feasibility i the model, we assume that = d i. The model of the semi-ap is show as model () (Barr 976): m Mi x j= S.t x = di j= i =,..., m, m x = j =,...,, x 0, () where is the ost of assigig aget j to task group i. Eah group should be assiged to d i agets, ad eah aget should be assiged to oly oe task. Or we a osider the problem as a produtio lie worker alloatio problem. Assume there are i produtio lies ad eah lie should be assiged d i workers, ad eah work should be assiged 390

2 to oly oe produtio lie. Their objetive futios are all to determie how all the assigmet should be alloated to miimize the total ost. Due to a semi-ap, whih is a totally uimodular property of ostrait matrix, it a be treated both as a liear programig, ad also as a 0 iteger programmig problem or a etwork problem. With that property, some partiular algorithms have bee developed whih a solve a semi-ap very effiietly. Sie may methods for obtaiig optimal solutios to the semi-ap have bee proposed, sesitivity aalysis is usually arried out for osiderig the uertaity prie flutuatio. With our algorithm we a keep the same assigmet ad still got the miimum ost but the objet futio hage absolutely. Whe degeeray ours, usig the ovetioal sesitivity aalysis may mislead the deisio maker. For avoidig that problem, there are three defiitios of sesitivity aalysis rages show up. Aordig to Li ad We (2003) ad Hadigheh ad Terlaky (2006), we summarize three types of sesitivity aalysis for liear Programmig whih is based o differet propose. Let B be oe of the optimal bases of the assigmet problem with matrix C. Defie ω = {(i, j) x =, (i, j) B } the orrespodig optimal assigmet whih is also alled support set. Type I (Basis Ivariay): Type I sesitivity is the traditioal sesitivity aalysis, as implemeted i all ommerial LP pakages, ad be itrodued i may operatioal researh related text books. We deote the perturbed rage i (2-) as [L, U ] suh that B remais the same. Type I sesitivity aalysis uses the simplex method ad is based o the odegeeray assumptio of the optimal solutio. It is worthwhile to metio that whe a problem has multiple optimal or/ad degeerate solutios, the depedig o the basis ivariay make o sese o optimality rages, ad thus bewilder the poliy-maker. Type II (Support Set Ivariay): Takig multiple optimal or(ad) degeerate solutios ito osideratio, Type II sesitivity aalysis deals with fidig a perturbed rage [L, U ] suh that maitai the same ω (support set). It meas that the perturbed problem should have a optimal solutio with the same support set. Type III (Optimal Partitio Ivariay): I this type of sesitivity aalysis, we wat to determie those values of model parameters for whih the rate of hage of the optimal objetive futio value is ostat. I this paper, we ivestigate Type II sesitivity aalysis for the semi-ap. Degeeray of optimal solutios auses osiderable diffiulties i sesitivity aalysis by usig traditioal sesitivity aalysis method. Therefore, we wat to develop advaed sesitivity aalysis of the semi-ap to solve the problem. Also, there are may suessful ases exted the theory to pratial i the semi-ap. Aordig to the study of Duffuaa et al. (994), he metioed that i the well-kow ar ompay GM, semi-assigmet formulatios are used to assiged miroproessors to tasks or futios that meas a proessor a hadle more tha oe task. Besides, the semi-ap a also be applied to mapower plaig (assigig persoel to jobs), shedulig (assigig subassemblies to tasks) quotig from Petio (2007). For solvig those pratial problems, there are may differet algorithms developed. The first algorithm speially desiged for solvig the semi-ap was developed by Barr et al (976) ad was alled the alteratig basis method. The major differee oept betwee the alteratig basis method algorithm ad the primal simplex method is that it osiders a subset of bases, alled alteratig path bases, for the purpose to obtai a optimal solutio. The, Duffuaa et al. (994) laimed the total solutio time of the algorithm developed by them alled a shortest augmetig path algorithm is faster tha alteratig path bases method. Shortest augmetig path algorithm is developed to maitai dual feasibility ad omplemetary slakess ad works toward satisfyig primal feasibility. Volgeat (996) developed a modify algorithm base o shortest augmetig path proedure ad said the ode is superior to other odes for the semi-ap. Some assigmet problem a be solved by deompositio to the semi-ap suh as a vehile routig problem desribed by Toth ad Vigo (2002). Moreover, some approahes get the rages whih the optimal assigmet remais optimal by osiderig all optimal basis suh as (Gal 986), but the method is troublesome ad tedious. 39

3 May of methods ivestigatig the solutio proedure, ad very few sesitivity aalysis methods are proposed. 2. The Advaed Sesitivity Aalysis Algorithm I the setio, we propose two theories for simplifyig our algorithm proedure ad demostrate our algorithm. 2. Type II Sesitivity Aalysis of the Semi-AP Theorem shows that the lower boud of Type II rage for a assiged ell is ubouded i the semi-ap, ad the Theorem 2 shows the upper boud of Type II Rage for a uassiged ell is also ubouded i the semi-ap. Before we start to prove the theorem, we eed to kow that if the optimal solutio of a dual problem is ifeasible, the it optimal solutio of the primal problem is ubouded proposed by Bazaraa el al. (200). Theorem Suppose ω is the urret optimal assigmet of C as model (2), ad C as model (3) is the perturbed matrix as follows. If (p, q) ω, the the lower boud of Type II Rage L. II C = (2) C = p q + q p (3) Proof: Theorem is proved with model (4) ad model (5), ad L II is the optimal of model (2) Mi s.t. u i + v j + Q = if (i, j) (p, q) ;,,m, j=,, u p + v q + Q = ; (p, q) ω Q = 0 if (i, j) ω ;,,m, j=,, Q 0 if (i, j) ω ;,,m, j=,,. (4) Model (5) is the dual problem of model (4) 392

4 Max t j= s.t. t = 0, 2,..., m j= t t = 0 j=, 2,..., = t 0 if( i, j) ω ; i, j =...,. (5) Sie t = ad t pj 0, j q; oe a idue j= t pj < 0. This otradits with the ostrait j= t pj = 0. Hee, model (5) is ifeasible, ad we a olude that model (4) is ubouded. The proof has ompleted. Theorem 2 Suppose the upper boud of Type II RageU. ω is the urret optimal assigmet of C ad C is the perturbed matrix. If (p, q) II ω, the Proof: Theorem 2 is proved with model (6) ad model (7), adu is the optimal value of model (6): Max s.t. ui + vj + Q = if (i, j) (p, q) ;,,m, j=,, up + vq + Q = if (p, q) ω Q = 0 if (i, j) ω ;,,m, j=,, Q 0 if (i, j) ω ;,,m, j=,,. II (6) Model (7) is the dual problem of (6). Mi t j= s.t. t = 0, 2,..., m j= t t = 0 j=, 2,..., = t 0 if (, ) ω ;,..., m. j=,...,. (7) Sie equatio model (6) is ubouded. t = otradits with t 0 if (p, q) ω, problem model (7) is ifeasible. Hee, problem After provig the two theorems, we a olude that the lower boud for ad assiged ell ad the upper boud for a uassiged ell are ifiite. Hee, we just eed to alulate the upper boud of a assiged ell ad the lower boud of a uassiged ell i our algorithm. That a simplify our proedure. 393

5 2.2 T he Proposed Algorithm Whe we wat to alulate a Type II rage of a semi-ap through the algorithm, we should iput a matrix Q ew. The iput matrix is modify from Q whih represets urret optimal redue osts (u i + v j ) of a optimal problem, ad it is show at model (8). After we use - to replae the redue ost of a assiged ell (i, j), Q is trasferred to a ew matrix Q ew whih is show as model (9) Q = (8) 2 Q ew 0 3 = (9) The algorithm osists of three phases: perturbig phase, labelig phase ad modify phase, ad the proedure is show o Figure. First, we explai the purpose of the four phases. After that, we talk about the detail steps i eah phase. Let q deotes elemets of Q ew matrix i row i ad olum j, ad let a parameter t= to deote ell (i, j) is a assiged ell ad usig t = 0 to deote ell (i, j) is a uassiged ell to make the algorithm be oded easily. Here, we explai the followig phase by let ell (p, q) as a perturbed ell. The perturb phase: It ours whe a ell s ategory is equal to oe. Suppose t is equal to oe ad is a perturbed variatio we ve metioed before. After some modify of redue ost for feasibility, we set all q iq - 0 (i p) for the optimal oditio, ad we have equatios q iq (i p) Derive itersetio of these iequalities, we got a upper boud ours at ell (i, q)= Mi {q iq i p} The we let pi, ad deide it as the pivot ell (p, q). Hee, we a get the same pivot ell by hoosig the miimum redue ost of all uassiged ells of olum j diretly. The labelig phase: It is used to fid the leavig or eterig loop, ad the detail is at below. Let R matrix to deote if the row is marked or ot, ad r i is the elemets of R matrix i row i. C ad J are used for the olum i the same purpose. o Let r p = o If r i = (,, m), let b i. The, let j = if q bj = -(j=,, ). Breakig here, if there are o more rows a be labeled o If j = (j=,, ), let a j. The, let r i = if q ia = 0(,, m). Breakig here, if there are o more olums a be labeled. 394

6 Start N t= Y Labelig Perturb C Modify N = Y Ed Figure : Flow hart of the algorithm The modify phase: Whe we aot fid a loop after performig labelig phase, the modify phase ours. A matrix M is give to deote how may times a ell has bee labeled, ad modifig the Q ew as the idea of Hugarium algorithm. m represet elemets of M matrix i row i olum j. The proedure is show below. o If r i = (,..., m), let bi ad m bj m bj +. o If j = 0 (j=,, ), let aj ad m ia m ia +. o s=mi {q ew m = 0}( s is a parameter used for reord the miimum umber of all q uder the oditio the matrix M of a ell(i, j) is equal to zero.) o If m = 0,let q ew q ew - s. ( ~m, j= ~) If m = 2, let q q + s. ( ~m, j= ~) After we desribe the speifi otet of the four phases, we start to explai the proedure of the algorithm whih is give i Figure.. There are four mai steps iluded i the proedure as the followig 395

7 Labelig proedure for determie the type II rage of the semi-assigmet problem o Step: We iput the data Q ew, ad distiguishig their t of eah ell (i, j). If t is equal to oe, go to step2. If t is equal to zero go to step3. o Step2: Perform the perturb phase ad got the pivot ell (p, q),the go to step3. o Step3: Perform the algorithm phase. If q is equal to oe, we got the Type II rage here, ad the rage is [-,q ], whe t is equal to oe. While the rage is [-q, ] for t is equal to zero. If q is ot equal to oe, go to step4. o Step4: Perform Modify phase ad go to Step2 if t is equal to oe. If t is equal to zero, go to step3. 3. Computatioal Results I this hapter, the differee perturbed rage of Type I ad Type II is demostrated ad ruig time of 2 ases by usig the algorithm is displayed. 3. The Compariso betwee the Type I ad Type II Rages Type II rage a be larger or equal tha Type I rage as we metioed before, ad the two differet rages is ompared for support our view. Here, Type I rages are obtaied by usig Ligo pakage software, ad Type II rages are aquire by usig the proposed algorithm with the same ost oeffiiets value ad RHS values. The Type I ad Type II rages is show at Table. Table : Type I ad Type II Rages Type Method Perturbatio rages of Δ [- 5, ] [- 3, ] [-7, ] [- 3,2] [- 6, ] [2, 3] Type I Type II LINGO Labelig algorithm [- 3, ] [- 5, ] [, 2] [2, 3] [- 7, ] [- 3, ] [, 3] [, 3] [- 2, ] [- 4, ] [, 6] [- 3,2] [- 5, ] [- 3, ] [-0, ] [- 3, ] [- 6, ] [, 3] [- 3, ] [- 5, ] [, 5] [, 3] [- 7, ] [- 3, ] [, 3] [, 3] [- 5, ] [- 7, ] [, 6] [- 3, ] Regardig to the result, all Type II rages are all bigger or equal to Type I rages without exeptio, ad the upper boud ad the lower boud is ifiite i uassiged ell ad assiged ell separately as the two theories we proposed. 3.2 The Result of Test Problem Twelve ases are geerated with the ombiatio of four differet problem size ad three ost itervals. Results are give for problem sizes , , ad , ad ost oeffiiets is deided radomly betwee ost itervals -50, -00 ad The average times, ad stadard deviatios are average of te problems i a sigle ase. The results are give as Table 2. Table 2 shows the eah average time of three problems i the same problem size are substatially o differet; however, we still a fid the average time i -50 ost rages is relatively smaller tha the wilder oe i almost ases. The reaso is the smaller ost rages is more likely to ause alterative solutios whih result i more zero value redue osts; therefore the average time of a ase i smaller ost rages is more likely faster tha others, beause the yle a be foud more easily. Figure2 is give as follows to display the tred of the average time more learly. 396

8 Problem Size (umber of rows) Table 2: The average time (mi) of test problems Cost Rages The, the stadard deviatio is give i Table 3 to show the dispersio exists from the average. The tred of stadard deviatio from the smaller problem size to the bigger problem size beomes higher tha the prior oe i almost ases. That meas there are more uertaity i the bigger size problem suh as the size of the yle we fid ad the umber of modifyig times eeded for fidig a yle. Time(mi) Average time Cost Rages Figure 2: The tred of average time i eah problem size Problem Size (umber of rows) Table 3: The stadard deviatio of test problems Cost Rages

9 4. Colusios This paper oetrates o the sesitivity aalysis of the semi- AP. Cosiderig the speial struture of the semi-ap, oly the ost oeffiiets perturbed raged are oered. Moreover, the semi-ap is a highly degeeray problem as all the other etwork problem, ad hee, the traditioal sesitivity aalysis aot be applied to the problem. The algorithm we proposed here a attai the Type II rage without the eed of redue osts of all basis by usig the revised labelig method. I this study, we perturb oly oe ost oeffiiet at a time that is suitable whe the hage of a ost oeffiiet is about the aget or workers. however, the ost hagig of a produtio lie may ours by the reovatio or totally mahie update of oe produtio lie that may result i ost oeffiiets hagig i the same degree i oe produtio lie. Therefore, the algorithm a be exteded by osiderig how to perturb oe row at a time. Referees Barr, R., F. Glover, D. Kligma, A New Alteratig Basis Algorithm for Semi-Assigmet Networks, Computers ad Mathematial Programmig, Gaithersburg, 976. Bazaraa, M.S., J.J. Jarvis, ad H.O. Sherali, Liear Programmig af Network Flowers, 4th editio, Wiley-Itersiee, New York, 200. Duffuaa, S.O., ad M.S. Ghassab, A Suessive Shortest Path Algorithm for the Semi-Assigmet Problem, Tehial Report, 994. Gal,T., Shadow Pries ad Sesitivity Aalysis i Liear Programmig uder Degeeray-State-of-the-Art-Survey, OR Spektrum, vol. 8, o. 2, pp. 59-7, 986. Hadigheh, A.G ad T. Terlaky, Sesitivity aalysis i liear optimizatio: Ivariat support set itervals, Europea Joural of Operatioal Researh, vol. 69, o. 3, pp , Li, C.-J. ad U.-P. We, Sesitivity aalysis of the optimal assigmet, Europea Joural of Operatioal Researh, vol. 49, o., pp , Petio, D.W., Assigmet problems: A golde aiversary survey, Europea Joural of Operatioal Researh, vol. 76, o. -3, pp , Toth, P ad D. Vigo, The vehile routig problem, Soiety Idustrial ad Applied Mathematis, Philadelphia, Volgeat, A., Liear ad Semi-Assigmet problems: A ore orieted approah, Computers & Operatios Researh, vol. 23, o. 0, pp ,