Heuristic Approaches for Solving the Multidimensional Knapsack Problem (MKP)

Transcription

1 Heuristic Approaches for Solvig the Multidimesioal Kapsack Problem (MKP) R. PARRA-HERNANDEZ N. DIMOPOULOS Departmet of Electrical ad Computer Eg. Uiversity of Victoria Victoria, B.C. CANADA Abstract: - There are exact ad heuristic algorithms for solvig the MKP. Solutio quality ad time complexity are two mai differeces amog exact ad heuristic algorithms. Solutio quality is a measure of how close we are from the optimal poit ad time complexity is a measure of the required time to reach such poit. The purpose of this paper is to report the solutio quality obtaied from the evaluatio of three heuristic approaches preseted else i the literature ad to report the solutio quality obtaied from the evaluatio of modified versios of those approaches. A geetic algorithm ad a at system are two of the heuristic evaluated. Key-Words: - Kapsack problem, At System, Geetic Algorithm. Itroductio As it is kow, the Multidimesioal Kapsack Problem (MKP) ca be expressed as follows x r i b i p max = { } = subect to ri x bi i =,..., m () x 0, =,..., px represets the -variable, represets the required i-resource by the x variable, represets the total amout of i-resource, represets the value of the x variable. Exact ad heuristic approaches have bee proposed for solvig the MKP, some of them are [], [6], [8] ad [9]. A excellet review of the MKP ad its associated exact ad heuristic algorithms is give i [3]. As it is poited out i [3], for the same m, as icreases the problem become harder ad take more time to solve. Likewise, if m icreases while fixig, the difficulty icreases as well. I the problems evaluated i this paper, is the variable that icreases. For real time applicatios, the closer to the optimal poit ad the faster that poit is reached the better. It is always possible that a optimal poit may ot be foud withi a reasoable amout of computatioal effort. Because heuristic algorithms are faster tha exact algorithms, heuristic approaches are eeded for solvig problems that are too large to be solved, i a fiite amout of time, by optimal solutio procedures. [3] ad [8] (GA ad MKHEUR) are kow heuristic approaches for solvig the MKP. The heuristic approach called At Coloy System (ACS) appears to be iterestig as well; the idea behid it was proposed by Dorigo [4][5]. To the best of our kowledge there is ust oe reported work o at systems applied to the MKP [7]. The heuristic approaches MKHEUR, GA ad ACS were tested ad their performace compared. Ufortuately, the code was ot made available to us; the algorithms were coded based o the descriptio of the methods outlied i the correspodig papers. Modificatios to the metioed heuristics were proposed ad coded as well. Comparisos ad evaluatio performace, based o solutio quality ad time complexity, are give for the modified approaches. The umerical results were obtaied usig MATLAB. The paper is orgaized as follows. I sectio 2, descriptios of MKHEUR, GA ad ACS are give; the performace produces by each approach is reported i the same sectio. I sectio 3 modificatios to the origial approaches are suggested ad evaluated. Fially, coclusio from this work is give i sectio 4.

2 2 Heuristic approaches: MKHEUR, GA ad ACS 2. Multikapsack Heuristic: MKHEUR The procedure discussed i [8] uses a surrogate costrait to determie the order i which variables are fixed equal to oe. The surrogate problem ca be defied as follows max = m m i i i i = i= i= { } x 0, =,..., px (2) subect to w r x w b w={w,...,w m } set of surrogate multipliers. The surrogate problem (2) is usually solved to obtai a upper boud o the origial MKP. Usig m u = p / wr i= i i a pseudo-utility ratio is obtaied ad used to determie the order of fixig equal to oe variables. The MKHEUR procedure is as follows:. Determie a set of surrogate multipliers. 2. Calculate the pseudo-utility ratios. Sort ad reumber variables accordig to decreasig order of these ratios. 3. Fix variables equal to oe accordig to the order determied i Step 2. If fixig a variable equal to oe causes violatio of oe of the costraits, fix that variable to zero ad cotiue. Deote the feasible solutio determied i this step as X For each variable fixed equal to oe i X 0, fix the variable equal to zero ad repeat Step 3 to defied a ew feasible solutio. Deote these feasible solutios as X z ( z={,..., q}; q equals the umber of variables equal to oe i X 0 ) As Pirkul poited out, the first solutio obtaied i Step 3 (X 0 ) is geerally ot optimal. This observatio ad the observatio that optimal solutios differ from X 0 by oly a few variables, led to Step 4. This last step attempts to capture optimal solutios by forcig variables, which have values oe i X 0 to zero. The X z ={x,..., x } z {0,..., q} that max px = is the solutio uder MKHEUR. 2.2 Geetic Algorithm: GA I this approach a heuristic operator that utilizes problem-specific kowledge is icorporated ito the stadard geetic algorithm. The basic steps of a GA are. Geerate a iitial populatio. 2. Obtai value of idividuals i the populatio. 3. Repeat: select parets from populatio recombie parets to produce a child obtai value of the child replace populatio by the child Util: a satisfactory solutio has bee foud. I [3] a -bit strig represetatio, is the umber of variables i the MKP, is used. A value of 0 or at the -bit implies that x =0 or i the solutio, respectively. A bit strig represets a member or idividual i the populatio. If a bit strig represets a ifeasible solutio, a repair operator is used to covert a ifeasible solutio ito a feasible oe. The repair operator cosists of two phases. The first phase is called DROP. It examies each variable i icreasig order of the pseudo-utility u ad chages the variable from oe to zero if feasibility is violated. The pseudo-utility is give by m u = p / λ r (3) i i i= λ i Lagrage multipliers of the LPR-MKP solutio LPR-MKP stads for Liear Programmig Relaxatio of the MKP. The LPR-MKP is expressed by () but usig the relaxatio 0 x =,... (4) The secod phase is called ADD. It reverses the process by examiig each variable i decreasig order of u ad chages the variable from zero to oe as log as feasibility is ot violated. The obective of the DROP phase is to obtai a feasible solutio from a ifeasible oe. The ADD phase improves, if possible, the solutio quality of a feasible solutio. The first step i the GA approach is the creatio of a iitial feasible populatio. The, a touramet selectio is performed i order to select the parets who will have a child i the GA. After the child is created, it suffers a mutatio. I the mutatio, some radomly selected bits are chaged. If the child after the mutatio is ifeasible, the repair operator is used.

3 If the child is a o-duplicated child, i.e., the child is ot a member of the populatio, the it replaces the populatio member that mi px =. O the other had, if the child is the same as a member of the populatio it is discarded ad the touramet selectio is performed agai. This process teds to improve the solutio quality of the populatio as time goes by. 2.3 At Coloy System: ACS Dorigo tried to reproduce the at behaviour i the ACS algorithm. Real ats are able to fid a path from a food source to a destiatio est by exploitig pheromoe iformatio. While walkig, ats deposit pheromoe o the groud, ad follow, i probability, pheromoe previously deposited by other ats. As it is explaied i [7], i the MKP there are ot liks or paths to be followed, istead there are variables that seems to be idepedet. The ACS works as follows: each at geerates a complete solutio (local solutio) by acceptig variables accordig to a probabilistic state trasitio rule. The rule is give by β ( ) τ( ) η( ) / ( ) η( ) β Pk x = x x r x x x Jk (5) x P k (x ) τ(x ) η(x ) β J k feasible -variable probability of variable x to be take by at k pheromoe value i variable x pseudo-utility value of x variable relative importace of pheromoe versus pseudo-utility set of all feasible variables that ca be take by at k There is ot a sigle rule to assig a value for τ ad η. I our implemetatio, the iitial value of τ(x ) equals the value of x give by the LPR-MKP solutio; η(x ) equals the pseudo-utility value give by (3). Before acceptig a variable, the at evaluates the variables that have ot bee take by the at yet, ad geerates a set of feasible variables. A feasible variable is oe that satisfies the at resources costrait. The iitial amout of at resources is give by b=[b,..., b m ]. Whe the at accepts the x variable the amout of its resources decreases by r=[r,..., r m ] (see ()). Based o the pheromoe ad pseudo-utility value, a probability value usig (5) is geerated for each of the feasible variables; the a radom selectio, based o the probability obtaied, is performed. It is evidet, from (5), that ats prefer to accept variables that have a high pseudo-utility value η(x ) ad a high amout of pheromoe τ(x ). Whe a at accepts a variable, it decreases the pheromoe amout o that variable by applyig a local updatig rule. The idea is to make the variable less desirable ad therefore it will be chose with a lower probability by the other ats. A local solutio is reached whe a at either allocates its iitial resources or, although there are available resources, it is ot feasible to accept more variables, i.e., resources are ot eough. After each at reaches a local solutio, a at-value is geerated usig at. value r = p x r =,..., R ( ) R represets the umber of ats (6) = The at with the highest at.value(r) is chose, ad the pheromoe amout of the variables take by this at is icreased. This is doe by applyig a global updatig rule. The goal of the global updatig rule is to make the variables, which belog to the best solutio, more attractive. The variables take by this at are recorded i a global memory. This complete oe cycle of the searchig process. A ew cycle is performed ad, agai, the at with the highest value is chose. If this value is higher tha the oe obtaied i the last cycle, the the global memory is updated with the ew variables. This process is doe cycle after cycle ad stops util either the umber of maximum cycles is reached or a certai criterio performace is obtaied. 2.4 Numerical results As suggested i [8], the dual variables from the LPR-MKP solutio were used as the surrogate multipliers. The performace of MKHEUR, GA, ad ACS is give i Tables ad 2. The approaches were tested i 4 sets of problems. Each set has 0 problems. The problems were take from the OR- Library (address The OR-Library test problems evaluated are cotaied i the files: mkapcb7 (first 0 problems: Set ), mkapcb8 (first 0 problems: Set 2) ad mkapcb9 (first 20 problems: Sets 3 ad 4). The umber of variables () goes from 00 to 500 ad the umber of costraits is 30 i all cases. For a i

4 detail descriptio about how the problems were geerated ad how they are orgaized i the metioed files, see [3] ad OR-Library web site. The heuristics were tested i a Petium II 400 MHz Persoal Computer ruig Liux OS. The performace of the approaches is measured by the percetage gap betwee the best solutio foud by the heuristic ad the optimal LPR-MKP solutio, i.e., gap=00*(optimal LPR value - best solutio value) / (optimal LPR value). Table. Heuristic Performaces (gap) MKHEUR GA ACS Set () % gap mea value (00) (250) (500) (500) Table 2. Heuristic Performaces (time) MKHEUR GA ACS Set () time (secs) (00) 0.50 >3000 <25 2 (250).47 >3000 <00 3 (500) 4.50 >3000 <300 4 (500) 5.68 >3000 <500 Because the GAP ad ACS are probabilistic algorithms, dispersio values are give i Table 3. GA ad ACS were ru 00 times for each set. For the GA, each time 0 6 o-duplicated childre were allowed. 00 feasible members were created for the iitial populatio. For the at system, each time 0 ats were used ad 50 iteratios performed. β equals log(0e2)/log(max(p.utility.value)/mi(p.utility.valu e)). Parameters eeded i the local ad global updated rules were set equal to 0. Table 3. Heuristic Performaces (dispersio values) MKHEUR GA ACS Set () Variace (00) (250) (500) (500) From Table is clear that GA is the best heuristic approach based o solutio quality. Nevertheless, GA is the heuristic that takes the logest (see Table 2). For small size problems (<250 variables) ACS appears to be a good choice. If time matters, as it usually does i real time applicatios, the MKHEUR appears to be the optio. 3 Modified Heuristic Approaches The results obtaied i the last sectio shows that MKHEUR is a good heuristic based o solutio quality ad time complexity. Nevertheless, GA produces better solutio quality if greater computatio time is allowed. GA performace ca be speeded up if more kowledge about the behaviour of the MKP solutio is icorporated ito the algorithm. GA starts with the geeratio of a iitial populatio. Throughout the algorithm the populatio performace is improved. After a certai performace is reached, almost all the idividuals of the populatio have the same characteristics (variables amog populatio members have almost the same values). The differeces are amog a subset of variables. The pseudo-utility values of this subset fall aroud a bad i betwee the highest ad lowest value of the pseudo-utility. This meas that almost always all the variables with the highest pseudo-utility are take ad almost all the variables with the lowest pseudoutility are reected. The MKHEUR solutios (Step 4 sectio 2.) could be used to geerate the iitial populatio i the GA algorithm. The populatio will take the highly desirable characteristics from the MKHEUR solutios. This will save the time origially eeded, i the GA algorithm, to start improvig the populatio performace. The solutio behaviour of the kapsack problem was origially otice by Balas ad Zemel, [2], while workig o sigle-costrait kapsack problems. It was oticed that the solutio obtaied by packig the kapsack i the decreasig order of utility ratios differed from the optimal solutio by oly a few variables. Differig variables were always aroud the poit the heuristic solutio switched from oes to zeros (after reumberig variables i decreasig value of their ratios). A core problem was defied as "the problems i those variables, whose cost/weight ratio falls betwee the maximum ad miimum c/a ratio for which x has a differet value i a optimal solutio to KP from that i a optimal solutio to LKP" (KP- sigle costrait problem; LKP-liear programmig relaxatio of KP). A similar statemet for the MKP core problem is give i [8].

5 This kowledge about the characteristics of the solutio could be used i the ext heuristic as follows. Reduce the origial problem. As it was metioed before, the variables with the lowest pseudo-utility values are almost always reected; these variables represet approximately 5% of the origial problem size. After the reductio is doe, MKHEUR is applied i the ew problem. The a re-mappig is performed i order to get the solutio, so far, i terms of the origial variables. 2. Apply a heuristic rule to get a estimatio of the core problem size ad 3. Apply GA oly o the core problem 3. MKHEURv2 For the part cocered to the reductio of the origial problem, the followig preprocessig steps eed to be doe:. Determiate a set of Lagrage multipliers 2. Calculate the pseudo-utility ratios. Sort ad reumber variables accordig to decreasig order of those ratios. 3. Fix variables equal to oe accordig to the order determied i the last step. If fixig a variable equal to oe causes violatio of oe of the costraits, fix that variable equal to zero ad cotiue. Deote the feasible solutio determied as X. Calculate the pseudo-utility mea value of X ( x% ). 4. Create a set of variables called Pseudo Reected Variables (PRV) (last 5% of the origial MKP variables with the lowest pseudo-utility ratio). 5. Merge PRVs based o their pseudo-utility ratio. Create ew variables made of packed PRVs. The umber of PRVs eeded to create a ew variable depeds o the pseudo-utility value of each PRV. The pseudo-utility value of each PRV is added up util the total pseudo-utility of the ew variable beig packed is greater tha x%. The total pseudo-utility value obtaied is the pseudo-utility value of the ew variable. I the same way, the amout of resources eeded by each PRV beig packed is added up. The total amout of resources is the oe required by the ew variable. It has to be oticed that a ew problem has bee created. The size of this oe is smaller tha that of the origial problem. The ew problem dimesio is -rv+v, i.e., origial dimesio mius umber of reected variables plus ew variables. The ext step is to apply MKHEUR to the ew problem. Fially, a mappig is performed i order to obtai the solutio foud i terms of the origial variables. As it is clearly evidet, the complexity of this modified versio (MKHEURv2) is greater tha that of MKHEUR, but as the umber of variables icreases the complexity pays off. The time used by MKHEURv2 to solve large-size problems is lesser tha the oe eeded usig MKHEUR. Moreover, MKHEURv2 seems to produce better solutio quality tha MKHEUR. A compariso performace is give i Table 4. Table 4. MKHEUR ad MKHEURv2 performace MKHEUR MKHEURv2 Set () % gap time (sec) % gap time (sec) (00) (250) (500) (500) MKHEURv2+GA MKHEURv2 was used to create the iitial populatio eeded for the GA algorithm. Because the populatio has highly desirable characteristics, the umber of o-duplicated childre for GA was reduced from 0 6 to 0 3. However, before mergig MKHEURv2 ad GA the dimesio of the core problem has to be defied. The core problem dimesio was set equal to the dimesio of the origial problem mius 80% of the first cosecutive variables equal oe i solutio X (sectio 3. Step 3) ad mius the variables used as PRVs. The results obtaied are give i Table 5. Table 5. GA, MKHEURv2 ad MKHEURv2+GA performace GAP (0 6 ) MKHEURv2 MKHEURv2 + GA (0 3 ) Set () % gap (00) (250) (500) (500) (000) For this sectio a ew set of problems was created. The set has 0 problems, each oe has 30 costraits ad it is made of Set 3 plus Set 4. Each set was evaluated 00 times. It is iterestig to otice that, although the umber of o-duplicated

6 childre i the MKHEURv2+GA algorithm has bee reduced dramatically, the performace of MKHEURv2+GA is similar as GA allowig 0 6 o-duplicated childre (problems with 250 variables or less). 3.3 ACS + MKHEUR It is also possible to merge ACS ad MKHEUR i order to try to obtai a approach that produces a better performace tha ay oe of these approaches aloe. I sectio 2.3 is reported that the LPR-MKP solutio is eeded for the pseudo-utility ratios estimatio; this is the mai step i the MKHEUR algorithm. The MKHEUR missig step ca be implemeted at the ed of the ACS algorithm. Whe a cycle is completed i the ACS, the best local solutio is recorded i memory. I the ACS+MKHEUR approach, the best local solutio is cosidered to be X 0 from Step 3 of MKHEUR (see 2.), ad the Step 4 is applied over this X 0 give by the ats. This was doe ad the solutio quality obtaied is slightly better the the oe reported for ACS aloe (about % better). Because the improvemet was miuscule the ACS + MKHEUR approach was ot further explored 4 Coclusio The MKP ca be used to express iterestig problems such as resource allocatio. The decisio about what algorithm use to solve the MKP is problem depeded. For real time applicatios, time is a critical factor to be cosidered ad heuristic methods are used more ofte over exact methods. MKHEUR produces a good solutio quality without requirig a excessive computatioal time. The LPR-MKP solutio part takes more tha 90% of the time eeded by MKHEUR. Therefore, ay improvemet i that directio is welcome. Usig kowledge about the solutio behaviour of the MKP, a mappig of the problem is suggested ad doe, ad the dimesio of the MKP is reduced. The, MKHEUR is applied ad the a re-mappig of the problem is performed i order to express the solutio i terms of the origial variables. Because MKHEURv2 works over a smaller dimesio of the origial problem, it is faster ad the solutio quality seems to be better tha that of MKHEUR. It has to be poited out that for small size problems, MKHEURv2 is slower because the time eeded to perform the mappig is greater the the time eeded for solvig the LPR-MKP. As the problem size icreases, MKHEURv2 becomes a better cadidate for solvig the MKP. If more computatioal time is allowed, the the GA algorithm ca be applied o a defied core sectio ad further improvemet could be obtaied. Heuristic approaches are divided i two sets, the probabilistic oes ad the o-probabilistic. GA ad ACS are probabilistic approaches, therefore dispersio values must be cosidered if a heuristic approach is to be chose. It is difficult to obtai reliable dispersio values for probabilistic approaches due to its ature. GA ad ACS were ru 00 times o the same problems. It appears that ACS produces a smaller dispersio value tha GA whe the latter is allowed to geerate up to 0 3 oduplicate childre. Refereces: [] Balas, E. ad Marti, C., Pivot ad Complemet A Heuristic for 0- programmig, Maagemet Sciece, Vol. 26, 980, pp [2] Balas, E., ad Zemel, A algorithm for large Zero-Oe kapsack problems, Operatios Research, Vol. 28, 980, pp [3] Chu, P., ad Beasley, J., A Geetic Algorithm for the Multidimesioal Kapsack Problem, Joural of Heuristics, Vol. 4, 998, pp [4] Dorigo, M., Optimizatio, learig ad atural algorithms. PhD Thesis. Politecico di Milao. Italy [5] Dorigo, M., ad Maiezzo, V., ad Colori, A., The at system: optimizatio by a coloy of cooperatig agets, IEEE Trasactios o Systems, Ma, ad Cyberetics - Part B, Vol. 26, 996, pp [6] Gavish, B. ad Pirkul, H., Efficiet Algorithms for Solvig the Multicostrait Zero-Oe Kapsack Problem to Optimality. Mathematical Programmig, Vol. 3, 985, pp [7] Leguizamo, G., ad Michalewicz, Z., A ew versio of at system for subset problems. Cogress o Evolutioary Computatio, Vol. 2, 999, pp [8] Pirkul, H., A Heuristic solutio procedure for the Multicostrait Zero-Oe Kapsack Problem, Naval Research Logistics, Vol. 34, 987, pp [9] Toyota, Y., A simplified algorithm for obtaiig approximate solutio to Zero-Oe programmig problems, Maagemet Sciece, Vol. 2, 975, pp