An Iterated Beam Search Algorithm for Scheduling Television Commercials. Mesut Yavuz. Shenandoah University

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1 An Iterated Beam Searh Algorithm for Sheduling Television Commerials Mesut Yavuz Shenandoah University The Harry F. Byrd, Jr. Shool of Business Winhester, Virginia, U.S.A. POMS 19 th Annual Conferene La Jolla, California, U.S.A. May 9 to May 12, 2008 Abstrat This paper is onerned with the problem of sheduling ommerial video-lips for television. The literature of the problem inludes a 40-instane test-bed and implementations of branh and bound (B&B) and simulated annealing (SA) methods. Currently, optimal solutions of 24 out of 40 instanes are known. In this paper, we propose an iterated beam searh (IBS) method that is based on the priniples of dynami programming (DP) and B&B, and an be used for exat or heuristi solution of the problem. Our experiments on the existing test-bed show that the proposed IBS method (i) finds the exat solution of one more instane, (ii) outperforms SA in a majority of the instanes, and (iii) improves the best known lower bounds and hene narrows 1

2 the optimality gap for the instanes that are not solved to optimality. The results are promising for the solution of the response time variability problem, as well. 1. Introdution Sheduling problems onstitute a signifiant portion of the operations researh literature, and there are numerous different types of sheduling problems with various appliation areas [1]. Our fous is on sheduling in the television broadasting industry. Earlier sheduling studies for television have been based on programs, see [2-4] and the referenes therein. A more reent line of researh fouses on sheduling ommerial video-lips on television [5-10]. Bollapragada and Garbinas [5] are onerned with assigning ommerials to slots in a number of breaks suh that ommerials of ompeting advertisers are not aired during the same break, and eah ommerial reeives its fair share of the first and last slots in a break, sine they are more valuable than the ones in the middle. Bai and Xie [7] and Kimms and Muller-Bungart [9] are onerned with simultaneously aepting and sheduling ommerials to maximize revenue. Zhang [8] onsiders a broader problem whih onsists of first assigning ommerials to shows and then sheduling ommerials in eah show. Bollaparagada et al. [6] onsider a sequene of slots with no breaks. They formulate a problem in order to spae subsequent appearanes of a ommerial as evenly as possible. The authors propose a branh and bound (B&B) algorithm for the exat solution of the problem, and also several heuristis for approximate solution. They also present a 40-instane test-bed for the problem. Bruso [10] onsiders the same problem and uses the test-bed established by Bollapragada et al. He proposes an enhaned B&B algorithm and a fast simulated annealing heuristi. His methods improve the best-known solutions of several instanes in the test-bed. 2

3 This paper is onerned with the problem proposed by Bollapragada et al. [6] and proposes a novel iterated beam searh (IBS) algorithm for its solution. The remainder of the paper is organized as follows. Setion 2 presents a mathematial model of the problem. Setion 3 briefly reviews the literature on beam searh and speifies our IBS algorithm. Setion 4 presents the results obtained by IBS on the same test-bed and ompares them to those of Bollaparagada et al. [6] and Bruso [10]. Setion 5 presents a losely related problem, namely, the response time variability problem (RTVP), and disusses how IBS an be used for its solution. Finally, Setion 6 onludes the paper with a summary and disussion of future researh. 2. A mathematial model Suppose ommerial, = 1, 2,.., C, should be aired n times throughout the planning C n = 1 horizon, whih onsists of N = slots. A ommon analogy used in the literature is to onsider ommerials as balls with different olors. That is, a total of N balls to be plaed into N slots. Let X i, be the slot oupied by ball i of olor, i = 1, 2,.., n, = 1, 2,.., C. In an ideal shedule, balls of the same olor are evenly spaed. Sine there are N slots in the sequene, the ideal distane between two onseutive balls of olor is N q =. Let Y i,, k = 1 if ball i of olor n is assigned to the k th slot of the sequene, and Y 0 otherwise. The following optimization model is formulated for the problem: i,, k = 3

4 Minimize subjet to C n = 1 i= 2 X X q i, i 1, (1) X i, X i 1, 1, i = 1, 2,.., n, = 1, 2,.., C (2) N n k= 1 i= 1 Yi,, k = 1, = 1, 2,.., C (3) N X i, kyi,, k = 0, i = 1, 2,.., n, = 1, 2,.., C (4) C n = 1 i= 1 N Y i, k k = 1 i k = 1 Y, = 1, k = 1, 2,.., N (5) i, k, = 1, i = 1, 2,.., n, = 1, 2,.., C (6) X, { 1,2,.., N}, { 0,1} Y i, k, i = 1, 2,.., n, = 1, 2,.., C (7), i = 1, 2,.., n, = 1, 2,.., C, k = 1, 2,.., N (8) The objetive funtion (1) is the summation of absolute differenes between the atual and ideal distanes between subsequent balls of eah olor. Note that it does not inlude a term for the first ball of a olor. Also note that the objetive funtion formulated above is non-linear, but ould be easily replaed by a linear equivalent [6]. Constraint (2) defines the order between balls of the same olor. Constraint (3) guarantees that all balls are plaed. Constraint (4) ties deision variables X i, and Y i, k, together. More speifially, the binary assignment variable i, k Y, is used to determine the value of the shedule variable that exatly one ball is plaed into a slot and eah ball is plaed into a slot in the sequene. Finally, onstraints (7) and (8) define the deision variables. 4 X i,. Constraints (5) and (6) jointly assure

5 3. An iterated beam searh algorithm 3.1 Beam searh Beam searh (BS) is essentially a trunated B&B method. It starts with a null solution and onstruts omplete solutions by fixing one variable per stage. In eah stage partial solutions are extended using a branhing strategy. All the newly reated partial solutions are evaluated and only the best β (i.e., beam width) of them are kept for further branhing. In other words, the searh tree is pruned at eah stage in order to keep it at a manageable size. An optimal solution may not be reahed if all the partial solutions leading to it are pruned at intermediate stages, whih makes BS a heuristi method. Strengths of the BS method are its speed and intelligene that an be inorporated into the evaluation of partial solutions. The fastest implementation of BS an be obtained by letting β = 1 and evaluating partial solutions only based on loal information. This is the well-known greedy method. BS is a broad family of methods ranging from an extremely fast greedy heuristi to an enumerative exat method, haraterized by the beam width and the partial solution evaluation strategy. BS has been implemented on a variety of problems, and has been enrihed by adding new omponents to improve its speed and/or auray. Two most ommon implementations are filtered beams searh (FBS) and reovering beam searh (RBS). FBS uses a simple evaluations strategy to limit the number of partial solutions at eah stage, that are evaluated via a more ompliated and aurate strategy. RBS is a further improvement that uses loal searh on partial solutions in order to reover from previously made poor hoies. We refer the reader to [11-14] and the referenes therein for examples of reent FBS and RBS implementations. 5

6 3.2 Our IBS approah Existing BS implementations mentioned above are all single-pass methods. That is, partial solutions reated at a stage of the tree are either further evaluated immediately, or disarded. With a single pass, some good partial solutions may be trunated due to the poor performane of the evaluation strategy employed. In this paper, we aim to improve BS by overoming this defiieny via multiple passes (iterations). In the ore of BS is a tree, whih an be searhed either partially or ompletely. An empty sequene is the root of the tree, and eah slot in the sequene adds a level to the tree. In our problem, we have C different ommerials, hene, at most C branhes emanating from any non-leaf node in the tree. The number of leaf nodes is C = 1 N! ( n!). The large number of leaf nodes, i.e., number of feasible solutions, renders expliit enumeration impratial for larger-sized instanes of the problem. Dynami programming (DP) is known as an impliit enumeration method exploiting the problem struture to redue the size of the searh tree. Consider a partial solution where 1 < m < n balls of olor, = 1, 2,.., C, are sheduled. Let M = C m = 1 be the length of the partial sequene. In order to extend this partial solution, we must know the objetive funtion value of the partial sequene, numbers of balls already sequened (m ) and the position of the last ball in the partial sequene (say, l ) for eah olor. Preeding balls an be sequened in ( ) ( M C)! Ψ m, l = different ways. That is, Ψ( m, l) C ( ( m 1 )! ) = 1 distint intermediate nodes an be represented by only one node in a DP network. We take advantage of this fat in our IBS method. 6

7 We onstrut a DP network of N stages with (possibly) multiple nodes in eah stage and ars pointing from stage t-1 to t only, for t = 1, 2,.., N. A node at the N th stage is a leaf node, i.e., a omplete solution. Eah node is represented by m and l vetors and has a lower bound (LB) value. Nodes are explored in forward passes. When a node is explored, all the branhes from that node are onsidered, i.e., at most C nodes at the next stage are reated, and the node itself is eliminated. Note that only one value (m and l ) in eah vetor differs between the parent and a hild node. Calulation of LB is disussed later in the paper. Designing a multiple-pass method requires speial attention to not reate the same node multiple times. We ahieve this by using two linked lists (of nodes) at eah stage. The first one ontains unexplored nodes and it is ordered from smallest to largest in the LB value. The seond one ontains already explored (and eliminated) nodes, and is unordered. When a new node is reated, it is first ompared to the nodes in the list of explored nodes. If it is already in the list, it is simply eliminated. Otherwise, it is ompared to the nodes in the list of unexplored nodes. If it is not in the list, it is added to the list in the order of its LB. If it is already in the list, the LB values of the two are ompared and the better one is kept in the list. The lowest lower bound in eah stage is easily found in the head node of the ordered list in that stage. Let LLB be the lowest of all lower bounds. LLB is initially equal to the lower bound of the root node, and is updated at the end of eah iteration, i.e., forward pass. A forward pass starts at the earliest stage with a non-empty set of unexplored nodes. The first β nodes in the list are explored. We inorporate an adaptive strategy at this point and explore all the nodes with an LB equal to LLB. That is, at eah stage at least β nodes are explored and no node remains with the lowest lower bound. This adaptive strategy ensures that the LLB is inreased at eah iteration. At the end of eah iteration, the objetive funtion value 7

8 of the best new omplete solution is ompared to that of the previously known best solution, and the upper bound (UB) is updated if neessary. If the UB is updated at the end of an iteration, a quik pass is made over the lists of unexplored nodes to prune out the ones that annot lead to an optimal solution. Calulation of the LB for a newly reated node is an involved proess that requires solving an assignment problem. First, let us fous on one olor,. In a partial solution, m balls th of olor are already plaed into the first M positions, with the last of them being in the l position. Minimization of the total deviation from the remaining n -m balls requires keeping the distanes between them as lose to the ideal distane N/n as possible. Obtaining this lower bound does not require sheduling the remaining balls, and, hene, an be performed in O(1) time. Furthermore, if we assume sequening m +1 st ball of olor in position t > M, the total deviation from the remaining n -m -1 balls an be alulated in O(1) time, as well. Let us denote that lower bound by F (m +1,t). Also let the total deviation resulting from the first m balls of olor already sequened be P (m,l ). Then, LB (m,l,t) = P (m,l ) + t - l - q + F (m +1,t) is a lower bound for all the balls of olor, if the next one is sequened in position t. Note that this lower bound an be obtained in O(1) time. Moreover, an ideal position t* for the next ball of olor an easily be obtained. Ideal positions of the next opies of different olors may overlap. Therefore, simply summing lower bounds LB (m,l,t* ) gives a loose lower bound. A better lower bound is obtained by identifying a set of andidate positions for the next balls of different olors, alulating the lower bound for eah olor and eah andidate position, and finally solving an assignment problem. The size of the andidate positions list is O(C), and the resulting assignment problem is solved in O(C 3 ) time. 8

9 In our initial study of the IBS method, we tried simpler and faster LB alulation strategies. The strategy desribed above yielded the best results, and hene, we dropped the others from further onsideration. Our node evaluation strategy is solely based on the LB. The node with smallest LB at a stage is the first one to be explored. This serves our purpose to improve the best known LB for a problem, while searhing for a good solution. We finally note that we use omputation time as the only termination ondition. That is, IBS is allowed to run until a designated amount of time passes. As we have disussed throughout this setion, our IBS method has a simple struture and it takes only two parameters, namely β and run time. 4. Results We use the 40-instane test-bed developed by Bollapragada et al. [6] to test the performane of our IBS method. We oded the method in Mirosoft C# and onduted the experiments on a desktop omputer with Intel Dual Core2 CPU at 1.86 GHz and 3GB of RAM. Our preliminary experiments showed that β = 10 was a good hoie. When implemented the IBS method, we fed the best UB reported in [6] as input for eah instane. This allows IBS to find good (if not optimal) solutions faster. In order to explore the performane of the method to quikly find good solutions, we allowed a maximum run time of 120 seonds. We present the results for all 40 instanes in Table 1. The first olumn shows the instane number. The next three olumns are the best known results from the literature. That is, LB is from [6], UB is from [10], and gap is the ratio of their differene to LB (gap = (UB-LB) / LB). The last four olumns in the table present the performane of IBS. Time is reported in seonds. LB, UB, and gap values in bold are the ones improved over the best known values. 9

10 Table 1 Computational results Instane Best Known IBS # LB UB Gap Time LB UB Gap

11 The table shows that IBS is suessful in finding optimal solutions of all 24 instanes (#1 to 23, and 27) previously solved in [6] and [10]. In addition, #26 is solved to optimality in 107 seonds. Optimal solutions of 15 instanes (#24, 25, and 28-40) remain unknown. However, we see from the table that IBS provides improved lower bounds for them. As to the upper bounds, we see that IBS yields an improved solution in nine instanes, repeats the best known solution in one, and does not reah the best known solution in the remaining five. The optimality gap is improved in 12 of the instanes and remained approximately the same (due to rounding) in the other three. Overall, the IBS method demonstrates good performane, and appears promising for future researh on this and other hard optimization problems. 5. The response time variability problem In many real-life sheduling problems, it is aimed to find a fair alloation of resoures to ompeting produts, ustomers, et. Response time is defined as the distane between two onseutive times a resoure is alloated to a produt or ustomer. In the ase of sequening ommerial video-lips, response time is the number of slots between two appearanes of a ommerial. The RTVP is first formulated by Corominas et al. [15]. Their formulation is different from the problem addressed in this paper in two ways. First, the RTVP is onerned with a repetitive sequene. That is, slot N of the preeding exeution of the sequene is slot 0 of the sueeding exeution. Therefore, a distane is defined for the first ball of a olor and it is inluded in the objetive funtion. Seond, the objetive funtion that Corominas et al. use sums 11

12 the squared deviations between the ideal and atual response times, rather than the absolute values. This approah puts more emphasis on larger deviations. Using the sum of squared deviations approah in the sheduling of ommerial videolips is addressed by Bruso [10]. The author states that the branh and bound method used to solve the problem runs muh slower in that ase. In this regard, it would be interesting to use the absolute value approah in the RTVP and see if that modified problem is easier to solve. IBS an be implemented to the RTVP with relative ease. The node struture should be enhaned to inlude the first position allotted to a ommerial in a partial sequene. This is needed to alulate the distane between the last and first positions, and to ompute its objetive funtion ontribution. The onnetion between the beginning and the end of a sequene gives it a yli struture, whih an be used to fix the first position. That is, without loss of generality, a ommerial of type one an be assigned to the first position in the sequene. This makes the root node a partial sequene of length one instead of an empty sequene, and redues the searh spae. 6. Conlusions This paper presents a novel iterated beam searh method for sheduling ommerial video-lips in television broadasting. The proposed method is shown to be both fast and aurate. Furthermore, the method improves the best known lower bound in all test instanes with an unknown optimal solution. The IBS method an be improved by adding a few new omponents. Loal searh is a omponent that is often used to enrih meta-heuristi methods. In the ase of IBS, it an be used at the very end to improve the best solution found throughout all the iterations. Alternatively a 12

13 loal searh an be performed at the end of eah iteration, on the best solution found in that iteration. The latter would be more time onsuming, but also expeted to yield better upper bounds earlier. Partial solutions are evaluated solely based on their lower bound values in the proposed method. If a fast method an be developed to omplete a partial solution, it an provide an upper bound for eah solution. Then, those upper bounds an be used to evaluate partial solutions. These improvements are future researh diretions on IBS in solving the problem addressed in this paper. A third and final potential future researh diretion we point at is using IBS to solve the response time variability problem. Referenes [1] Leung, J. Y.-T., 2004, Handbook of sheduling: algorithms, models, and performane analysis, Chapman & Hall/CRC omputer and information siene series, Boa Raton: Chapman & Hall/CRC. [2] Reddy, S. K., Aronson, J. E. and Stam, A., 1998, SPOT: Sheduling programs optimally for television, Management Siene, 44(1), [3] Danaher, P. J. and Mawhinney, D. F., 2001, Optimizing television program shedules using hoie modeling, Journal of Marketing Researh, 38(3), pp [4] Ytrenerg, E., 2002, Continuity in environments: The evolution of basi praties and dilemmas in Nordi television sheduling, European Journal of Communiation, 17(3), pp

14 [5] Bollapragada, S. and Garbinas, M., 2004, Sheduling ommerials on broadast television, Operations Researh, 52(3), [6] Bollapragada, S., Bussiek, M. R. and Mallik, S., 2004, Sheduling ommerial videotapes in broadast television, Operations Researh, 52(5), [7] Bai, R. and Xie, J., 2006, Heuristi Algorithms for Simultaneously Aepting and Sheduling Advertisements on Broadast Television, Journal of Information and Computer Siene, 1(4), pp [8] Zhang, X., 2006, Mathematial models for the television advertising alloation problem, International Journal of Operational Researh, 1(3), [9] Kimms, A. and Muller-Bungart, M., 2007, Revenue management for broadasting ommerials: the hannel s problem of seleting and sheduling the advertisements to be aired, International Journal of Revenue Management, 1(1), [10] Bruso, M. J., Sheduling advertising slots for television, Journal of the Operational Researh Soiety, forthoming. [11] Valente, J. M. S. and Alves, R. A. F. S., 2005, Filtered and reovering beam searh algorithms for the early/tardy sheduling problem with no idle time, Computers & Industrial Engineering, 48(2), [12] Ghirardi, M. and Potts, C. N., 2005, Makespan minimization for sheduling unrelated parallel mahines: A reovering beam searh approah, European Journal of Operational Researh, 165(2),

15 [13] Esteve, B., Aubijoux, C., Chartier, A. and T Kindt, V., 2006, A reovering beam searh algorithm for the single mahine just-in-time sheduling problem, European Journal of Operational Researh, 172(3), [14] Erel, E., Gogun, Y. and Sabunuoglu, I., 2007, Mixed-model assembly line sequening using beam searh, International Journal of Prodution Researh, 45(22), [15] Corominas, A., Kubiak, W. and Palli, N. M., 2007, Response time variability, Journal of Sheduling, 10(2),