Srinivas Bollapragada GE Global Research Center. Abstract

Transcription

1 Sheduling Commerial Videotapes in Broadast Television Srinivas Bollapragada GE Global Researh Center Mihael Bussiek GAMS Development Corporation Suman Mallik University of Illinois at Urbana Champaign Abstrat This paper, motivated by the experienes of major US based broadast television network, presents algorithms and heuristis to shedule ommerial videotapes. Major advertisers purhase several slots to air ommerials during a given time period on a broadast network. We study the problem of sheduling advertiser's ommerials in the slots it purhased when the same ommerial is to be aired multiple times. Under suh a situation, the advertisers typially want the airings of a ommerial to be as muh evenly spaed as possible. Thus, our objetive is to shedule a set of ommerials on a set of available slots suh that multiple airings of the same ommerial are as muh evenly spaed as possible. A natural formulation of this problem is a mixed integer program that an be solved using third party solvers. We also develop a branh and bound algorithm based on a problem speifi bounding sheme. Both approahes fail to solve larger problem instanes within a reasonable timeframe. We present an alternative mixed integer program that lends itself to effiient solution. For solving even larger problems, we present multiple heuristis. Various extensions of the basi model are disussed. Published: 2002 URL: pdf

2 Sheduling Commerial Videotapes in Broadast Television Srinivas Bollapragada GE Global Researh Center 1 Researh Cirle, Shenetady, NY Mihael Bussiek [email protected] GAMS Development Corporation 1217 Potoma Street NW, Washington DC Suman Mallik 1 [email protected] Department of Business Administration, University of Illinois 350 Wohlers Hall, 1206 South Sixth Street, Champaign, IL November 2002 Abstrat This paper, motivated by the experienes of major US-based broadast television network, presents algorithms and heuristis to shedule ommerial videotapes. Major advertisers purhase several slots to air ommerials during a given time period on a broadast network. We study the problem of sheduling advertiser's ommerials in the slots it purhased when the same ommerial is to be aired multiple times. Under suh a situation, the advertisers typially want the airings of a ommerial to be as muh evenly spaed as possible. Thus, our objetive is to shedule a set of ommerials on a set of available slots suh that multiple airings of the same ommerial are as muh evenly spaed as possible. A natural formulation of this problem is a mixed integer program that an be solved using third party solvers. We also develop a branh-and-bound algorithm based on a problem speifi bounding sheme. Both approahes fail to solve larger problem instanes within a reasonable timeframe. We present an alternative mixed integer program that lends itself to effiient solution. For solving even larger problems, we present multiple heuristis. Various extensions of the basi model are disussed. Keywords: Sheduling, Mixed integer programming, heuristis, broadast television 1 The order of authors merely reflets alphabeti sequene

3 1. INTRODUCTION The objetive of this paper is to develop heuristis and algorithms for sheduling ommerial videotapes in broadast television. Our work is motivated by the problem faed by National Broadasting Company (NBC), one of the leading firms in the television industry. Major advertisers suh as Protor & Gamble and General Motors buy hundreds of time slots to air ommerials on a network during any broadast season. The atual ommerials to be aired in these slots are deided at a later stage. During the broadast season the lients ship videotapes of the ommerials to be aired in the slots that they had purhased. Eah tape has a single ommerial and has a ode written on it for identifiation. These odes are alled Industry Standard Commerial Identifiation (ISCI) odes. The ommerials are sheduled by ISCI odes by the TV network personnel aording to the instrutions given by the advertiser. The advertisers most often speify the following guideline. Whenever a ommerial is to be aired multiple times within a speified period (for example: a month), the advertiser wants these airings to be as muh evenly spaed as possible over that time period. Thus, a lient has ertain number of advertising slots that it had purhased during a speified time period. It also has a set of ommerials to be sheduled in its time slots. The question naturally arises: how to shedule the ommerials on the available advertising slots suh that two airings of the same ommerial are as muh evenly spaed as possible. We propose to study this problem. Stated in more formal terms, we have N balls (ommerials) out of whih n 1 are of olor 1 (ISCI ode 1), n 2 are of olor 2 (ISCI ode 2), and so on. We want to put the N balls into N slots suh that balls of any one olor are as muh evenly spaed as possible, i.e. the distane between subsequent balls of olor is as lose as possible to N/ n. This paper presents algorithms and heuristis for aomplishing this. We will all this problem to be the basi ISCI Rotator problem or, simply, the ISCI problem. To motivate the disussion we provide the following data from the said firm. Table 1(a) shows the individual ommerials of an advertiser with their ISCI odes and the number of times eah ommerial to be aired. In this example, N = 17, the total number of slots, while n 1 = n 5 = 3, n 2 = 5, and n 3 = n 4 = n 6 =2. Table 1(b) shows the 17 advertisement slots purhased by the advertiser between May 11 and May on the Pax Network, 2

4 whih is partially owned by NBC. The olumn labeled Pod refers to a ommerial break within a program. Thus, the first entry in the table denotes that the lient purhased an advertising slot in the 5 th ommerial break (pod 5) for Touhed By An Angel on May 11. The last olumn, labeled Shedule represents an optimal solution. The literature on sheduling problems in the television industry mainly deals with sheduling programs (rather than ommerials) for the television to optimize some speified riteria (the viewership ratings or a network s share of audienes). Typial examples are Goodhardt et al. (1975), Headen et al. (1979), Henry and Rinnie (1984), Webster (1985), Rust and Ehambadi (1989). Reddy, Aronson, and Stam (1998) desribe strategies for optimal prime time TV program sheduling. Strategies for sheduling advertisements (though not neessarily the television ommerials) have also been studied in marketing literature. This literature is onerned with whether the advertising should be steady or pulsed (i.e. turned on and off) so that the effetiveness of the advertising is maximized. Some examples are Simon (1982) and Mahajan and Muller (1986). Lilien, Kotler, and Moorthy (1992) provides for a review of these models. There is a vast array of OR-based literature on sheduling. Lawler et al. (1993) provides for a omprehensive review of this literature. The work that omes losest to our urrent work is the problem of obtaining optimal level shedules for mixed model assembly lines in JIT systems studied by Miltenburg (1989) and Kubiak and Sethi (1991). They onsider C produts with demands n 1, n 2,.., n C to be produed during a speified time horizon. Eah produt takes a unit of time to be produed so that the speified time horizon is N = C n = 1. Define r = n /N. The objetive is to keep the proportion of umulative prodution of produt to the total prodution as lose to r as possible. Miltenburg (1989) proposes a quadrati integer programming formulation of the problem and several approximate solutions to it. Kubiak and Sethi (1991) show that the same problem an be transformed into an assignment problem and, hene, an be solved effiiently. We differ from these studies in the following way. In the ISCI problem there is no fixed demand shedule determined by r. Rather than minimizing the deviation from this fixed demand shedule, the plaement of a ball in the ISCI problem is based on the deviation of the distane of subsequent balls of the same olor to the ideal 3

5 distane. Nevertheless, to ompare our work with the literature, we desribe a heuristi solution based on the work by Kubiak and Sethi (1991) in Setion 4. It is evident from the previous disussion that while sheduling programs and effetiveness of various advertising poliies are well studied, atual sheduling of ommerials in the time slots purhased by a lient has reeived very little attention in literature. Reently, Bollapragada et al. (2002) have developed a math programming based algorithm to rapidly generate near-optimal sales plans that meet advertiser requirements. A sales plan onsists of a omplete shedule of ommerials to be aired for an advertiser to meet its requirements. The requirements inlude budget goals, audiene demographis, and the mix of shows, ommerial lengths and the weeks during the broadast year that the lient is interested in. They implemented the sales planning and demand predition algorithms in a system that is urrently being used by NBC generating more than $50 million in additional revenues annually. They also introdued the ISCI problem and presented a simple heuristi. The objetive of our work is to analyze the ISCI problem using a formal optimization framework. Note that the fous of this paper is to provide effiient solutions for the sheduling problem disussed. In line with our observation in the industry, we treat the objetive (i.e. to have airings of the same ommerial evenly spaed) as given. Thus, we are not onerned with whether this is the best advertising strategy for the advertiser. The remainder of this paper is organized as follows. Setion 2 desribes an intuitive approah based on a mixed integer programming formulation. In Setion 2.1 we present a branh-and-bound algorithm based on a problem speifi bounding sheme. In Setion 3 we develop an alternative mixed integer programming model for the ISCI problem. We explore several heuristi solutions for large ISCI instanes in Setion 4. All omputational results are presented in Setion 5. Setion 6 desribes two extensions of the basi model while the summary and the onlusions are presented in Setion MODEL FORMULATION Consider a set of N balls out of whih n 1 are of olor 1, n 2 are of olor 2, and so on. We want to plae these N balls into N slots suh that the balls of any one olor are as 4

6 muh evenly spaed over the slots as possible. We define the following notation to formulate our first intuitive model. 2.1 Notation n N = index on olor, = 1, 2,, C = number of balls of olor = total number of balls = i = index on balls of olor, i = 1, 2,, n, j, k = index on slots, j, k = 1, 2,, N n, also equals the total number of slots q = ideal distane between any two balls of olor = N / n Z i = slot number of ball i of olor (deision variable) Y i k, 1, if ball i ofolor is assigned toslot k = 0 otherwise We formulate the basi ISCI Rotator problem as an integer program with non-linear objetive funtion. Problem P1 Minimize Z Z 1 q (1) Subjet to: i i i Zi 1 Zi 1, i (2), Yi k = n, i, k (3) Z i = ky, i, (4) k i k Yi k = 1, k i, (5) Yi k = 1, i, (6) k 1 N, i binary (7) Z i Y k 5

7 Note that we defined q to be the ideal distane between any two balls of olor so that the balls of olor are evenly spaed over the slots. This distane need not be an integer. However, the slot numbers (indexed by k) are always integers. Our objetive is to have the spaing between any two balls of olor to be as lose to q as possible. The quantity Z i i 1 Z q is the deviation of the distane between (i-1) th and i th ball of olor from its ideal spaing. The objetive funtion (1) in the above formulation, thus, is the sum of deviations from the ideal spaing for eah ball of eah olor. Constraint (2) ensures an ordered arrangement of the balls. Constraint (3) ensures that all balls are used. Constraint (5) ensures that a slot an hold only one ball, while (6) ensures that eah ball an be plaed in only one slot. Note that the integrality of the variables Zi is guaranteed through onstraint (4), whih desribes the relationship between the Y and Z variables. The nonlinear objetive of P1 an be transformed to a linear problem using a standard tehnique desribed below. Problem P2 i + Minimize ( π ν ) (8) i Subjet to: Constraints (2), (3), (4), (5), (6) i π ν = Z Z q, i (9) i i i i 1, π 0, ν 0, i (10) i i, In the above formulation we have introdued two additional sets of variables π and ν to aommodate the positive and negative part of the distane alulation. Together with the right diretion of π and ν in the objetive funtion, we transformed the nonlinear (absolute value) funtion into a linear representation. The following orollary follows diretly from our model. Corollary 1: The problem P2 an be solved trivially when n = n,, i.e. when we have the same number of balls of eah olor. 6

8 The proof follows trivially. Consider an arrangement of balls in whih the sequene of olors 1, 2,, C are repeated n times. This is an optimal arrangement as it gives rise to an objetive funtion value of zero. In other words, when n = n,, we an obtain the optimal arrangement of balls for an N-slot-problem by solving a problem with N/n slots (ontaining one ball of eah olor) and repeating the arrangement n times. Under suh a situation, we say that solving a problem of size N/n is equivalent to solving a problem of size N. However, a generalization of Corollary 1 is not possible. We state this result as Corollary 2. Corollary 2: If there is a ommon fator, α (with α > 1), among the n s, then solving a problem of size N/α is not neessarily equivalent to solving a problem of size N. Proof: The proof is by onstrution. Consider the following example: N = 14, with n 1 = 6, n 2 = n 3 = 4. Here, α = 2, q 1 = 2.33, q 2 = q 3 = 3.5. We want to show that optimal arrangement of balls for this problem annot be obtained by solving a problem with N = 7, and n 1 = 3, n 2 = n 3 = 2. It an be verified (using any ommerial MIP solver) that the optimal objetive value for the original problem with N = 14 is Also note that in any feasible solution to this problem, the ontribution to objetive funtion of equation (7) from the balls of olor 2 and 3 will be at least 1.5 eah (sine q 2 = q 3 = 3.5, any arrangement of ball of olor 2 or olor 3 will ontribute at least 0.5 x (4-1) = 1.5 to the objetive). Thus, in an optimal arrangement, the ontribution to objetive funtion from olor 1 balls will be at most Note that q 1 = This means that ontribution to objetive funtion from olor 1 balls will be minimum when the spaing between the balls is 2. Thus, we an have only 4 possible arrangements of olor 1 balls in the slots: (1,3,5,7,9,11), (2,4,6,8,10,12), (3,5,7,9,11,13), and (4,6,8,10,12, 14), eah of whih will give a ontribution of 1.67 to the objetive funtion. Therefore, in an optimal solution, we must have one of these 4 arrangements for olor 1 balls. Note however, that none of these arrangements an be obtained by juxtaposing optimal solutions for the problem with N = 7 twie (as none of the 4 arrangements of olor 1 balls are symmetri about halfway). Therefore, it is not possible to find the optimal arrangement of balls for the original problem of size 14 by solving a problem of size 7. 7

9 We implemented the mixed integer programming (MIP) model P2 in GAMS (see Brooke et. al., 1988) and solved with MIP solver GAMS/CPLEX (2002). The implementation details are disussed in Setion 5. We used 40 test problems with size ranging from 8 slots to 500 slots to test our model. The test problems are presented in Table 2 and are disussed at length in Setion 5. Problems with slots are typial in the broadast industry. Suh problems need to be solved in a few minutes as shedulers use the algorithm in an interative mode to shedule videotapes for eah of the several hundred lients sequentially. Using GAMS/CPLEX on model P2, we were only able to solve 13 test problems within one minute. Twenty-four out of the 40 test ases did not solve to optimality in ten minutes of CPU time. We, therefore, explored alternative solution methodologies to obtain optimal or lose to optimal solutions. In the next subsetion we present an algorithm that takes advantage of the struture of the problem to obtain the optimal solution. 2.1 A Modified Branh and Bound Algorithm The linear programming relaxation of P2 at the nodes of the branh and bound tree does not result in tight lower bounds. As a result we developed a different bounding sheme to be used with a branh and bound tree. We refer to this algorithm as the modified branh and bound algorithm. This algorithm onstruts the solution by plaing one ball at a time starting with the first slot. At the root node, we deide the olor of the ball to be assigned to slot 1. Sine there are C olors, there are at most C branhes emanating from this node, one for eah olor ball. At level 2, we deide the ball to be plaed in slot 2, given that slot 1 is already filled. Thus the depth of the omplete tree to be evaluated is N. Let S = s, s,..., s ) be the vetor of assignments of balls in the slots. Thus, s k = implies ( 1 2 N that a ball of olor is plaed in slot k ( s = 0 implies that slot k is empty, with no balls k assigned). Also, let M = m, m,..., m ) denote the vetor of yet unassigned balls. Thus, m ( 1 2 C = λ implies that there are λ number of olor balls yet to be assigned to slots. At level i of the searh tree, all sk for whih k < i are fixed and a deision on si has to be made. Visiting all nodes in the tree is equivalent to evaluating all the feasible solutions. 8

10 However, if we an ompute a good lower bound on the solution at eah node, the searh spae ould be pruned. We use the following sheme to obtain the lower bounds Computing the Lower Bound At level i of the searh tree: Set urrent lower bound (LB) equal to the objetive funtion ontribution from the first i-1 slots For eah olor 1. Plae the remaining m as lose to q balls in the remaining N-i open slots to ahieve spaing as possible between these balls (i.e. assume that all of the remaining N-i slots are available to eah olor). Let OB be the ontribution to objetive funtion from these m balls. 2. Update lower bound by setting LB = LB + OB The Algorithm Let Obj(S) and LB(S) denote, respetively, the objetive funtion value and the lower bound (alulated using the proedure desribed in Setion 2.1.1) for a vetor of assignments S. The modified branh and bound algorithm involves the following steps. 1. Start with any feasible solution. Let B denote the urrent best objetive value. 2. Set s = 0, k = 1, 2,, N k 3. At any branh level k, k = 1, 2,, N-1 a. For any olor with m > 0, set s k = b. If LB(S) < B, then set B = Obj(S). Go to Step 3, repeat steps 3 for k = k+1 (i.e. go to the next level of the tree). Otherwise, go to Step 3(a), repeat for the next olor with m > 0. The modified branh-and-bound algorithm runs signifiantly faster than GAMS/CPLEX an solve model P2 at the ost of exhanging an off-the-self produt with a ustom build branh and bound ode. The number of instanes solved to optimality inreases from 16 to 20. (refer to Setion 5 for the full omputational results.) 9

11 However, the algorithm failed to solve the last 20 test problems within 10 minutes and does not produe near optimal solutions. As a result we developed an alternative and less intuitive integer-programming model for the ISCI problem that lends itself to a more effiient solution. We all this model the flow formulation of the ISCI problem. We desribe this model in the next setion. 3. FLOW FORMULATION OF THE ISCI PROBLEM We define the following notation in addition to our notation of Setion 2. 1, if ball of olor is assigned toslot j p j = 0, otherwise 1, if ball ofolor is shipped from slot f jk = 0, otherwise j toslot k Problem P3 Minimize k j q Subjet to: 0< j< k N f jk (11) pj = 1, j (12) 1 j N 0 k< j 0< k N j< k p j = n, f kj = f jk + f, j,0,, j< k N (13) j (14) f, 0, k = 1, (15) fjk + f, j,0 = pj,, j (16) f, p binary (17) In the above formulation, for eah olor, we have a set of ars as shown in Figure 1. Slot 0 represents an artifiial soure and sink in the network. Ars go from slot j (inluding 0) to all other slots k, j < k N, as well as bak from j to 0. The network ars 10

12 f (, j, k) ship one ball of olor from slot 0 through a number of slots bak to 0. At eah slot and for eah olor the flow onservation onstraint holds (equation 14). In addition, a flow-arrying ar f(,j,k) an leave position j if and only if the position j is olored (equation 16). This oupled with (13) ensures that the hain of ars arrying a ball of olor goes through exatly n slots. Equation (12) ensures that a slot is oupied by only one ball while (15) ensures that only one ball of eah olor leaves the soure 0. Traversing an ar f(,j,k) represents the plaement of onseutive balls of olor in positions j and k; hene we have a ost of k j q for using that ar. The objetive in (11) is to find flows of minimum ost that obey the onstraints of the problem. The number of variables in formulation P3 is muh larger than that of formulation P2. The main advantage of this formulation, as the reader will see in Setion 5, is that the relaxation provides a deent bound on the problem. Note that the underlying graph of this formulation (Figure 1) is very dense as we have ars going out from every slot j to every other node k (with k > j), and bak to slot 0 resulting in a model with a large number of variables. Ars that are unlikely to arry flow may be removed (at the risk of loosing the optimal solution for P3) to redue the size of the model. Good andidates for removal are ars jk with ost k j δ. Choosing appropriate small values ofδ that make the graph sparse but retain the optimal solution is the main hallenge in this approah. Using formulation P3, we were able to solve more than half of the test problems to optimality within one minute using GAMS/CPLEX. We were also able to solve three more problems to optimality within ten minutes of CPU time and found feasible solutions for five more problems. Table III desribes the evolution of our optimal solution strategies for the ISCI problem and ompares them. While our best optimal solution tehnique (solving P3 using GAMS/CPLEX) was able to solve the problems we usually enountered at NBC, the gap between quikly solvable and unsolvable is extremely small (e.g. ompare instanes 23 and 24 in Table IV). Therefore, we explored several heuristi solutions desribed in the next setion. q 11

13 4. HEURISTIC SOLUTIONS FOR THE ISCI PROBLEM We desribe four heuristis for the ISCI problem in this Setion. The first heuristi is based on a simple greedy searh algorithm, while the seond and third utilize the effiieny of the flow formulation P3 desribed earlier. The fourth heuristi is motivated by the work of Kubiak and Sethi (1991). All omputation results will be presented in Setion Greedy Heuristi Under a greedy heuristi we fill the slots sequentially. For eah slot, we hoose a ball olor (from the pool of available balls of different olors) that gives the least ontribution to the objetive funtion defined by equation (1). The speifi steps are desribed below. 1. Plae a ball of olor 1 (or of any other olor) in slot Update the number of balls left for the urrent olor 3. Proeed to the next slot. Among the available balls, selet that olor whih gives minimum ontribution to objetive funtion based on the arrangement thus far. 4. Stop if this is the last slot, else go to Step P3 Delta Heuristi The P3 Delta heuristi is diretly derived from the flow model P3 with a sparse underlying network. Ars with high osts (i.e. k j q δ = max( 20,0.05 q ) have been removed from the network. 4.3 P3 Bathing Heuristi We developed this heuristi with the large problems in mind. We divide a large problem into several smaller sub-problems (or bathes) so that eah bath an be solved effiiently by using formulation P3. The speifi steps are desribed below. 1. Divide a problem into several bathes whenever N > 60. Therefore, a problem with N 60 is solved in one bath, while a problem with 60 < N 120 is solved in two bathes and so on. For example, our test problem 36 onsists of 400 balls with 229 red, 149 blue, 11 white, 8 green and 3 yellow balls. This problem will be divided into 7 bathes. 12

14 2. Distribute the balls of eah olor evenly over the bathes. For the example disussed above, the first five bathes will have 33 red balls eah and the remaining bathes will have 32 red balls eah. Similarly, the first 2 bathes will have 22 blue balls eah and the remaining bathes will have 21 blue balls eah. The first bath will have 2 green balls while the remaining bathes will have only one. Only bathes 2, 4, and 7 will have one yellow ball eah. Finally, bathes 1,2,4, and 6 will have 2 white balls eah while bathes 3,5, and 7 will have one white ball eah. A simple GAMS program aomplishes this distribution. 3. Solve model P3 using a sparse network with δ = max( 20,0.2 q ) in the following steps a. Solve P3 for first 60 slots, and using the first bath of balls only. b. Solve model P3 for first 120 slots using balls from bath 2 and the solution for the first 60 slots obtained from step 3a. Treat the solution for first 60 slots as fixed (unhangeable).. Next, solve model P3 for first 180 slots and treating the solution for first 120 slots as fixed. Repeat till we have overed all slots. The philosophy in this heuristi is similar to divide and onquer. The hoie of the bath size 60 is driven by the tradeoff of avoiding many bathes and being able to solve eah bath effiiently. 4.4 Assignment Heuristi Miltenberg (1989) and Kubiak & Sethi (1991) onsider the problem of obtaining optimal level shedules for mixed model assembly lines in JIT systems. We have defined their problem in Setion 1. Letting x k denote the total umulative prodution of produt in periods 1 through k, their problem an be formulated as the following integer program. N C C 2 max ( x kr ) xk = k, k; 0 xk x, k 1 1, ; x k= 1 = 1 = 1 k k, 0 Integer Kubiak & Sethi (1991) show that above problem an be transformed into an assignment (18) problem with deision variable x jk (binary variable; equals 1 when j th unit of produt is 13

15 produed in k th period, and is zero otherwise). The ost γ jk (for assigning j th unit of produt to k th period) depends on the deviation of position k from the perfet position j Z = ( 2 j 1)/ 2r of the j th unit of produt. The solution of the assignment problem min = γ x st. x 1, k; x = 1, j,, x 0 jk jk jk jk (19) jk j, k, j, k an be easily transformed into a solution of the original problem (18). For details see Kubiak & Sethi (1991). The key for using this assignment approah for the ISCI problem is to find a strong relation between the perfet positions and the ideal distane. Consider the following trivial example with nine slots and three olors n 1 =n 2 =n 3 =3. The perfet positions Z j are 2,5, and 8 for j=1 to 3 and all olors. Sine Z j are the same for all olors, the ost γ jk are the same and therefore the set of optimal solutions of (19) inludes solutions that are not optimal with respet to the ISCI objetive. For example, the sequene has the same ost in (19) as whih is learly favored under the ISCI objetive. The problem is that there are multiple perfet positions for an ideal distane. In our example the positions (1,4,7), (2,5,8), as well as (3,6,9) are perfet with respet to the ideal distane 9/3=3. If we would hange the perfet positions to 1 = (1,4,7 Z ), 2 = (2,5,8 Z ), and Z 3 = (3,6,9) together with the ost γ jk, model (19) would produe the optimum sequene In general, we try to find for eah olor one of the shifted perfet positions P ε ) = { Z + ε j = 1.. n } with { N / 2n + 1,..., (2n 1) N / 2n } ( j ε that make the optimal solution of (19) likely to be a good solution for the ISCI problem. Our suggestion is to find positions P ε ) suh that the number of multiple perfet positions is minimized. More formally, we try to find ( ε * U = arg min {1.. N} \ { P ( ε )}. Finding suh an ε * ε vetor is in general diffiult but an be easily found for a small number of possible ε by integer programming tehniques used for similar set partitioning and overing models. After finding jk ( P ( * ) k) 2 γ = ε (20) * ε, we define the ost 14

16 to be the quadrati deviation from the perfet position and solve (19). The resulting solution an be evaluated in terms of the ISCI objetive. 5. COMPUTATIONAL RESULTS We used 40 test problems to test the effetiveness of the heuristis and the optimal algorithms. A test problem with slots is indiative of the usual size of the problem faed by the broadast television industry. Test problems are onsidered large. We onstruted these test problems to hek the range of effetiveness of our algorithm and heuristis. All omputations were run on a personal omputer with Intel Pentium IV proessor at 1.6GHz. Table II desribes our test problems. We have speified the total number of slots (N) and the number of olors (C) against eah problem. The olumn labeled Color Details in Table II desribes the numbers of balls of eah olor. Thus, problem 1 has a total of 8 balls of 2 olors: 5 red balls and 3 blue balls. Computations for a test problem were terminated after 10 minutes (600 seonds) of CPU time usage. Table IV has for eah test instane the omputational results for all models and algorithms desribed so far. Eah setion of the table inludes the olumn Obj for objetive value and CPU for CPU time usage in seonds. For setion GAMS/CPLEX P2 and GAMS/CPLEX P3 we also list the best-known linear programming-based lower bound from the CPLEX branh-and-ut algorithm. If a model/algorithm produed the best solution among all algorithms, the Obj olumn of the orresponding setion is shaded, i.e. the more shading, the better the method. The last setion, titled Best Choie takes the minimum objetive value over all setions and the maximum bound of setion GAMS/CPLEX P2 and GAMS/CPLEX P3 and also lists the relative optimality gap, i.e. (best objetive best bound)/best bound. The development of effiient optimal and near optimal solution methods for the ISCI problem has been the fous of this paper. Computational results are the essential tools for providing evidene for the relevane of our analysis. Most important is the reproduibility of our experiments. Therefore, we made the GAMS soure ode for all models available at 15

17 The remainder of this setion is organized into two sub-setions. Setion 5.1 desribes the omputational results for the optimal solution approahes, i.e. model P2, the modified branh and bound algorithm and the flow model P3. Setion 5.2 desribes the omputations for the four heuristis. 5.1 Computations for the Optimal Solution Approahes The formulation P2 was modeled using GAMS and the test problems were solved using GAMS/CPLEX solver (CPLEX version 8.0). We were able to obtain optimal solutions for problems 1-14, 19, and 20 within 600 seonds of CPU time. In addition, feasible integer solution was obtained for problems 15-18, 21-25, while no feasible solution was found for the rest of the problems. Along with the size, the omputation time is also influened by the struture of the problem. For example, we were able to get optimal solution for problems 19 (45 slots) while we were unable to get optimal solutions for problems 15-18, whih are smaller. This is due to the speial struture of problem 19 where we have almost equal number of balls of eah olor. The modified branh and bound algorithm has a feast-or-famine behavior. We were able to obtain optimal solutions for the first 20 test problems using the algorithm within the allowable time. We ould solve up to a maximum of 50-slot problem within the said time. The solution times are signifiantly improved ompared to model P2. Unfortunately, it did not improve upon the starting feasible solution within the allowable time for instanes higher than 20. With formulation P3 we were able to obtain optimal solutions for problems 1-23 (up to 60 slots), and problem 27 within 600 seonds of CPU time. Feasible integer solution was obtained for all but instanes 34 and 40. Although the gap between objetive value and best bound is signifiantly large, this represents is a substantial improvement over model P2 and the modified branh-and-bound algorithm and is able to solve any typial broadast television problem. 5.2 Computations for the Heuristis The greedy heuristi performs reasonably well for smaller test problems (up to problem 11). The elegane of this heuristi lies in its simpliity and easy omputability. 16

18 However, the deviation from the best-known solution inreases, in general, as the size of the problem inreases. Nevertheless, the solutions produed by the greedy algorithm an help to hot-start the CPLEX branh-and-ut algorithm for P2 and P3 as well as the modified branh-and-bound algorithm. The removal of ars unlikely to arry flow resulted in the P3/Delta Heuristi. δ has been seleted in a way that the underlying graph had signifiantly fewer ars but maintained the paths representing the optimal solution. For all but two instanes P3/Delta Heuristi found a feasible solution and provided in 28 of the 40 ases the best-known solution. Reall that in the P3 Bathing Heuristi we are solving large problems in bathes of 60 slots. Therefore, we are solving problems with less than 60 slots in one bath. This explains the why we are getting the optimal solutions for problems By omparing setion GAMS/CPLEX P3 and P3/Bathing Heuristi the reader will observe that the latter runs somewhat faster for these test problems. This is beause we are using a sparse network to P3/Bathing Heuristi. Overall, the P3 Bathing Heuristis performs very well whih is also expressed in the largest number of shaded table entries. Furthermore, this heuristi an be modified to speed up the overall proess by reduing the bath size from 60 to smaller number, whih results in faster solution times for the P3 sub-models but also potentially worse solutions. We next take a look at the Assignment Heuristi. The running times for the Assignment Heuristi are almost as good as for the Greedy Heuristi. But the Greedy Heuristi outperforms the Assignment Heuristi based on speed and quality (always faster, better in 26, worse in 10 of the 40 ases). We also experimented with an absolute value form of the ost funtion in (20). However, that results in worse performane of the heuristi ompared to the squared form of ost funtion. Nevertheless, the Assignment Heuristi represents a first step in asting the ISCI problem into an assignment framework, whih may result in the development of better optimal and near optimal algorithms. The last setion Best Choie in Table IV summarizes our efforts in solving the ISCI problem to optimality. For 24 out of the 40 models, the solution of model P3 provided the optimum solution. For the remaining ases the relative gap between best 17

19 know solution and best bound varies between 4% and 210%. It appears that the struture of the problem, in addition to its size, plays a key role in determining the auray of the heuristis as well as the quality of the lower bound. 6. EXTENSIONS We onsider two extensions to the basi ISCI problem. We all the first extension the ISCI Rotator problem with pre-plaed balls, and the seond extension the ISCI Rotator problem with equal time intervals. We desribe the two extensions in the following two subsetions. 6.1 The ISCI Rotator Problem with Pre-plaed Balls An advertiser often wants a speifi slot for a speifi ommerial. For example, the lient desribed in Setion 1 might want the ommerial with ISCI ode TOPS9016 to be aired at 9:00 pm on May 12. Television networks typially aommodates these requests while still maintaining the overall objetive of plaing the ommerials of an ISCI ode as muh evenly spaed as possible. Given that the exat identity of the desired slot is known, this situation an readily be aommodated in our modeling framework by assuming that some slots out of N possible slots ontain pre-plaed balls. We all this extension as the ISCI Rotator problem with pre-plaed balls. Stating in more formal terms, we now have N slots out of whih p slots have balls pre-plaed in them and the remaining m slots are empty. Therefore, N = p + m. Out of the p pre-plaed balls, p 1 are of olor 1, p 2 are of olor 2 and so on. Similarly, out of the m balls available for plaement, m 1 are of olor 1, m 2 are of olor 2 and so on (total number of balls of olor is given by n = p + m ). We want to plae the m available balls into m slots suh that the balls of any speifi olor are as muh evenly spaed as possible in the final arrangement of N balls. Note that the pre-plaed balls are sarosant, i.e., neither the position nor the olor of the balls in the pre-plaed slots an be altered. This extension an easily be aommodated in formulations P2 and P3 desribed in Setion 2. Let θ (, k ) denote the set of pre-plaed balls. We add the following onstraint to problem P2 to aount for the pre-plaed balls. 18

20 Y 1,, k i (, k) k = θ (21) i Similarly we add the following onstraint to the problem P3 to aount for the pre-plaed balls. p j = 1,, j θ(, j) (22) In (21) and (22) we have simply set the binary variable(s) orresponding to the pre-plaed balls equal to one. The ISCI problem with pre-plaed balls an thus be formulated and solved using enhaned versions of model P2 and P3. The modified branh and bound algorithm an also be extended to handle the preplaed balls. Reall that we defined S = s, s,..., s ) to be a solution vetor for the ISCI ( 1 2 N problem in Setion 2.1. Our algorithm does not require S to be an empty vetor to begin with. Thus, we an pre-speify the position and olor of the pre-plaed balls in the vetor S and apply our algorithm to solve the ISCI Rotator problem with pre-plaed balls. In addition, the heuristis an similarly be modified to handle pre-plaed balls. A detail disussion inluding omputational results for ISCI problem with pre-plaed balls would exeed the sope of this paper. We refer to the ISCI Model web page for details. 6.2 The ISCI Rotator problem with Equal Time Intervals In this extension, our objetive is to shedule a set of ommerials on a set of available slots suh that the ommerials with the same ISCI ode are as muh evenly spaed in time as possible during time period under onsideration. We found this to be another riteria that the advertisers often speify. This problem is not the same as the basi ISCI Rotator problem. For example, the first available slot might be at time 1, while the seond available slot might be at time 7, and the third available slot might be at time 8 and so on. For the data in Table 1(a), the time period under onsideration is between May 11 and May 27, The reader an see that the positions of the available ommerial slots are not evenly spaed with respet to time. We define the following quantities to formulate the problem. Note that the definitions and notations of Setion 2.1 still hold. T = the time horizon under onsideration (in appropriate time units) 19

21 τ = ideal distane (in time units) between any two ommerials of olor = T / n t(k) T i j = time loation for slot k = time of airing of ommerial i of olor j The following two formulations desribe the ISCI problem with equal time intervals. As before, formulation P4 is the natural formulation while formulation P5 is the flow formulation. Problem P4 Minimize T T 1 τ (23) Subjet to: Constraints (3), (5), (6) i i i Ti 1 Ti 1, i, j (24) T i = t( k) Y, i, (25) k i k Y k i binary (26) Problem P5 Minimize t ( k) t( j) τ f (27) 0< j< k N Subjet to: Constraints (12), (13), (14), (15), (16), (17). jk Note that the total number of available slots, N, and the time loation of eah slot, t(k), is known. The formulation P4 is similar to P1 (and an also be made linear) while P5 is similar to P3. We have simply redefined one variable and introdued an additional parameter. The objetive funtion in (27) is a salar transformation of (11). Thus, all heuristis of Setion 4, and the modified branh and bound algorithm of Setion 2.1 an be used with minor modifiations to solve P4 and P5. 20

22 7. SUMMARY AND CONCLUSION In this paper, we developed effiient solution methodologies to shedule ommerial videotapes in broadast television. Our objetive was to make the airings of same ommerial as muh evenly spaed as possible over a speified time period. We first modeled this problem as a mixed integer program that minimizes the sum of deviations from the ideal spaing of the ommerials. We also developed a branh and bound like algorithm that uses the struture of the problem to obtain the optimal solution. Both models/algorithms did not solve problems of pratial size in the set time limit. We then presented an innovative formulation that was able to improve upon the performane. We also desribed four heuristis for quikly solving the ISCI problem. All models and algorithms handle extensions of the basi ISCI problem, the problem with pre-plaed balls and with equal time intervals. A variation of one of the heuristi algorithms desribed in Setion 5 was implemented in a sheduling system for plaing ommerial videotapes at the Pax network. The algorithm resulted in signifiant improvement in sheduling produtivity. It is interesting to note that while we developed the models and the algorithms presented here with one speifi problem in mind, similar situations arise in other business senarios. One suh example is the problem of designing sales atalogs. While designing these atalogs, the atalog merhants often want the similar produts to be as muh spread out as possible through the atalog. Considering the sales atalog to be a olletion of slots to hold desriptions for different types of produts, this problem an be transformed into the models desribed in this paper. Our future researh will address some extensions to this problem. One suh extension is to have different lasses of balls. In our urrent work we are minimizing the sum of deviations from the ideal spaing between any two balls of same olor. In a multiple lass senario, it is more important to have even spaing for balls of one olor over balls of another olor. Thus, minimizing the weighted sum of deviations from the ideal spaing will be an appropriate objetive. 21

23 Aknowledgement We would like to thank Hans Kellerer from Graz University who pointed out the similarity of the ISCI problem and the problem studied by Miltenburg (1989) and Kubiak and Sethi (1991). The first author would like to thank the National Broadasting Company for introduing the problem and funding the initial work. The last author would like to aknowledge the support of Campus Researh Board, University of Illinois at Urbana-Champaign through grant #

24 8. REFERENCES Bollapragada, S., H.Cheng, M. Phillips, M. Sholes, T. Gibbs, M. Humphreville NBC's Optimization Systems Inrease its Revenues and Produtivity. Interfaes. 32(1). F Bollapragada, S., Bussiek, M.R., Mallik, S The ISCI Model Web Page: Brooke, A., Kendrik, D., and Meeraus, A GAMS: A Users Guide. The Sientifi Press, Redwood City, CA. GAMS/CPLEX GAMS The Solver Manuals. Washington D.C. Garey, M.R., D.S. Johnson Computers and Intratability: A guide to the Theory of NP-Completeness. Freeman, New York. Goodhardt, G.J., A.S.C. Ehrenberg, M.A. Collins The Television Audiene: Patterns of Viewing. Saxon House, Westmead, England. Headen, R.S., J.E. Klompmaker, R.T. Rust The Dupliation of Viewing Law and Television Media Sheduling Evaluation. Journal of Marketing Researh. 16, Henry, M.D., H.J. Rinne Prediting Program Shares in New Time Slots. Journal of Advertising Researh. 24(2), Lawler, E.L., J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys Sequening and Sheduling: Algorithms and Complexity. In Handbooks in OR & MS. Vol. 4, S.C. Graves et al. (Eds.). Elsevier. Kubiak, W., S. Sethi A note on Level Shedules For Mixed-Model Assembly Lines in Just-in-Time Systems. Management Siene. 37(1), Lilien, G.L., P. Kotler, K.S. Moorthy Marketing Models. Prentie Hall, NJ. Mahajan, V., E. Muller Advertising Pulsing Poliies for Generating Awareness for New Produts. Marketing Siene. 5(2), Miltenberg, J Level Shedules For Mixed-Model Assembly Lines in Just-in-Time Systems. Management Siene. 35(2), Reddy, S.K., J.E. Aronson, A. Stam SPOT: Sheduling Programs Optimally for Television. Management Siene. 44(1), Rust, R.T., N.V. Ehambadi Sheduling Network Television Programs: A Heuristi Audiene Flow Approah to Maximizing Audiene Share. Journal of Advertising. 18(2),

25 Simon, H ADPLUS: An advertising Model with Wear out and Pulsation. Journal of Marketing Researh. 19, Webster, J.G Program Audiene Dupliation: A Study of Television Inheritane Effets. Journal of Broadasting & Eletroni Media. 29(2),

26 Table I(a): The Individual Commerials to be Aired ISCI Code Airings TOPS MABH TOPS MABT MAIT MAGM Table I(b): Slots Purhased by An Advertiser plus optimal shedule Show Day of Week Air Date Air Time Pod Shedule TOUCHED BY AN ANGEL THU 5/11/2000 9:00:00 PM 5 TOPS9004 TOUCHED BY AN ANGEL FRI 5/12/2000 9:00:00 PM 2 MAIT0206 TWENTY -ONE SAT 5/13/2000 9:00:00 PM 2 MAGM0205 PAX THREE-HANKY MOVIE SAT 5/13/ :00:00 PM 2 MABH7503 CHRISTY SUN 5/14/2000 5:00:00 PM 3 TOPS9016 SHOP 'TILL YOU DROP MON 5/15/2000 6:30:00 PM 4 MABT6903 SCARECROW & MRS. KING TUE 5/16/2000 4:00:00 PM 2 MABH7503 TREASURES IN YOUR HOME TUE 5/16/ :35:00 PM 3 MAIT0206 DR QUINN MEDICINE WOMAN WED 5/17/2000 3:00:00 PM 3 TOPS9004 IT'S A MIRACLE THU 5/18/2000 8:00:00 PM 3 MABH7503 IT'S A MIRACLE FRI 5/19/2000 8:00:00 PM 2 TOPS9016 DIAGNOSIS MURDER FRI 5/19/ :00:00 PM 3 MAGM0205 DIAGNOSIS MURDER MON 5/22/ :00:00 PM 5 MABH7503 JACK HANNA TUE 5/23/2000 5:30:00 PM 2 MAIT0206 IT'S A MIRACLE WED 5/24/ :05:00 PM 3 MABT6903 EIGHT IS ENOUGH SAT 5/27/2000 6:00:00 PM 2 MABH7503 D JACK HANNA SAT 5/27/2000 8:00:00 PM 2 TOPS9016 Figure1: The underlying Graph of Flow Formulation P3 25

27 Table II: Test Problems ID N C Color Details ID N C Color Details R, 3B R, 16B, 13W R, 2B, 2W R, 12B, 10W, 2G R, 3B, 2W R, 23B, 12W R, 3B, 2W R, 25B, 14W, 9G R, 4B, 2W R, 25B, 21W, 15G, 6Y R, 3B, 3W, 2G R, 35B, 17W, 9G R, 4B, 4W R, 31B, 30W, 29G, 29Y R, 5B, 4W R, 67B, 53W, 3G R, 3B, 3W, 2G R, 47B, 39W, 30G, 23Y R, 6B, 4W R, 63B, 54W, 31G, 11Y R, 7B, 5W R, 83B, 17W, 9G R, 7B, 3W, 2G R, 95B, 77W, 19G, 7Y R, 5B, 5W, 4G, 1Y R, 68B, 65W, 62G, 58Y R, 6B, 6W, 5G R, 81B, 63W, 54G, 53Y R, 6B, 5W, 4G, 3Y R, 102B, 92W, 71G R, 8B, 7W, 5G R, 149B, 11W, 8G, 3Y R, 7B, 6W, 5G, 3Y R, 107B, 86W, 73G, 33Y R, 10B, 8W, 5G R, 95B, 82W, 79G, 56Y R, 15B, 14W R, 150B, 90W, 90G R, 13B, 12W R, 112B, 111W, 70G, 64Y Key: R = red, B = blue, W = white, G = green, Y = yellow. Table III: Evolution & Comparison of Optimal Solution Strategies Solution Method Optimal Solution Integer Solution Comment GAMS/CPLEX P2 Up to 25 slots Up to 100 slots Typial size of a broadast television industry problem slots Modified B&B Up to 50 slots N/A GAMS/CPLEX P3 Up to 60 slots Up to 200 slots Solves larger problems meeting most real world needs, forms basis for heuristi solutions 26

28 Table IV: Computational Results Modified B&B GAMS/CPLEX P2 GAMS/CPLEX P3 Greedy P3/Delta P3/Bathing Assignment Best Choie Algorithm Heuristi Heuristi Heuristi Heuristi Id Obj Bound CPU Obj CPU Obj Bound CPU Obj CPU Obj CPU Obj CPU Obj CPU Obj Bound Gap * * * * * * * * * * * * * * * * * * * * * * * 600 * * * 600 * * * * * 600 * * * * 600 * * * * 600 * * * 600 * 600 *

29 28