TALENT HOLD COST MINIMIZATION IN FILM PRODUCTION

. This paper investigates the talent scheduling problem in ﬁlm production, which is known as rehearsal scheduling in music and dance perfor- mances. The ﬁrst lower bound on the minimization of talent hold cost is based upon the outside-in branching strategy. We introduce two approaches to add extra terms for tightening the lower bound. The ﬁrst approach is to formulate a maximum weighted matching problem. The second approach is to retrieve structural information and solve a maximum weighted 3-grouping problem. We make two contributions: First, our results can fathom the matrix of a given partial schedule. Second, our second approach is free from the requirement to schedule some shooting days in advance for providing anchoring information as in the other approaches, i.e., a lower bound can be computed once the input instance is given. The lower bound can ﬁt diﬀerent branching strategies. Moreover, the second contribution provides a state-of-the-art research result for this problem. Computational experiments conﬁrm that the new bounds are much tighter than the original one.

specifies the daily hold cost c i of actor a i . In the context of concert rehearsal scheduling, each day is referred to as a piece of music and is additionally associated with a duration specifying the length of this scene. The results presented in this paper can be easily generalized to the setting with durations.
The planning of shooting days consists of two types of talent cost, namely base cost and hold cost. The base cost of actor a i is equal to the number of shooting days he needs to be present times his daily pay cost, i.e., c i × n j=1 t i,j . The base cost is fixed regardless of how the shooting days are arranged. The hold cost is the extra cost for retaining an actor on location although the actor is not required. To illustrate the problem definition, consider the instance shown in Figure 1. If the default shooting sequence (d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , d 7 , d 8 , d 9 , d 10 ) is adopted, then hold days will occur to the actors. For example, actor a 3 has to be present from the third day to the eighth day, which incurs three hold days: d 4 , d 6 , and d 7 . The base cost of actor a 3 is 3 × 1, 100 and the extra cost under the shooting sequence is 3 × 1, 100. In the given instance with the default shooting sequence, the hold days for all the actors are indicated by the underlined zeros.
Through film-editing in post-production, the various components of the film are not necessarily filmed in the same order as they appear in the final version of the film. Let σ be a permutation of the shooting days. With respect to sequence σ, let e i (σ) and l i (σ) be the first and last working day of actor a i , respectively. Actor a i must be present from the beginning of day e i (σ) to the end of day l i (σ). Any day between e i (σ) and l i (σ) that does not require actor a i incurs a hold day to him. The number of hold days of actor a i is given by h i (σ) = (l i (σ)−e i (σ)+1)− li(σ) j=ei(σ) t i,j . Therefore, the total hold cost of sequence σ for the film is the sum of the hold costs of all the actors, i.e., H(σ) = m i=1 c i × h i (σ). Let Π be the set of all the permutations of the shooting days. The problem is to find σ * ∈ Π such that H(σ * ) = min σ∈Π {H(σ)}. Off-the-shelf software, such as Sunfrog Film Scheduling 2 and Gorilla 3 , is available for managing film production. Still, analytic approaches are required to be embedded in the decision engines to compose effective schedules in an automatic manner. Cheng et al. [3] presented an outside-in approach for exploring the enumeration tree in the development of a branch-and-bound algorithm. They proposed a lower bound to prune the unnecessary branching. Nordström and Tufekçi [10] proposed a genetic algorithm for producing approximate solutions. We notice that the studied model is related to the serialization problem, for which Adelson et al. [1] designed an O(mn2 n ) dynamic programming algorithm to minimize the number of enclosed zero's. Based upon the lower bound of Cheng et al. [3], Lin [9] gave an idea to improve the lower bound. de la Banda et al. [2] proposed several dominance properties that help the dynamic program and the branch-and-bound algorithm to curtail the unnecessary recursion and conducted extensive experiments to test the performance of the new properties. The main contribution of this study is to provide a systematic development of lower bounds for the talent scheduling problem.
We organize the remaining part of the paper as follows: In Section 2 we review the lower bound provided by Cheng et al. [3]. In Section 3 we introduce an approach to enhance the lower bound. In Section 4 we present another approach to enhance the lower bound. In Section 5 we present and discuss computational results on evaluating the merits of the proposed lower bounds. We conclude the paper and suggest topics for future research in Section 6.
2. First lower bound. One of the potential uses of upper or lower bounds in dealing with optimization problems is to facilitate the design of efficient branchand-bound algorithms. To construct a search tree for solving a sequencing problem, the most commonly adopted approach is to iteratively assign items to positions of the sequence starting from the front end. For some problems, like scheduling to minimize the total tardiness, sequences can be constructed from the rear end so as to achieve a tighter lower bound. To facilitate the design of lower bounds for use in branch-and-bound algorithms to solve the talent scheduling problem, Cheng et al. [3] proposed the outside-in strategy that assigns days to the front end and the rear end simultaneously.
Denote P as a partial sequence, in which the first k 1 days and the last k 2 days of the shooting plan have been determined. If actor a i has to show up on at least one of the first k 1 days, then let ε i (P ) = 1; otherwise, ε i (P ) = 0. If actor a i has to be present on at least one of the last k 2 days, then λ i (P ) = 1; otherwise, λ i (P ) = 0. For ε i (P ) = 1, denote e i,1 (P ) (respectively, e i,2 (P )) as the start day (respectively, end day) of actor a i among the first k 1 days. Similarly, for λ i (P ) = 1, denote l i,1 (P ) (respectively, l i,2 (P )) as the start day (respectively, end day) of actor a i among the last k 2 days. In the following equations, P denotes the set of the remaining unscheduled days. Figure 2 gives a visual depiction of the zones in which hold days might or might not be included. Note that the hold days within the purple shaded zones are calculated for they are inevitable subject to the partial schedule P . The potential hold days among the remaining days of P are not included because P could be optimally scheduled without any hold days. Depending on the scenarios of each actor in the partial schedule P , the number of hold days of actor a i is calculated as follows: Case 1: ε i (P ) = 1 and λ i (P ) = 1: Case 2: ε i (P ) = 1 and λ i (P ) = 0: Case 3: ε i (P ) = 0 and λ i (P ) = 1: Case 4: ε i (P ) = 0 and λ i (P ) = 0: In this case, no hold days are incurred if (P ) is optimally scheduled. Incidentally, our new developments will improve this situation.
Combining the above analysis, the first lower bound for any complete schedule derived from the partial schedule P is given by As depicted in Figure color-matrix, the first three cases correspond to the purple shaded areas in segments (a), (b), and (c). The lower bound LB 1 comprises the inevitable hold cost in the purple shaded area. Consider the partial schedule P , in which σ 1 = d 2 , σ 2 = d 3 , σ 9 = d 6 , and σ 10 = d 7 , given in Figure 3 as an example. The lower bound is calculated as LB 1 (P ) = (4 × 800 + 3 × 550 + 3 × 610) + (1 × 700 + 1 × 750) = 8, 130.
3. Maximum weighted matching. Attempting to identify more inevitable hold days from among the unscheduled days, we focus on the green shaded areas in segments (b) and (c) in Figure 2. With regard to segment (b) in Figure 2, define Φ(P ) = {a i |ε i (P ) = 1 and λ i (P ) = 0}, i.e., Φ(P ) contains actors who are required by at least one of the first k 1 days, but not by any of the last k 2 days. Similarly, define set Λ(P ) = {a i |ε i (P ) = 0 and λ i (P ) = 1} corresponding to segment (c). Using the instance in Figure 1, we assume that days d 2 , d 3 , d 6 , and d 7 are fixed in the partial schedule P . Grouping the actors according to the colour segments, we have the partial matrix shown in Figure 3, with Φ(P ) = {a 2 , a 3 , a 6 , a 10 } and Λ(P ) = {a 9 }. We discuss the hold cost that will be inevitably incurred by the actors of Φ(P ).
For actors a i1 , a i2 ∈ Φ(P ), let α i1, i2 (P ) denote the set of unscheduled days that require actor a i1 but not actor a i2 , i.e., α i1, i2 For the actors of Φ(P ) = {a 2 , a 3 , a 6 , a 10 }, we have the subsets: Through these subsets, we have the following property.

TAI CHIU EDWIN CHENG, BERTRAND MIAO-TSONG LIN AND HSIAO-LAN HUANG
Lemma 3.1. Given partial schedule P , actors a i1 , a i2 ∈ Φ(P ) together incur a hold cost of at least for the unscheduled days of α i1,i2 (P ) ∪ α i1, i2 (P ) in any complete schedule derived from P .
Proof. Consider the illustrative arrangement of the unscheduled days in Figure 4 for actors a i and a j . If in some optimal schedule derived from the partial schedule P , the last day of α i1,i2 (P ) ∪ α i1, i2 (P ) is from α i1,i2 (P ), then actor a i2 is idle for at least |α i1, i2 (P )| days, resulting in a cost of c i2 |α i1, i2 (P )|. The counterpart case leading to a cost not less than c i1 |α i1,i2 (P )| of actor a i1 can be similarly analyzed.
Applying the same line of reasoning to the actors of subset Λ(P ), we obtain the following theorem.
Theorem 3.2. Given partial schedule P , is a lower bound on the optimal solution derived from P .

4.
Maximum weighted 3-grouping. In the development of lower bounds LB 1 and LB 2 , some shooting days are already scheduled at the front part and/or at the rear part as sentinels, which provides anchoring information on zeros enclosed between two or more ones on a row. In the following we improve the lower bound LB 2 by considering the yellow shaded area (d) in Figure 2. The contribution of the new development is not only an enhancement of the lower bound LB 2 , but also the establishment of a new lower bound itself. The new development provides a viable way to calculate a lower bound based on the input instance only, i.e., we can derive a lower bound at the root node without the need to fix any days in specific positions.

Arrangement
Minimum hold cost Next, we proceed to analyze the second term y i1,i2,i3 . The unscheduled shooting days of the four disjoint subsets β i1, i2, i3 , β i1i2 i3 , β i1, i2,i3 , and β i1,i2,i3 are considered. We analyze the cases corresponding to the combinations of the first day and the last day. Referring to Table 2, we discuss cases y-1, y-2, y-5, and y-8. Table 2. Analysis of y i1,i2,i3 for actors a i1 , a i2 , and a i3 .
Case y-5: The first day is from β i1, i2,i3 and the last day is from β i1,i2, i3 (Figure 7(b)). The unscheduled shooting days of β i1, i2, i3 may become hold days to actors a i2 and a i3 . The first and the last days, together with any shooting day of β i1,i2,i3 scheduled somewhere in between, will result in actor a i2 being held for r days, say, and in actor a i3 being held for |β i1, i2, i3 | − r days, say. Therefore, the minimum cost of this case is min{c i2 , c i3 }|β i1, i2, i3 |. However, the minimum cost of this case will vanish when set β i1,i2,i3 is empty because hold days will not be necessarily incurred by actors a i2 and a i3 .
Since the subsets used for the analysis of the x-type cases and those used for the analysis of the y-type cases are disjoint, the lower bound z i1,i2,i3 = x i1,i2,i3 + y i1,i2,i3 on the hold cost incurred by actors a i1 , a i2 , and a i3 is established.
The lower bound given in Lemma 4.1 is calculated for three actors. Similar to the development of the lower bound LB 2 (P ), we can find an optimal (maximum) 3grouping of all the actors to obtain a tighter bound. For every three actors a i1 , a i2 , and a i3 , define a cost z i1,i2,i3 . The Max-3-Grouping(P ) problem is to partition the actors into groups of three such that the total grouping cost is maximized. The discussion of the third lower bound is summarized in the following theorem.
Theorem 4.2. Given partial schedule P , is a lower bound on the optimal solution derived from P .
In combinatorial optimization, there are several NP-hard 3-grouping problems, including 3-Partition and 3-Dimensional Matching. Nevertheless, whether Max-3-Grouping(P ) is NP-hard is unknown. An alternative is to use heuristic algorithms, like greedy selection, to find an approximate solution for 3-Dimensional Matching.

5.
Evaluation of lower bounds. We present in this section a computational study to assess the merits of the lower bounds. We formulate the Max-3-Grouping problem as an integer program and solve it using the commercial software CPLEX. We use the binary variable Z i,j,k , i = j = k, to indicate if the group of actors a i , a j , and a k is accepted or not. That is, if actors a i , a j , and a k are grouped to define a weight z i,j,k , as given in Lemma 4.1, then Z i,j,k = 1; otherwise Z i,j,k = 0. The right-hand side of constraints (5) is set to 1 to avoid double or triple counting of each group.

IP-3-Grouping:
To generate the test instances, we set m = 50 actors and n = 50 days. We randomly generated the entries of the matrix by a control variable density, which was 0.1, 0.2, or 0.3. We then generated a random number for each entry t i,j such that if the random number is greater than density, then t i,j = 0; otherwise, t i,j = 1. We used another control parameter k ∈ {1, 3, 5, 10} to indicate that k days are scheduled in the first k positions and k days are scheduled in the last k positions. This reflects the outside-in branching scheme. In the experiments, for each combination of k and density, we generated ten partial schedules in random. We invoked the three lower bounds to compute the lower bound values for the ten instances. The results, averaged over each ten instances, are shown in Table 3. The sub-columns entitled "ratio" contain the ratios of LB2 LB1 and LB3 LB1 . As evinced from the numerical values, the largest ratios are due to the setting where k = 1 and density = 0.1, while the lowest ratios are due to the setting where k = 10 and density = 0.3. This is attributed to the fact that when there are more days scheduled on both sides of the schedule and the actors are more likely to participate in the film, then the hold costs of most actors are fathomed by the first lower bound. Nevertheless, when the information gained from the partial schedules is sparse, i.e., few days are scheduled, the new lower bounds are from 10% to 400% tighter than the first lower bound.
We conducted another set of experiments to examine the effect of adopting the commonly used sequential branching scheme that fills in the front-end positions of a schedule one by one. The computational results are shown in Table 4. The effectiveness of the new lower bounds is more significant for this conventional branching scheme. A very interesting phenomenon is that the ratios are higher as density increases. This is especially evident for k ∈ {1, 3, 5}. When there are more 1's in the scheduled days, more actors will be involved in the calculation of LB 2 , thus increasing the value of LB 2 . Before closing this section, we note that the new bounds require more computing effort than the previous ones. Therefore, we do not champion the efficiency of constructing the new lower bounds for use in branch-and-bound algorithms but their efficacy in terms of their tightness and applicability in different scenarios.
6. Concluding remarks. We provide three lower bounds on the total hold cost of the talent scheduling problem. The new lower bounds can be deployed to support different branching schemes. The computational study reveals that the new lower bounds are much tighter than the original one. As the zero-one matrix possesses different interpretations and/or applications in other areas, our results have a wide range of applications. For further study, it will be interesting to ascertain the optimality gaps between the proposed lower bounds and optimal solutions. Another direction for further research could be applying the model to treat the problem of optimal linear arrangements of objects that are characterized by several attributes. Objects (days) are individuals or products, and the matrix T describes if each object possess a certain attribute (actor). A linear arrangement (permutation) describes a spectrum of the concerned objects and its objective value gauges the inappropriateness of the permutation. Extending the model by replacing days with scenes and introducing daily working capacities such that the total length of the scenes included in each single day cannot exceed the capacity, Wang et al. [13] incorporated bin packing into the decision. In a similar vein, future research may consider problem settings that closely reflect real-life practice.