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In this paper, the problem of program performance scheduling with accepting strategy is studied. Considering the uncertainty of actual situation, the duration of a program is expressed as a bounded interval. Firstly, we decide which programs are accepted. Secondly, the risk preference coefficient of the decision maker is introduced. Thirdly, the min-max robust optimization model of the uncertain program show scheduling is built to minimize the performance cost and determine the sequence of these programs. Based on the above model, an effective algorithm for the original problem is proposed. The computational experiment shows that the performance’s cost (revenue) will increase (decrease) with decision maker’s risk aversion.

In the planning stage of a performance, many programs will sign up for the performance, and each program bring different revenue and different cost. The program group needs to comprehensively consider the performance revenue and cost, and decides which programs to be accepted. Each program requires multiple performers, and a performer can also participate in multiple programs. If the programs which the same performer participates in are not consecutively arranged, the waiting cost will occur. Therefore, after determining the set of the accepted programs, the reasonable scheduling of these accepted programs to increase the performance revenue is the main concern of the decision maker.

Scholars have done extensive researches on program performance scheduling and the similar film production scheduling problem. Cheng et al. [

Based on the above research results, it can be concluded that the film shooting or program rehearsal duration of the predecessors is assumed to be a certain value. In the actual program performances, due to factors such as staff absence, equipment failure, performance effects, etc., the duration of each program is uncertain. Therefore, the results obtained by the deterministic research method are greatly deviated from the actual situation. Zhen et al. [

Robust optimization is an effective method to solve the uncertain problem and has been widely used. Wang and Tang [

This paper considers the uncertain duration program performance scheduling problem under accepting strategy (recorded as P0), the remainder is organized as follows. In Section 2, we use a simple example to describe the program performance scheduling problem and introduce the application of the min-max robust optimization method in this paper. In the case of determining the set of the accepted performance programs, the decision maker’s risk preference coefficient [

In the actual performance, due to limitations of time or layout and so on, decision maker needs to consider program revenue and cost in a comprehensive manner, select a part of the programs from many registration programs to perform, and arrange the performance sequence of these accepted programs. Accepting a program will generate revenue, and rejecting a program will occur penalty cost. After determining the set of the accepted performance programs, multiple performers will participate in the performance. If there is no comprehensive arrangement for a performer who participates in different programs, the performer will generate waiting time and waiting cost. The objective of this paper is to find a solution that maximizes the overall revenue of the program group, including determining the set of the accepted performance programs and scheduling these accepted performance programs. The overall revenue of the program group consists of the following two parts: 1) the revenue value of the accepted programs and the penalty cost of the rejected programs. 2) the performance cost of the performers at the performance scene, including the appearance fees and the waiting cost.

Let’s look at the following example. A program group receives 6 programs { s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } . These programs are performed by 3 performers { a 1 , a 2 , a 3 } , and the performers participation list is shown in

Assume that the program group decides to accept the programs { s 1 , s 3 , s 4 , s 6 } to perform, the performance revenue from these accepted programs { s 1 , s 3 , s 4 , s 6 } is 635, and the penalty cost for these rejected programs { s 2 , s 5 } is 45. After determining the set of the accepted programs, in order to improve efficiency and reduce cost, we assume that performers show up on time before the first program they play starts, leave immediately after the last program they play finishes. In this paper, the robust optimization method is used to arrange the sequence of these accepted programs { s 1 , s 3 , s 4 , s 6 } , which is based on the principle of “min-max” [_{1} is idle in t_{3}, waiting time is the performance duration of s_{4}. a_{2} and a_{3} are idle in t_{2}, waiting time is the performance duration of s_{3}. The

s_{1} | s_{2} | s_{3} | s_{4} | s_{5} | s_{6} | |
---|---|---|---|---|---|---|

a_{1} | √ | × | √ | × | √ | √ |

a_{2} | √ | √ | × | √ | √ | × |

a_{3} | √ | √ | × | √ | × | √ |

scheme 1 | t_{1} | t_{2} | t_{3} | t_{4} | scheme 2 | t_{1} | t_{2} | t_{3} | t_{4} |
---|---|---|---|---|---|---|---|---|---|

s_{1} | s_{3} | s_{4} | s_{6} | s_{1} | s_{4} | s_{6} | s_{3} | ||

a_{1} | √ | √ | × | √ | a_{1} | √ | × | √ | √ |

a_{2} | √ | × | √ | a_{2} | √ | √ | |||

a_{3} | √ | × | √ | √ | a_{3} | √ | √ | √ |

maximum performance cost under scheme 1 is calculated as c_{1}. In the scheme 2, only a_{1} is idle at t_{2}, and the waiting time is the performance duration of s_{4}, and the maximum performance cost is calculated as c_{2}. Our robust optimization method is to calculate the maximum performance cost of all feasible solutions, then pick the solution with the least cost in the maximum performance cost to determine the performance sequence of these accepted programs. Based on the above, the overall revenue of the program group is obtained by comprehensively considering the revenue of the accepted programs and the penalty cost of the rejected programs.

1) n performers A = { a 1 , a 2 , ⋯ , a n } participate in k programs. If the performer a i participates in the program s j , defining w i j to be 1, 0 otherwise, ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , k } . The program group needs to select several programs from k registration programs to perform. Suppose that the number of accepted programs is m (m is part of our decision). T = { 1 , 2 , ⋯ , m } represents the set of time series, arranged from small to large, each program occupies a time series.

2) The duration of program q j is an uncertain number, q j ∈ [ l q j , u q j ] , l q j , u q j value is known.

3) Unit time waiting cost of performer a i is l c i , unit time appearance fee of performer a i is u c i , ∀ i ∈ { 1 , 2 , ⋯ , n } .

4) Performer a i shows up on time before the first program he play starts, leave immediately after the last program he play finishes.

5) The revenue of the accepted program is p j , the penalty cost of the rejected program is f j , ∀ j ∈ { 1 , 2 , ⋯ , k } . The objective function of this paper is the revenue of accepted programs minus the penalty cost of rejected programs, and then minus the performance cost of these accepted programs. The performance cost of the accepted programs is the appearance fees and waiting cost of the performers at the scene.

After determining the set of the accepted performance program, it is important to properly schedule these accepted programs and reduce the performance cost so that the program group can achieve a better overall revenue. Without loss of generality, let’s assume that the accepted programs are { s 1 , s 2 , ⋯ , s m } , the decision variables are as follows:

x j t = 1 if program s j performs in time series t, 0 otherwise.

y i t = 1 if performer a i performs in time series t, 0 otherwise.

a i t = 1 if performer a i arrives at the scene before time series t (including t), 0 otherwise.

l i t = 1 if performer a i leaves the scene after time series t (including t), 0 otherwise.

d i t = 1 if performer a i waits at the scene in time series t, 0 otherwise.

Let θ j = q j − l q j u q j − l q j be the degree which the performance duration q j of

program s j deviates from the lower bound l q j , θ j ∈ [ 0 , 1 ] , ∀ j ∈ { 1 , 2 , ⋯ , m } . In this paper, the idea of Bertsimas and Sim [

coefficient μ of decision maker is introduced, ∑ j = 1 m θ j = ∑ j = 1 m q j − l q j u q j − l q j ≤ μ ,

0 ≤ μ ≤ m , indicating that up to μ programs which duration reaches the upper bound at the same time. The μ value is given by the decision maker in advance, and the more conservative the decision maker is, the larger the μ value is. Therefore, the uncertain set of the program duration q j is expressed as:

{ q j | l q j ≤ q j ≤ u q j , ∑ j = 1 m q j − l q j u q j − l q j ≤ μ , ∀ j ∈ { 1 , 2 , ⋯ , m } } (1)

We build the following min-max robust performance scheduling model for the accepted programs (RPSM):

min { max { ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ ∑ j = 1 m q j ⋅ x j t + ∑ i = 1 n u c i ∑ j = 1 m q j ⋅ w i j } } (2)

s.t.

∑ t = 1 m x j t = 1 , ∀ j ∈ { 1 , 2 , ⋯ , m } (3)

∑ j = 1 m x j t = 1 , ∀ t ∈ T (4)

y i t − ∑ j = 1 m x j t ⋅ w i j = 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (5)

a i t + l i t − h i t ≤ 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (6)

a i t − a i , t + 1 ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (7)

l i , t + 1 − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (8)

− a i t − l i t ≤ − 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (9)

y i t − a i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (10)

y i t − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (11)

h i t − d i t − y i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (12)

l q j ≤ q j ≤ u q j , ∀ j ∈ { 1 , 2 , ⋯ , m } (13)

∑ j = 1 m q j − l q j u q j − l q j ≤ μ (14)

x j t , y i t , h i t , a i t , l i t , d i t ∈ { 0 , 1 } , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , m } , ∀ t ∈ T (15)

Equation (2) is the minimum performance cost in the worst case of the uncertain set, including the waiting cost and appearance fees of the performers at the scene. Equation (3) and Equation (4) force to schedule only one program at one time series and assign each program to only one time series. Equations (5)-(9) indicate the presence of performers in each time series. Equation (10) and Equation (11) indicate that each performer shows up on time before the first program he play starts, leave immediately after the last program he play finishes. Equation (12) determines if the performer is in a wait state at a time series. Equation (13) and Equation (14) constitute an uncertain set of program performance duration. Equation (15) indicates that the decision variables are a 0 - 1 variable.

This paper studies the uncertain duration program performance scheduling problem under accepting strategy (P0). Firstly, we decide to accept some programs from the registration programs; secondly, we build a robust performance scheduling model for the accepted programs (RPSM) to get the program performance sequence and performance cost; Finally, based on the above, we consider the revenue of the accepted programs and the penalty cost of the rejected programs comprehensively, and determine a feasible scheduling scheme for P0 to maximize the performance revenue.

We observe the Equations (2)-(15) of RPSM, finding that only the objective function (2), Equation (13) and Equation (14) are affected by the uncertainty of duration q j . Therefore, RPSM can be expressed as a two-stage robust optimization model. The decision variables of the first stage are x j t , y i t , h i t , a i t , l i t , d i t , the decision variable of the second stage is q j . Then the two-stage robust optimization model can be expressed as:

min F ( { q j } j ∈ { 1 , 2 , ⋯ , m } ) (16)

The constraints are Equations (3)-(12) and Equation (15).

And:

F ( { q j } j ∈ { 1 , 2 , ⋯ , m } ) = max { ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ ∑ j = 1 m q j ⋅ x j t + ∑ i = 1 n u c i ∑ j = 1 m q j ⋅ w i j } (17)

s.t.

l q j ≤ q j ≤ u q j , ∀ j ∈ { 1 , 2 , ⋯ , m } (18)

∑ j = 1 m q j − l q j u q j − l q j ≤ μ (19)

Noticing that (17)-(19) is a linear programming problem for q j , which is equivalent to the following form:

max { ∑ j = 1 m q j ⋅ ( ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ x j t + ∑ i = 1 n u c i ⋅ w i j ) } (20)

s.t.

− q j ≤ − l q j , ∀ j ∈ { 1 , 2 , ⋯ , m } (21)

q j ≤ u q j , ∀ j ∈ { 1 , 2 , ⋯ , m } (22)

∑ j = 1 m q j u q j − l q j ≤ μ + ∑ j = 1 m l q j u q j − l q j (23)

Since the feasible domain of the linear programming problem is bounded, according to the strong dual theory [

min ∑ j = 1 m ( − l q j ) ⋅ ρ j + ∑ j = m + 1 2 m u q j − m ⋅ ρ j + ρ 2 m + 1 ⋅ ( μ + ∑ j = 1 m l q j u q j − l q j ) (24)

s.t.

− ρ j + ρ m + j + 1 u q j − l q j ⋅ ρ 2 m + 1 = ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ x j t + ∑ i = 1 n w i j ⋅ u c i , ∀ j ∈ { 1 , 2 , ⋯ , m } (25)

ρ j ≥ 0 , ∀ j ∈ { 1 , 2 , ⋯ , 2 m + 1 } (26)

Bringing the Equations (24)-(26) to the RPSM to get the equivalent model RPSM1:

min ∑ j = 1 m ( − l q j ) ⋅ ρ j + ∑ j = m + 1 2 m u q j − m ⋅ ρ j + ρ 2 m + 1 ⋅ ( μ + ∑ j = 1 m l q j u q j − l q j ) (27)

s.t.

∑ t = 1 m x j t = 1 , ∀ j ∈ { 1 , 2 , ⋯ , m } (28)

∑ j = 1 m x j t = 1 , ∀ t ∈ T (29)

y i t − ∑ j = 1 m x j t ⋅ w i j = 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (30)

a i t + l i t − h i t ≤ 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (31)

a i t − a i , t + 1 ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (32)

l i , t + 1 − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (33)

− a i t − l i t ≤ − 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (34)

y i t − a i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (35)

y i t − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (36)

h i t − d i t − y i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (37)

− ρ j + ρ m + j + 1 u q j − l q j ⋅ ρ 2 m + 1 = ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ x j t + ∑ i = 1 n w i j ⋅ u c i , ∀ j ∈ { 1 , 2 , ⋯ , m } (38)

ρ j ≥ 0 , ∀ j ∈ { 1 , 2 , ⋯ , 2 m + 1 } (39)

x j t , y i t , h i t , a i t , l i t , d i t ∈ { 0 , 1 } , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , m } , ∀ t ∈ T (40)

It is observed that the Equation (38) contains nonlinear part ∑ i = 1 n ∑ t = 1 m l c i ⋅ d i t ⋅ x j t .

In order to convert the nonlinear constraints into linear constraints, we introduce variable Δ i t j , let Δ i t j ≥ d i t + x j t − 1 , Δ i t j ∈ { 0 , 1 } , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , m } , ∀ t ∈ T . Then we can obtain the following 0-1 mixed linear programming model RPSM2:

min z 1 = ∑ j = 1 m ( − l q j ) ⋅ ρ j + ∑ j = m + 1 2 m u q j − m ⋅ ρ j + ρ 2 m + 1 ⋅ ( μ + ∑ j = 1 m l q j u q j − l q j ) (41)

s.t.

∑ t = 1 m x j t = 1 , ∀ j ∈ { 1 , 2 , ⋯ , m } (42)

∑ j = 1 m x j t = 1 , ∀ t ∈ T (43)

y i t − ∑ j = 1 m x j t ⋅ w i j = 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (44)

a i t + l i t − h i t ≤ 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (45)

a i t − a i , t + 1 ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (46)

l i , t + 1 − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ { 1 , 2 , ⋯ , m − 1 } (47)

− a i t − l i t ≤ − 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (48)

y i t − a i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (49)

y i t − l i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (50)

h i t − d i t − y i t ≤ 0 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ t ∈ T (51)

− ρ j + ρ m + j + 1 u q j − l q j ⋅ ρ 2 m + 1 − ∑ i = 1 n ∑ t = 1 m l c i ⋅ Δ i t j = ∑ i = 1 n w i j ⋅ u c i , ∀ j ∈ { 1 , 2 , ⋯ , m } (52)

x j t + d i t − Δ i t j ≤ 1 , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , m } , ∀ t ∈ T (53)

ρ j ≥ 0 , ∀ j ∈ { 1 , 2 , ⋯ , 2 m + 1 } (54)

x j t , y i t , h i t , a i t , l i t , d i t , Δ i t j ∈ { 0 , 1 } , ∀ i ∈ { 1 , 2 , ⋯ , n } , ∀ j ∈ { 1 , 2 , ⋯ , m } , ∀ t ∈ T (55)

It is observed that the difference between the above two models is the Equation (38) in RPSM1 and the Equation (52), Equation (53) in RPSM2. However, RPSM1 and RPSM2 have the same optimal solution and objective function value. The reason is the coefficients symbol of ρ j , ρ m + j , ρ 2 m + 1 in Equation (41) and Equation (52) are the same (positive or negative simultaneously). So the change of the variable value in Equation (52) will have the same effect in Equation (41). From Equation (53), it can be known that when at least one of d i t and x j t is 0, Δ i t j can be 1 or 0. However, in the process of finding the optimal solution, Δ i t j should be 0 as much as possible, because if Δ i t j = 1 , then in the Equation (52), ρ j decreases or ρ m + j , ρ 2 m + 1 increases, in the corresponding Equation (41), ρ j decreases or ρ m + j , ρ 2 m + 1 increases, the objective value increases(When other variables remain unchanged). The performance of this solution is worse than the solution corresponding to Δ i t j = 0 . So when at least one of d i t and x j t is 0, Δ i t j = 0 , only when d i t = x j t = 1 , Δ i t j = 1 . In summary, variable Δ i t j is introduced to transform the nonlinear programming RPSM1 into linear programming RPSM2 successfully.

We denote p j ¯ = p j l q j + u q j 2 , and sort the registration programs from large to

small according to p j ¯ value. Without loss of generality, we assume that the program sequence is { s 1 , s 2 , ⋯ , s k } . The algorithm H for solving P0 is described as follows:

Algorithm H:

Step 1. Select a set of program number { m 1 , m 2 , ⋯ , m x } that may be accepted, 1 ≤ m 1 , m 2 , ⋯ , m x ≤ k . τ = 1 .

Step 2. The number of the accepted programs is m τ , the set of performance program is { s 1 , s 2 , ⋯ , s m τ } .

Step 2.1. Build a robust performance scheduling model RPSM of these accepted programs. Through the method in Section 4.1, we can get the performance sequence of { s 1 , s 2 , ⋯ , s m τ } and the performance cost z 1 .

Step 2.2. Calculate the difference between the revenue of m τ accepted programs

and the penalty cost of ( k − m τ ) rejected programs, namely z 2 = ∑ j = 1 m τ p j − ∑ j = m τ + 1 k f j ,

and get the performance revenue z ( m τ ) = z 2 − z 1 .

Step 3. τ = τ + 1 . If τ ≤ x , go back step 2, otherwise go step 4.

Step 4. Denote m ¯ = arg max { z ( m 1 ) , z ( m 2 ) , ⋯ , z ( m x ) } , select program set { s 1 , s 2 , ⋯ , s m ¯ } to perform, get the maximum performance revenue z ( m ¯ ) and the performance sequence.

We use Matlab 2017b software for numerical experiments. The experiment was carried out under the Windows 10 Professional 64-bit i5-3230M 8GB RAM operation environment. RPSM2 is a 0 - 1 mixed linear programming model, so it can be solved by the Intlinprog function in Matlab 2017b.

Although RPSM2 contains ( n + 1 ) m 2 + 5 m n + 2 m + 1 variables, n m 2 + 8 m n + 3 m − 2 n constraints, after observation, we find that the zero element in the coefficient matrix of the constraints is the majority, which is a sparse matrix [

( 2 n + 4 ) m 2 + ( 18 n + 3 ) m − 4 n [ ( n + 1 ) m 2 + 5 m n + 2 m + 1 ] [ n m 2 + 8 m n + 3 m − 2 n ] .

And as the values of m and n increase, the density of the RPSM2 coefficient matrix decreases sharply. For sparse matrices, Matlab only stores non-zero element values and their positions. Therefore, we use the sparse feature of RPSM2 coefficient matrix to reduce the variable storage space of the computer and improve the running speed of the procedure.

The parameter setting of the problem P0 is as the following:

1) There are 15 registration programs. These programs require 30 performers to participate in. The relationship between the performers and the programs is a

0 - 1 matrix, defined as W = ( w i j ) 30 × 15 . The value of each element is generated by

Matlab according to the random uniform probability.

2) The duration q j of program s j is a bounded interval value, ∀ j = { 1 , 2 , ⋯ , 15 } . The lower bound l q j of q j obeys the random uniform distribution between the interval [3, 5], and the upper bound u q j obeys the random uniform distribution between the interval [6, 10].

3) The unit time waiting cost l c i obeys the random uniform distribution between the interval [10, 20], and the unit time appearance fee u c i obeys the random uniform distribution between the interval [50, 80].

4) The revenue p j of the accepted program s j obeys the random uniform distribution between the interval [5000, 20000], and the penalty cost f j of the rejected program s j obeys the random uniform distribution between the interval [800, 1000].

Due to time and layout restrictions, the decision maker decides to accept 8 to 13 programs, namely m ∈ { 8 , 9 , 10 , 11 , 12 , 13 } . In order to explain the influence of

decision maker’ risk preference on performance cost, μ = 0 , m 2 , m is selected to

conduct experiments, which represent that decision maker is extremely preferences, moderate risk preferences and very conservative.

The numerical experiment results are shown in

In

m | μ = 0 | μ = m 2 | μ = m | ||||||
---|---|---|---|---|---|---|---|---|---|

z | z_{1} | time/s | z | z_{1} | time/s | z | z_{1} | time/s | |

8 | 73,541 | 44,978 | 54 | 36,863 | 81,656 | 57 | 15,598 | 102,921 | 53 |

9 | 77,901 | 54,175 | 70 | 37,835 | 94,241 | 124 | 13,150 | 118,926 | 144 |

10 | 81,488 | 61,789 | 73 | 37,340 | 105,937 | 141 | 9158 | 134,119 | 283 |

11 | 84,608 | 70,084 | 152 | 37,433 | 117,259 | 194 | 4131 | 150,561 | 230 |

12 | 89,063 | 74,499 | 220 | 36,589 | 126,973 | 197 | −651 | 164,213 | 433 |

13 | 91,288 | 79,870 | 223 | 29,452 | 141,706 | 350 | −7779 | 178,937 | 437 |

maker, and the vertical axis is the performance cost z 1 . In the figure, m = 8 means that 8 programs are accepted, others and so on. It can be seen from this figure that in the case of determining the set of the accepted programs, the performance cost increases with the increase of μ , namely if the decision maker avoids the risk, the performance cost will increase and the performance revenue will reduce. Therefore, decision maker can obtain an ideal performance scheduling solution based on their own risk preference.

This paper studies a problem of uncertain duration performance scheduling under accepting strategy, the accepted programs will bring in revenue, and the rejected programs will produce corresponding penalty cost. After determining the set of the accepted performance programs, the decision maker’s risk preference coefficient is introduced, and the min-max robust performance scheduling model of these accepted programs is built, and then it is transformed into a 0 - 1 mixed linear programming model to minimize the performance cost. Based on this, we combined with the revenue of the accepted programs and the penalty cost of the rejected programs, the algorithm H for solving the performance scheduling problem under accepting strategy is proposed, which provides a reference for decision maker to choose the ideal program scheduling scheme.

This article does not limit the sequence of program performance, but in the actual world, the decision maker will arrange a program at the opening or finale. Or depending on the type of program, some programs must not be adjacent. In addition, multi-objective functions can also be studied, such as maximizing overall revenue on the basis of ensuring that the performance cost does not exceed the budget.

The authors declare no conflicts of interest regarding the publication of this paper.

Ding, H., Fan, Y.Q. and Zhong, W.Y. (2018) Robust Optimization of Performance Scheduling Problem under Accepting Strategy. Open Journal of Optimization, 7, 65-78. https://doi.org/10.4236/ojop.2018.74004