Knapsack Problem with Uncertain Weights and Values

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1 Noname manuscript No. (will be inserted by the editor) Knapsack Problem with Uncertain Weights and Values Jin Peng Bo Zhang Received: date / Accepted: date Abstract In this paper, the knapsack problem under uncertain environment is investigated, in which the weights and the values are assumed to be uncertain variables. With different criteria, three mathematical models are constructed for the problem, i.e., expected value model, chance-constrained programming model and uncertain measure model. Taking advantage of uncertainty theory, the first two models can be transformed to their corresponding deterministic forms. We also given a uncertain simulation algorithm to seek the approximate best solution for uncertain measure model. Finally, a numerical example is given to show the applications of the models. Keywords Knapsack problem Uncertainty theory Uncertain variable Uncertain measure Uncertain programming model 1 Introduction Knapsack problem (KP) is a typical problem in combinatorial optimization. Given a set of items, each with a weight and a value, our task is to determine the count of each item to be included in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. Belonging to a type of NP-complete problems, knapsack problem is easy to be described, but difficult to be worked out in the case of large scale. The KP has been studied for more than a century, with early works dating as far back as 1897 [21]. It is not known how the name knapsack problem originated, though the problem was referred to as such in the early works Jin Peng Institute of Uncertain Systems, Huanggang Normal University, Hubei , China Tel.: Fax: pengjin01@tsinghua.org.cn Bo Zhang School of Mathematics and Statistics, Huazhong Normal University, Hubei , China

2 2 Jin Peng, Bo Zhang of mathematician Dantzig ( ), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined. The quadratic knapsack problem (QKP) was first introduced by Gallo, Hammer, and Simeone [7] in After that, the QKP has been the subject of considerable research in recent. For example, Wang, Kochenberger, and Glover [27] studied the QKP with multiple constraints. As for other research of knapsack problem, interested readers may refer to [4 6, 9, 13, 22, 30], etc. In the above mentioned research, the knapsack problems were investigated in a deterministic environment, in which the weights and the values are positive crisp values. However, because of technical difficulties, the lack of history data, insufficient information or other reasons, the data for the problems are nondeterministic in many situations. In this case, it is not suitable to employ classical models and methods to study the KP. Some researchers introduced probability theory into the KP. For example, Schilling [26] provided some results on asymptotic value for the random knapsack problem. Lee and Oh [14] considered two growth models for multiclass stochastic knapsack problem. Beier and Vöcking [2] presented the first average-case analysis proving an expected polynomial running time for an exact algorithm for the 0-1 knapsack problem. Other researches for random knapsack problem, see Goyal and Ravi [8], Kleywegt and Papastavrou [11], and Kosuch and Lisser [12], for example. With the fuzzy theory proposed by Zadeh [28] in 1965, many researchers employed fuzzy theory to deal with nondeterministic for KP. Okada and Gen [23] presented a multiple choice knapsack problem with fuzzy coefficients. Abboud, Sakawa, and Inuiguchi [1] presented an interactive procedure for the multiobjective multidimensional 0-1 knapsack problem that takes into consideration the incorporation of fuzzy goals of the decision maker. Lin and Yao [15] investigated knapsack problems in which all of the weight coefficients are supposed to be fuzzy numbers. For more research of the fuzzy knapsack problem, we may consult Shih [25], Kasperski and Kulej [10], Chen [3], and so on. However, a lot of survey showed that there are some subjective information, especially those lack of or even without historical data, behave neither like randomness nor like fuzziness. In order to have a better mathematical tool to deal with this empirical data, uncertainty theory was founded by Liu [16] in 2007 and refined by Liu [19] in Up to now, the uncertainty theory has become a branch of mathematics for modeling human uncertainty and has gained considerable achievement. For example, in order to deal with differentiation and integration of functions of uncertain processes, uncertain calculus was initialized by Liu [17]. Cited by many literatures, Liu and Ha [20] gave the important formula of the expected value of function of uncertain variables. Peng and Yao [24] proposed a mean reversion uncertain stock model and also given the corresponding option pricing formulas. As a type of mathematical programming involving uncertain variables, uncertain programming was founded by Liu [18]. Recently, Zhang and Peng [29] presented some uncertain programming models for Chinese postman problem in uncertain environment.

3 Knapsack Problem with Uncertain Weights and Values 3 For exploring the recent developments of the uncertainty theory, the readers may consult [19]. The paper is organized as follows: Section 2 presents some necessary preliminary concepts and results selected from uncertainty theory. In Section 3, the uncertain knapsack problem (UKP) is introduced and some properties are investigated. In Section 4, three types of uncertain programming models for UKP are presented, which includes expected value model, chance-constrained programming model and uncertain measure model. Section 5 illustrates an example. The last section contains summary and conclusions. 2 Preliminaries Uncertainty theory provides an efficient tool of modeling UKP. In this section, we present some preliminaries from uncertainty theory. Let Γ be a nonempty set, and L a σ-algebra over Γ. For any Λ L, Liu [16] presented an axiomatic uncertain measure M{Λ} to express the chance that uncertain event Λ occurs. The set function M{ } satisfies the following three axioms: (i) (Normality) M{Γ } = 1; (ii) (Self-Duality) M{Λ} + M{Λ c } = 1 for any Λ L; (iii) (Countable Subadditivity) For every countable sequence of events {Λ i }, we have M { i Λ i } i M{Λ i }. The triplet (Γ,L,M) is called an uncertainty space and an uncertain variable is defined as a measurable function from this space to the set of real numbers (Liu [16]). An uncertain variable ξ can be characterized by its uncertainty distribution Φ : R [0, 1], which is defined by Liu [16] as follows Φ(x) = M { γ Γ ξ(γ) x }. Let ξ be an uncertain variable with uncertainty distribution Φ. Then the inverse function Φ 1 is called the inverse uncertainty distribution of ξ. The expected value of uncertain variable ξ is defined by Liu [16] as E[ξ] = M{ξ r}dr M{ξ r}dr provided that at least one of the two integrals is finite. As a useful representation of expected value, it has been proved by Liu [16] that E[ξ] = 1 0 Φ 1 (α)dα where Φ 1 is the inverse uncertainty distribution of uncertain variable ξ.

4 4 Jin Peng, Bo Zhang Liu [16] introduced the independence concept of uncertain variables. The uncertain variables ξ 1, ξ 2,, ξ m are independent if and only if { m } M {ξ i B i } = min M {ξ i B i } 1 i m for any Borel sets B 1, B 2,, B m of R. A real-valued function f(x 1, x 2,, x n ) is said to be strictly increasing if f(x 1, x 2,, x n ) f(y 1, y 2,, y n ) whenever x i y i for i = 1, 2,, n and f(x 1, x 2,, x n ) < f(y 1, y 2,, y n ) x j < y j for i = 1, 2,, n. Liu [16] introduced the following useful theorem to determine the distribution function of the strictly increasing function of uncertain variables. Theorem 1 (Liu [16]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with inverse uncertainty distribution Φ 1 (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). 3 Uncertain Knapsack Problem 3.1 Ordinary Knapsack Problem In the following, we have n kinds of items, labelled from 1 to n. Each kind of item i has a value v i and a weight w i, respectively. We usually assume that all values and weights are nonnegative. Without loss of generality, we can also assume that the items are listed in increasing order of weight in order to simplify the representation. The maximum weight that we can carry in the bag is W. The most common formulation of the problem is the 0-1 knapsack problem, which restricts the number x i of copies of each kind of item to zero or one. Mathematically, the 0-1 knapsack problem can be formulated as max v i x i (1) w i x i W x i {0, 1}.

5 Knapsack Problem with Uncertain Weights and Values Uncertain Knapsack Problem Assume that the item i has an uncertain value ξ i and an uncertain weight η i. Furthermore, ξ i are independent uncertain variables with uncertainty distributions Φ i, i = 1, 2,, n, respectively. Also, η i are independent uncertain variables with uncertainty distributions Ψ i, i = 1, 2,, n, respectively. For simplicity, the uncertain knapsack is denoted by K = (W, ξ, η), where W is the maximum weight that can be carried in the knapsack, the uncertain vector ξ = (ξ 1, ξ 2,, ξ n ) stands for the uncertain values, and η = (η 1, η 2,, η n ) stands for the uncertain weights. In this paper, we assume f(ξ) is the total value of maximum pack scheme of uncertain knapsack K, and the uncertainty distribution of f(ξ) is Φ. Obviously, f(ξ) = ξ i x i is still an uncertain variable. Formally, the uncertain knapsack problem can be described as max ξ i x i η i x i W x i {0, 1}. (2) However, it is not well defined model in natural. Note that there exist many uncertain variables in model (2) and they cannot be ranked directly. In order to optimize the objective, it is inevitable to rank uncertain variables according to some criteria. 4 Uncertain Knapsack Models 4.1 Expected Value Model Expected value is the average value of uncertain variable in the sense of uncertain measure, and represents the size of uncertain variable. The main idea of expected value model is to optimize the expected objective function, that is to say, the more optimal the expected value of objective function is, the better the corresponding solution is. Definition 1 Let K = (W, ξ, η) be an uncertain knapsack, K a feasible pack scheme. Then K is called expected maximum pack (EMP) if E[V ( K )] E[V ( K)]

6 6 Jin Peng, Bo Zhang holds for any pack scheme K, where V ( K ) stands for the total value of expected maximum scheme K and V ( K) stands for the total value of scheme K. If the decision maker wants to find a scheme in the sense of expected value, then the expected value model for UKP can be constructed as the follows: max E[ ξ i x i ] E[ η i x i ] W x i {0, 1}. Equivalently, it can be rewritten as 0-1 integer programming max E[ξ i ]x i E[η i ]x i W x i {0, 1}. In more detail, it can be represented as 1 max x i 0 1 x i 0 x i {0, 1} Φ 1 i (α)dα Ψ 1 i (α)dα W where Φ 1 i and Ψ 1 i are the inverse uncertainty distributions of ξ i and η i, respectively. (3) (4) (5) 4.2 Chance-Constrained Programming Model Chance-constrained programming is another method to deal with optimal problem in uncertain environment. Definition 2 Let K = (W, ξ, η) be an uncertain knapsack, K a feasible pack scheme. Then K is called α-maximum pack (α-mp) if max{v M{V ( K ) v} α} max{v M{V ( K) v} α} for any pack scheme K, and α (0, 1) is a predetermined confidence level.

7 Knapsack Problem with Uncertain Weights and Values 7 The meaning of α-maximum pack can be summarized as follows. Given an α (0, 1), we hope to get a maximum value v and a pack K, where uncertain variable V ( K ) is greater than v with confidence level α. If the decision maker prefers treating the knapsack problem under the chance-constraints, the model for UKP can be constructed as the follows: max v M{ ξ i x i v} α M{ η i x i W } β x i {0, 1} (6) where α and β are the predetermined confidence levels. Equivalently, it can be represented as max v x i Φ 1 i (1 α) v x i Ψ 1 i (β) W x i {0, 1} (7) where Φ 1 i and Ψ 1 i are the inverse uncertainty distributions of ξ i and η i, respectively. 4.3 Uncertain Measure Model The third criterion is to rank uncertain variables by way of uncertain measure, which results in so-called uncertain measure model. Definition 3 Let K = (W, ξ, η) be an uncertain knapsack, K a feasible pack scheme. Then K is called measure maximum pack (MMP) if M{V ( K ) v} M{V ( K) v} for any pack scheme K, where v is a predetermined value.

8 8 Jin Peng, Bo Zhang Given a value v, the measure maximum pack is the optimal pack scheme which is more than v with the largest uncertain measure. The uncertain measure model for UKP can be constructed as the follows: max M{ ξ i x i v } M{ η i x i W } β x i {0, 1} (8) where v is a given value and β is a predetermined confidence level. Equivalently, it can be transformed to max M{ ξ i x i v } Ψ 1 i (β)x i W x i {0, 1} (9) where Ψ i are the inverse uncertainty distributions of η i. Generally speaking, it may be very difficult for us to compute the uncertain measure M{ ξ i x i v } since we could not deduce the uncertainty distribution of n ξ ix i easily. In the following, we shall present a uncertain simulation algorithm to calculate the uncertain measure and then to obtain an approximation optimal solution for model (9). Algorithm Step 1. Input initial parameters n, v, W. Put the initial uncertain measure α = 0, the initial pack scheme S =. Step 2. Generate the pack scheme S = {i 1, i 2,, i p } {1, 2,, n} and check whether it is feasible. If the constrain go to Step 3; Otherwise, go back to Step 2. p k=1 Step 3. Compute the uncertain measure α = M{ Ψ 1 i k (β)x ik W is true, then p ξ ik x ik v } by uncertain simulation: Step 3.1 Input simulation times T, counter t = 0. Step 3.2 Randomly generate the observation value of uncertain variable ξ (j) i k, j = 1, 2,, T. k=1

9 Knapsack Problem with Uncertain Weights and Values 9 Table 1 List of values and weights i value ξ i weight η i i value ξ i weight η i 1 (12, 14, 16) (2, 3, 5) 7 (16, 17, 18) (5, 6, 7) 2 (12, 14, 15) (2, 3, 4) 8 (17, 19, 20) (5, 7, 9) 3 (14, 15, 16) (2, 5, 7) 9 (18, 20, 22) (5, 7, 8) 4 (13, 16, 18) (3, 5, 7) 10 (20, 21, 22) (6, 7, 8) 5 (15, 16, 18) (3, 5, 7) 11 (21, 23, 25) (6, 8, 9) 6 (15, 18, 20) (4, 6, 7) 12 (22, 23, 24) (7, 8, 9) Step 3.3 For j = 1 to T, if p k=1 ξ (j) i k x ik v, then t t + 1. Otherwise, next j. Repeat this step T times. Step 3.4 Put α t/t. Step 4. If α > α, then α α and S S. Step 5. Repeat the second to fourth steps for a given cycle. Step 6. Report the last S as the optimal solution and α as the optimal objective value. 5 Numerical Experiment In order to show the applications of the models as mentioned above, we shall present an example in this section. For the convenience of description, we summarize the problem as follows. Suppose that there are twelve items and a bag, in which we can carried the maximum weight is W. Our goal is to make a pack scheme such that the total weight is less than or equal to W and the total value is as large as possible. At the beginning of pack scheme, the decision maker needs to obtain the basic data, such as the weight and the value of each item. However, due to economic reason, technical difficulties, the lack of history data, insufficient information or other reasons, usually the decision maker cannot get these data exactly. In this case, we have to obtain the uncertain data by means of experts estimation. Assume that all uncertain variables are zigzag uncertain variables, which are listed in Table 1. And, we also assume that the maximum weight that we can carry in the bag is W = 50. Then the expected value model for the uncertain knapsack problem is equivalent to the following model: subject to max E[ξ i ]x i E[η i ]x i 50 x i {0, 1}, for i = 1, 2,, 12. (10)

10 10 Jin Peng, Bo Zhang Table 2 List of α-maximum pack when β = 0.9 α α-maximum pack v 0.95 (1, 2, 7, 9, 10, 11, 12) (1, 2, 7, 9, 10, 11, 12) (1, 2, 7, 9, 10, 11, 12) (2, 5, 6, 7, 10, 11, 12) (1, 2, 7, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) 133 With the help of mathematical software (e.g., LINGO), we can solve this 0-1 integer programming problem. Here we get the optimal solution of the model is (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, x 10, x 11, x 12 ) = (1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1), that is to say, in order to obtain the decision with maximum expected value, the decision maker should carry the items 1, 2, 3, 4, 5, 6, 10, 11, 12 to the bag, the corresponding pack scheme we denoted as (1, 2, 3, 4, 5, 6, 10, 11, 12). And the expected maximum total value of the items that we carry in the bag is Assume α = 0.95, β = 0.9, then the chance-constrained programming model for the uncertain knapsack problem is equivalent to the following model: max v subject to Φ 1 i (0.05)x i v Ψ 1 i (0.9)x i 50 x i {0, 1}, for i = 1, 2,, 12. (11) Also, for this 0-1 integer programming problem, we get the 0.95-maximum pack is (1, 2, 7, 9, 10, 11, 12) and the optimal total value of the items that we carry in the bag with confidence level 0.95 is Assume β = 0.9, for different α, we obtain the corresponding α-maximum pack, which are list in Table 2.

11 Knapsack Problem with Uncertain Weights and Values 11 Table 3 List of uncertain maximum pack when β = 0.9 v uncertain maximum pack α 124 (1, 2, 7, 9, 10, 11, 12) (2, 3, 6, 7, 10, 11, 12) (2, 3, 6, 7, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) (1, 2, 6, 9, 10, 11, 12) 0.50 Given a value v = 126, and assume β = 0.9. Now consider the following maximum uncertain measure model for the uncertain knapsack : 12 max M{ ξ i x i 126} Ψ 1 i (0.9)x i 50 x i {0, 1}, i = 1, 2,, 12. (12) According to the uncertain simulation algorithm, we obtain the approximate optimal solution is α = 0.95, and the corresponding measure maximum pack for uncertain measure model is (2, 3, 6, 7, 10, 11, 12). This means that the obtained measure maximum pack is such a solution that {f(ξ) 126} with the largest uncertain measure α = 0.95, where f(ξ) is the total value function. Assume β = 0.9, for different value v, we obtain the corresponding measure maximum pack, which are list in Table 3. 6 Conclusion In practical application, very often we are lack of observed data about the unknown state of nature due to economic reasons or technical difficulties. Usually, we obtain the uncertain data by means of experts estimation. This paper mainly studied uncertain knapsack problem, in which the weight and value of each item are supposed to be uncertain variables. According to different principles, expected value model, chance-constrained programming model and uncertain measure model are proposed for the knapsack problem. Taking advantage of properties of uncertainty theory, expected value model and chance-constrained programming model can be turned into the corresponding

12 12 Jin Peng, Bo Zhang deterministic forms. An uncertain simulation algorithm is given to obtain the approximate optimal solution for uncertain model. In addition, a numerical example was presented to show the application of the models and algorithm. It is worth mentioning that these models are constructed from the different points of view. We cannot conclude which model is the best in the process of decision making. The application of different models is completely based on the preference of the decision maker. It is worth pointing out that there are different types of nondeterministic environments in the real life, such as random uncertain environment, fuzzy uncertain environment, uncertain random environment, uncertain fuzzy environment, etc. This paper only studies the knapsack problem in uncertain environment, and the knapsack problem in more complex environments may become new topics in our further research. Acknowledgements This work is supported by the National Natural Science Foundation (No ), the Hubei Provincial Natural Science Foundation (No.2010CDB02801), and the Scientific and Technological Innovation Team Project (No.T201110) of Hubei Provincial Department of Education, China. References 1. N. Abboud, M. Sakawa, M. Inuiguchi, A fuzzy programming approach to multiobjective multidimensional 0-1 knapsack problems, Fuzzy Sets and Systems, Vol. 86, No. 1, 1-14, R. Beier, B. Vöcking, Random knapsack in expected polynomial time, Journal of Computer and System Sciences, Vol. 69, No. 3, , S. Chen, Analysis of maximum total return in the continuous knapsack problem with fuzzy object weights, Applied Mathematical Modelling, Vol. 33, No. 7, , M. Dyer, N. Kayal, J. Walker, A branch and bound algorithm for solving the multiplechoice knapsack problem, Journal of Computational and Applied Mathematics, Vol. 11, No. 2, , D. Fayard, V. Zissimopoulos, An approximation algorithm for solving unconstrained twodimensional knapsack problems, European Journal of Operational Research, Vol. 84, No. 3, , M. Fujimoto, T. Yamada, An exact algorithm for the knapsack sharing problem with common items, European Journal of Operational Research, Vol. 171, No. 2, , G. Gallo, P. Hammer, B. Simeone, Quadratic knapsack problems, Mathematical Programming Studies, Vol. 12, , V. Goyal, R. Ravi, A PTAS for the chance-constrained knapsack problem with random item sizes, Operations Research Letters, Vol. 38, No. 3, , J. Gorski, L. Paquete, F. Pedrosa, Greedy algorithms for a class of knapsack problems with binary weights, Computers & Operations Research, Vol. 39, No. 3, , A. Kasperski, K. Kulej, The 0-1 knapsack problem with fuzzy data, Fuzzy Optimization and Decision Making, Vol. 6, No. 2, , A. Kleywegt, J. Papastavrou, The dynamic and stochastic knapsack problem with random sized items, Operations Research, Vol. 49, No. 1, 26-41, S. Kosuch, A. Lisser, On two-stage stochastic knapsack problems, Discrete Applied Mathematics, Vol. 159, No. 16, , 2011.

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