# Knapsack Balancing via Multiobjectivization
## 2 Related Works  
In the literature, there exist many extensions to \(\mathcal{S}um\) KP problem formulation (see, e.g., [13; 14]). However, the knapsack problem with a product objective function has received limited attention. In a 2018 paper [15], the authors claim that they were the first to consider the knapsack problem with the product objective function. A 2022 survey [13] lists just three papers on this subject, namely [16], where it is shown that the problem is weakly \(\mathcal{NP}\)-hard; [15], where mixed-integer linear and nonlinear programming formulations of the problem and a dynamic programming algorithm for its exact solution are presented; and [17], where the first fully polynomial time approximation scheme for the problem was presented. The latter has been recently extended in [18] for a wide class of the knapsack problem generalizations.  
It is worth mentioning that the issue of balance of attribute values has emerged in behavioral decision theory early. [PERSON] ([19]) and [PERSON] ([20]) hypothesize that the attractiveness of an object is enhanced if it is an intermediate object in the choice set and is diminished if it is an extreme object, an effect they call _extremeness aversion_. The hypothesis is further elaborated in [21; 22], where it is argued that an object with equal attribute values will be perceived as the compromise object even when it is not the middle object. The hypotheses are supported by data studies but no formal model is offered.  
In [6], the aspect of balance (fairness) in combinatorial optimization is addressed. The authors consider four common measures of the balance of a set of numbers, and in the context of elements of a set _balance_ means _elements as equal as possible_. Based on case studies, the authors conclude that no measure of balance is systematically better than the others. Some general guidelines on what measure of balance to use depending on the given optimization problem are also presented.  
As mentioned, in the literature on the knapsack problem (cf., e.g., [13]), it is argued that a viable option to maximize \(\sum_{i\in S}p_{i}\) is to maximize  
\[\prod_{i\in S}p_{i}\,, \tag{2}\]  
i.e., to maximize the product of all profits of items selected for the knapsack (see, e.g., [15; 18]).  
Our inspiration to use function (2) as a tool to address the issue of attribute balance can be referred to the game theory. In that context, [PERSON] ([7]) proposed to use for the value function the product of attribute values, as an option to the sum of attributes. The latter is much more prone to equally assess alternatives with highly scattered attribute values and alternatives with concentrated attribute values than the former (Figure 1).  
In the Non-Linear Binary Programming (NLBP) context, function (2) has an equivalent form (cf. [15]):  
\[\mathcal{P}rod^{p}(x):=\prod_{i=1}^{n}p_{i}^{x_{i}}\,. \tag{3}\]This equivalency stems from the properties of the exponential function and the fact, that each subset of items \(S\subseteq N\) is represented by vector \(x\in\{0,1\}^{n}\), where \(x_{i}=1\) when \(i\in S\), and \(x_{i}=0\) otherwise, cf. problem (1).  
Function (3) is the binary form of the Cobb-Douglass function:  
\[a\prod_{i=1}^{k}x_{i}^{\alpha_{i}}\,,\]  
where \(x_{i}\) are continuous, \(a\) and \(\alpha_{i}\) are positive parameters, extensively used in the production theory (cf., e.g., [23]). Below, we use function (3) to define the knapsack problem with the product objective function.