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For an unbalanced cooperative game, its grand coalition can be stabilized by some instruments, such as subsidization and penalization, that impose new cost terms to certain coalitions. In this paper, we study an alternative instrument, referred to as cost adjustment, that does not need to impose any new coalition-specific cost terms. Specifically, our approach is to adjust existing cost coefficients of the game under which (i) the game becomes balanced so that the grand coalition becomes stable, (ii) a desired way of cooperation is optimal for the grand coalition to adopt, and (iii) the total cost to be shared by the grand coalition is within a prescribed range. Focusing on a broad class of cooperative games, known as integer minimization games, we formulate the problem on how to optimize the cost adjustment as a constrained inverse optimization problem. We prove NP -hardness and derive easy-to-check feasibility conditions for the problem. Based on two linear programming reformulations, we develop two solution algorithms. One is a cutting-plane algorithm, which runs in polynomial time when the corresponding separation problem is polynomial time solvable. The other needs to explicitly derive all the inequalities of a linear program, which runs in polynomial time when the linear program contains only a polynomial number of inequalities. We apply our models and solution algorithms to two typical unbalanced games, including a weighted matching game and an uncapacitated facility location game, showing that their optimal cost adjustments can be obtained in polynomial time. History: Accepted by Area Editor Andrea Lodi for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72022018 and 72091210]; the Research Grants Council of the Hong Kong SAR, China [Grant 16210020]; Hong Kong Polytechnic University [Grant P0032007]; and the Youth Innovation Promotion Association, Chinese Academy of Sciences [Grant 2021454]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0268. [ABSTRACT FROM AUTHOR]

The parameters of fractures, such as fracture geometries, placement, and conductivity, are critical for the production of horizontal wells in tight gas reservoirs. Current studies on fracture parameter optimization, especially those using the optimization algorithms, are often based on the assumptions that the fractures are uniformly placed with equal length and conductivity. However, non-uniform designs of fracture spacing have been proved to be effective in reducing fracture interference as well as increasing the well production, especially with strong rock heterogeneity. In addition, most of the current literature focuses on identifying optimal hydraulic fracture geometries without considering the fracture propagation process, which may oversimplify the evaluation of completion cost and result in fracture geometries that cannot be realized in practice. In this paper, an automatic optimization process, which could efficiently couple the fracture propagation, is proposed to optimize the fracture parameters. The genetic algorithm (GA) with penalty function method is used to optimize the constraint and non-linear problem. A modified PKN fracture propagation model is established to quickly predict the hydraulic fracture geometries. Fracture locations and fluid injection volume are sequentially optimized with consideration of heterogeneous rock properties. Model parameters are taken from typical tight gas horizontal wells in Jinqiu gas field in Sichuan Province, China. The results show the importance of permeability on both fracture length and production in tight gas well. The lower the permeability, the more significant the production increase after fracture location optimization. More fractures tend to be placed in area with small permeability and fewer fractures are required in highly permeable regions. With an increase in rock heterogeneity, the distribution of fractures becomes more uneven. Conversely, increasing the number of fractures can lead to a more even distribution of fractures. More fluid should be injected in low-permeability regions according to the optimization results. The optimization method is applied to a practical tight gas well in Sichuan Basin and the cumulative gas production can be increased by 6.5% to 8% when compared with the initial uniform designs. • A modified PKN model is proposed for quick prediction of fracture propagation, validated against numerical simulations. • GA is used to optimize fracture locations by considering rock heterogeneity's impact on fracture propagation and production. • Sequential optimization of fracture placement and injection volume is conducted with its reliability verified. [ABSTRACT FROM AUTHOR]