In this paper, the authors discuss methods of computing bounds on product graph pebbling numbers. \par Graph pebbling, first introduced by Chung in 1989, can be described as a two-person game. Given a connected graph $G$, the adversary (player 1) chooses a root vertex $r$ and an allocation of pebbles to vertices. In a pebbling move, player 2 chooses two pebbles at the same vertex, moves one to an adjacent vertex, and removes the other. Player 2 wins if she finds a sequence of pebbling moves that results in a pebble at the root vertex $r$. The pebbling number of graph $G$, denoted $\pi(G)$, represents the fewest number of pebbles such that, regardless of the initial configuration and root given by the adversary, player 2 has a winning strategy. The original motivation for graph pebbling was to solve a number-theoretic problem posed by Erdős and Lemke. Kleitman and Lemke gave a solution in the number-theoretic version, and Chung translated their technique into graph pebbling. Since then, the study of graph pebbling has proliferated in its own right, inspiring many applications and variations (see [G. Hurlbert, in {\it Handbook of graph theory}, 1428--1449, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2014; see MR3185588] for more details). \par Graham conjectured that $\pi(G \times H) \leq \pi(G)$ $\pi(H)$, if $G$ and $H$ are connected graphs. Graham's conjecture has been resolved for specific families of graphs including products of paths by Chung in 1989, products of cycles by Herscovici, Snevily, and Foster in 2000 and 2003, products of trees by Snevily and Foster in 2003, and products of fans and wheels by Feng and Kim in 2002. Graham's conjecture was also proved for specific products in which one of the graphs has the so-called 2-pebbling property by Chung, Snevily, Foster, Zou, Liu and Wang in 1989, 2000 and 2009. \par One of the major hurdles in tackling Graham's conjecture is the lack of tractable computational tools. Milans and Clark discussed the complexity of graph pebbling in 2006. Numerically verifying Graham's conjecture for specific graphs has been extremely difficult; as a result, there does not appear to be a discussion regarding whether or not the conjecture is true. A more practical intermediate goal is to improve the bounds on the pebbling numbers of product graphs both in general and in special cases. To this end, Asplund, Hurlbert, and Kenter have done some work in this direction. When seeking a counterexample to Graham's conjecture, it is natural to focus on small graphs that do not possess the 2-pebbling property. The Lemke graph was the first graph of this kind to be discovered. Since then, infinite families of examples have been constructed [see S.~S. Wang, Discrete Math. {\bf 226} (2001), no.~1-3, 431--438; MR1802615]. \par Integer programs (IPs) have been applied to pebbling before, mostly in the context of product graphs. In 2005, Hurlbert wrote an excellent review article titled ``Recent progress in graph pebbling" [Graph Theory Notes N. Y. {\bf 49} (2005), 25--37; MR2202298]. In that paper he wrote as follows: \par ``Typically, the parameter in question can be formulated in terms of an integer linear program, and the relaxation of the program gives rise to the fractional version of the parameter. In the case of graph pebbling, while it is routine to verify in polynomial time that a particular configuration reaches a particular root, no simple matrix condition exists to capture whether every configuration of a fixed size $t$ is solvable for a particular root. Thus minimizing $t$ as an integer program remains elusive, and so it is difficult to say at present whether or not a parameter exists that is dual to $\pi(G)$. Recently, A. Lourdusamy and S. Somasundaram [J. Discrete Math. Sci. Cryptography {\bf 4} (2001), no.~1, 1--15; MR1837596] found a proof that verifies Graham's conjecture for $C_5 \times C_5$. Their proof uses linear programming and lends credence to the above ideas while suggesting that further explorations into the connections between linear programming duality and pebbling numbers may be fruitful and worthwhile indeed. Because it is easy to verify a pebbling solution, the question of deciding whether a given configuration $C$ reaches a given root $r$ is in $\ssf{NP}$.'' \par The authors provide a brief description of Hurlbert's contribution for graph pebbling. Then they build an integer programming approach to pebbling. They directly incorporate the notion of 2-pebbling (that is, pebbling two or more pebbles at a time for a reduced cost). Intuitively, this seems to be a critical ingredient for improving bounds on the pebbling number of product graphs, considering the importance of the ``2-pebbling property'' in previous results. \par The first improvement, in which they extend the partial-pebbling approach so that $G$ and $H$ play symmetric roles in the model, accounts for most of the bound improvement. F.~H.~J. Kenter and D.~E. Skipper [in {\it Combinatorial optimization and applications}, 681--695, Lecture Notes in Comput. Sci., 11346, Springer, Cham, 2018; MR3893550] gave the modeling of 2-pebbling to discuss integer programming methods for pebbling graphs. In this paper, as a second extension, the authors generalize the modeling of 2-pebbling so that the IP applies to any pair of graphs $G$ and $H$ for which the pebbling numbers and 2-pebbling tables are known, with no special graph-specific modifications required. \par Symmetric treatment of $G$ and $H$ require modeling at the full level of granularity of $G \times H$, even though their constraints mostly model at the level of $G$ or $H$. Therefore, in order to bound $\pi(G \times H)$, one must bound the root-specific pebbling numbers for each choice of the root nodes in the vertex set of $G \times H$. In their third extension, the authors embed the IP in an algorithm that finds the overall bound of $\pi(G \times H)$ without exhaustively finding the best possible bound for every choice of root node. In Section 2, they give an overview of pebbling and introduce partial-pebbling. In Section 3, they develop an IP model based on partial-pebbling to bound $\pi(G \times H)$. In Section 4, they study an algorithm that uses the IP model to efficiently search all root nodes for the overall bound on the pebbling number. In Section 5, they enumerate computational results for a variety of product graphs. In Section 6, they list closing observations and discuss future directions.