The double-row layout problem (DRLP) is a well-known optimization problem that generally deals with the arrangement of machines or other such entities on the two sides along a corridor. Based on the placement of the machinery and equipment in a factory, this study analyzes the DRLP from a new perspective that addresses the arrangement of the given machines subject to the condition that certain designated positions are allocated to certain machines and specific relationships are enforced among the machines. This DRLP is termed as a constrained DRLP (cDRLP). In this study, considering the practical applications, four variations of the cDRLP have been developed for the first time and solved. Additionally, to solve this problem, some constraints (positioning, ordering, and relation constraints) have been defined and classified. Furthermore, mathematical models have been built with the objective of minimizing the total material handling cost. Next, the CPLEX software has been adopted to solve the above variations. Finally, a series of numerical instances have been employed to evaluate the efficiency of the proposed models and obtain the exact results.