# Raw Material Purchasing Optimization Using Column Generation
## 2 Literature Review  
Research on raw material purchasing can be generally divided into two categories: single-item procurement and multi-item procurement. The single-item raw material purchasing problem is the first studied problem [8]. However, general single-item procurement problems are NP-hard, so most solutions are heuristic [9]. [PERSON] and [PERSON] [16] examine several heuristics for the single-item uncapacitated lot-size version for single-item purchasing and show that the Silver-Meal heuristic can provide cost-effective performance superior to that of the W-W algorithm. [PERSON] [17] tests a heuristic called the part-period algorithm for the above problem and shows that the part-period algorithm is well-suited for industries whose demand forecast extends for a limited number of periods. [PERSON] et al. [18] studied cyclic ordering policies for single-item purchasing to minimize total periodic ordering costs consisting of fixed ordering costs, variable purchasing costs, and inventory holding costs. They show that their heuristic provides better results compared to other methods in the literature. [PERSON] [19] proposes a heuristic for supplier selection and purchase order sizing. The heuristic has been tested as a component of the advanced planner and optimizer (APO) of software of SAP. [PERSON] et al. [20] propose a branch and bound-based optimal approach for single-item purchasing with quantity discounts. Experimental results show the performance of the optimal approach in computational time measurement. [PERSON] et al. [21] propose an integration model for lot-sizing and safety strategy placement for handling demand uncertainty. The goal is to store a proper quantity of items to satisfy demand but concurrently avoid shortages and excess inventory. The multi-objective model is selected and a computation-effective method is used for the solution. There is much literature on single-item procurement and interested readers can refer to relevant publications. For the multi-item raw material purchasing problem, there are more literature studies on related issues. [PERSON] and [PERSON] [1] constructed a multi-objective linear programming (MOLP) model for RMP purchasing and supplier selection, where three optimization objectives, the minimization of the total purchasing cost, tardy-delivery ratio, and scrap fraction are used. The constraints considered include the purchasing budget, production demand, inventory capacity, and technology specifications. The model is solved using the point estimate weight-sums (PEWS) method. [PERSON] et al. [11] proposed a mixed integer linear programming (MILP) model for production lot size and raw materials purchasing based on the process industry background. The optimization objective is to minimize the raw materials costs, ordering costs, inventory holding costs, setup costs, and production costs. The constraints include purchasing, inventory control, packing tasks, and multipurpose storage tank control. The model is solved with the CPLEX solver. [PERSON] et al. [12] considered raw material purchasing with price fluctuations. Their objective is to maximize the net present value (NPV). The constraints include inventory movement and backorders. The model is solved by [PERSON]'s maximum principle. [PERSON] et al. [22] considered a multi-product and multi-period production system for minimizing the purchasing cost and inventory costs. They propose a genetic algorithm (GA) for the solution of the problem. [PERSON] and [PERSON] [23] proposed a multi-objective model that integrates project scheduling and raw material purchasing simultaneously and solved it with NSGA-II and Taguchi. [PERSON] et al. [13] proposed a multi-criteria decision-making (MCDM) approach called fuzzy axiomatic design (FAD) to select the best green supplier for a Singapore-based plastic manufacturing company for purchase and supply cycle to ensure the supplier of goods and meeting of green criteria. [PERSON] and [PERSON] [14] developed a data-driven approach for raw materials purchasing.