Raw Material Purchasing Optimization Using Column Generation

1. Introduction

Raw material purchasing is crucial for steel enterprises because it is related to the supply of raw materials and cost savings [1]. The raw material purchasing of large-scale steel enterprises is characterized by many varieties, large quantities, and high costs. The raw materials purchased by steel enterprises include three types: iron, coal, and auxiliary materials. Iron includes iron ore, pellets, sinter, pig iron, scrap steel, and so on. Coal includes all pulverized coal, lump coal, and coke. Auxiliary materials, also known as solvents, include limestone, quicklime, serpentine, etc. In a large-scale iron and steel enterprise, the purchase volume of raw materials reaches tens of millions of tons and the purchase expenditure reaches tens of billions of yuan. Therefore, it is very important to make the optimal raw material purchasing decisions.

The raw material procurement optimization is to determine the purchasing quantities of raw materials in some given periods or cycles, make the purchasing total cost minimal, and meet the production demand. The purchasing cost includes material price, procurement set-up cost (including negotiation and communication, transportation, etc.), and inventory cost (including capital occupation, management, maintenance, etc.). Raw material purchasing is often complex because it involves many factors, such as supplier selection, raw material price fluctuations, inventory control, product demand, etc., and its solution also involves many areas [2,3,4], and it has been a hot area of study in management and operations research [5,6,7].

Determining the purchase amount of raw materials is closely related to the number of varieties of raw materials, which affects the complexity of the problem. According to the number of varieties purchased, the raw material purchasing problems can be naturally divided into two categories: single-item procurement and multi-item procurement. Single-variety purchasing decisions determine the purchase of a single raw material for many periods. The famous economic order quantity (EOQ) model is a single-item purchasing example [8]. The EOQ assumes the demand is steady and constant, the price of raw materials is unchanged, and the supply of raw materials is sufficient. The EOQ model has an analytic solution. However, when the assumption of steady-state demand is dropped, the single-item procurement problem will become difficult to solve. W-W [9] problem is a dynamic version of the EOQ, it allows demand to vary from period to period and can be solved with dynamic programming, however, as the number of periods increases (e.g., 300), the solution is time-consuming. The W-W problem still assumes the satisfaction of requirements is unlimited. When there is a limitation on the availability of raw materials, the problem becomes NP-hard [10] called the single-item capacitated lot-size problem (SCLP). Most early research interest is focused on the kind of problems [11].

For the multi-item purchasing problem, since complex material structure and process relationships often need to be considered, mathematical programming models are the most common choice [1,12,13]. Mathematical programming models have the advantage of being able to handle complex constraints. There are also some other methods to study the multi-item procurement problem, such as the fuzzy mathematics-based methods and the data-driven methods [14,15]. Fuzzy mathematics based on fuzzy logic has been widely used in many fields, like industrial process control, production planning and scheduling, image processing, etc. [16]. Data-driven methods are a hot research topic nowadays and are used in almost all fields [7].

The main contributions of this paper are:

We present a mixed integer programming for this large-scale raw material procurement problem.

We present a column generation solution for the RMP problem.

The rest of this paper is organized as follows. Section 2 gives a literature review. Section 3 states the model and the solving method – the column generation. Section 4 gives the experimental results. Section 5 concludes this paper.

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 [8,9], however, general single-item procurement problems are NP-hard, so most solutions are heuristic [10]. Blackburn and Millen [17] 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. DeMatteis [18] 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. Ekici et al. [19] studied cyclic ordering policies for sing-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. Tempelmeier [20] proposes a heuristic for supplier selection and purchase order sizing. The heuristic has been tested as a component of the advanced planner and optimizer software of SAP. Hamid Mirmohammadi et al. [21] 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. Kania et al. [22] 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. Gao and Tang [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. Cunha et al. [12] 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. Arnold et al. [13] 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 Pontryagin’s maximum principle. Ahmed et al. [23] 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. Kazemi and Davari-Ardakani [24] proposed a multi-objective model that integrates project scheduling and raw material purchasing simultaneously and solved it with NSGA-II and Taguchi. Kannan et al. [14] 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. Muteki and MacGregor [15] developed a data-driven approach for raw materials purchasing. A partial least squares (PLS) regression is presented to extract the latent purchasing raw materials as input for a sequential quadratic programming (SQP) model that optimizes the raw materials purchasing.

To sum up, although there are some studies on the problem of multi-item procurement, there are very few studies on iron-steel raw materials, and the existing solution methods are mainly heuristic, or the optimal solution is only for small-scale problems. The model and solution proposed in this paper can effectively solve the large-scale iron-steel raw material purchasing problem that fills the gap.

3. Method

3.1. Model

The real-world RMP problem is from a certain large-scale iron and steel enterprise (below called M-Steel) in China. The M-Steel purchases about 15 million tons of raw materials per year with about 4.5 billion of CNY of purchase expenditure and the purchased number of raw material items is over 3000. The total raw materials purchasing cost accounts for more than 70% of the total operating cost of M-Steel. The purchasing cost is composed of raw materials price, purchasing set-up cost, and inventory cost. The optimization problem is to minimize the sum of the above three costs.

4. Assumptions

A1: a single period is set as a month; therefore, a one-year length includes 12 periods.

A2: the inventory of each item of raw materials in each period is known.

A3: the demand quantity of each item in each period is known.

The RMP model is as follows:

$$\min {\displaystyle \sum _{i=1}^N{\displaystyle \sum _{t=1}^T(p_{it}{x}_{it}+}{s}_{it}{Y}_{it}+h_{it}{I}_{it})}$$

(1)

Subject to

$$I_{i,t-1}+x_{it}+I_{it}=d_{it},i=1,...N,t=1,...,T,$$

(2)

$$x\le \mathrm{MY}{\mathrm{it}}_{}, \forall i,t,$$

(3)

$${\displaystyle \sum _{i=1}^N{r}_{ki}{x}_{it}}\le C_{kt},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=1,...T,k=1,...,K,$$

(4)

I_i0 = 0, I_iT = 0, i=1, …, N,

(5)

x_it ≥ 0, I_it ≥ 0, i=1, …, N, t=1, …, T,

(6)

The meanings of notations in the model are as follows.

Indices and sets:

i index of the item

t index of the period

k index of the resource

N number of items

T number of periods

K number of resources

Parameters:

d_it demand of item i in period t

r_1i unit capital budget absorption of item i

r_2i unit transportation capacity absorption of item i

s_it purchasing set-up cost of item i in period t

p_it unit base price of item i in period t

h_it unit inventory holding cost of item i in period t

C_1t capital budget in period t

C_2t transportation capacity in period t

Decision variables:

x_it purchase amount of item i in period t

I_it inventory level of item i at the end of period t

s_it purchasing set-up cost of item i in period t

${\mathrm{Y}}_{it=}\{\begin{array}{cc}\hfill 1,& \mathrm{if}\mathrm{item}i\mathrm{is}\mathrm{purchased}\mathrm{in}\mathrm{period}\mathrm{t}\hfill \\ \hfill 0,& \mathrm{otherwise}\hfill \end{array}$

The Model is a MILP. The objective (1) is to minimize the sum of raw materials price, purchasing set-up costs, and inventory holding costs. Constraint (2) represents supply-demand balance. Constraint (3) represents the incidence relation of Y_it and x_it, where, M is a sufficiently big number (e.g., M ≥$max{\displaystyle {\sum }_{k=t}^T{d}_{ik}}$), Y_it =1, if and only if x_it > 0. Constraint (4) represents for capacity absorption of each resource not exceeding the corresponding resource restriction available in each period. Constraint (5) without loss of generality assumes that the starting inventory of period 1 and the ending inventory of period T are 0. Constraint (6) represents all purchase quantity and inventory are non-negative.

The above MILP model is a single-level, multi-item capacitated lot-sizing problem (CLSP) [25,26]. However, it is a large-scale mixed integer programming problem and is thus difficult to solve.

4.1. Column Generation Solution

Solving a CLSP is NP-hard [10], therefore, the most existing solutions are heuristic [27,28,29], and those exact solutions are only for small-sized problems [30,31]. However. In the actual industrial production environment, the optimization problems are mostly large-sized, such as the RMP problem in this paper. We developed a column generation (CG) solution [32,33,34] for the RMP because it can handle large-sized problems. In the CG algorithm, we first convert the original problem to an equivalent set partitioning (SP) one and then use the column generation to solve the resulting SP problem.

4.1.1. Set-Partitioning Reformulation

We note that for the original MILP problem, if for each item i ∈ Ω, we introduce a set of schedules, denoted by Ω_i, where, a schedule is a T-dimension vector with the purchasing quantities in T periods for item i. Then, deciding the purchasing quantity of item i in each period t is equivalent to identifying a schedule j∈Ω_i for that item. we consider a specific set of schedules that satisfy the extreme flow conditions for each item i, i.e. I_it-1 · x_it = 0 [9]. These schedules sometimes also called W-W schedules which with the property the demand at a period is completely supplied by either inventory or purchasing, and/or a purchasing activity happens if and only if the purchased quantity satisfies the demands of an integral number of periods for any item. Based on the idea, the RMP model can be reformulated as a set partitioning problem:

$$\min {\displaystyle \sum _{i\in \Omega }^{\hspace{0.17em}}{\displaystyle \sum _{j\in {\Omega }_i}^{\hspace{0.17em}}{b}_{ij}{\theta }_{ij}}}$$

(7)

Subject to

$${\displaystyle \sum _{j\in {\Omega }_i}^{\hspace{0.17em}}{\theta }_{ij}=1},\forall i\in \Omega$$

(8)

$${\displaystyle \sum _{i\in \Omega }^{\hspace{0.17em}}{\displaystyle \sum _{j\in {\Omega }_i}\hspace{0.17em}}{R}_{ijt}^k{\theta }_{ij}\le }\stackrel{\hspace{0.17em}}{\hspace{0.17em}}{C}_{kt}^{\hspace{0.17em}},\forall t,k=1,\mathrm{...},K$$

(9)

$$\mathsf{\theta }\text{∈}\{0,1\text{}},\forall i\in \Omega ,j\in {\Omega }_i$$

(10)

Where, decision variable θ_ij equal to 1 when schedule j is selected, and 0 otherwise. K is the number of resources, parameter b_ij is the cost associated with the j^th schedule for item i and parameter $R_{ijt}^k$ is the k^th resource requirement in period t for schedule j, parameter $C_t^k$ is the k^th resource capacity available in period t. Let x_ijt, Y_ijt, I_ijt denote the purchasing quantity, purchasing set-up, and inventory variables associated with the schedule j for item i in period t, respectively. Consequently, hold:

$$b={\displaystyle \sum _{t=1}^{\mathrm{T}}(p_{it}{x}_{ijt}+s_{it}{Y}_{ijt}}+h_{it}{I}_{ijt})$$

(11)

$$R_{ijt}^k=r_{ki}{x}_{ijt}$$

(12)

In the SP model, the objective function (3.1) minimizes the total sum of purchasing costs, set-up costs, and inventory holding costs. The partitioning constraint (3.2) requires that for each item i, only a schedule can be selected. The constraint (3.3) requires the resource absorption of all schedules not to exceed the capacity of each resource in the period t available. Constraint (3.4) indicates the decision variable θ_ij is a 0-1variable.

The W-W schedules Manne called dominant schedules, because there are 2^T-1 dominant schedules per item i, generating them may be rather time-consuming or impractical and the resulting SP problem is far too large to solve. Therefore, a column generation algorithm is introduced to solve the SP problem, in which not all N·2^T-1 of schedules are generated explicitly, alternatively, many schedules are handled implicitly by column generation techniques [35], and thus, the large-sized SP problem is effectively resolved.

4.1.2. Column Generation Principle

column generation algorithms are used to solve a MILP problem with many many columns. For handling the huge number of variables (columns) of the MILP, the column generation algorithm instead of explicitly enumerating all columns in the constraint matrix, called the master problem (MP), only a very small subset of columns for the initial solution is used and the remaining columns can be added only when needed. We refer to an MP with only a subset of columns as the restricted master problem (RMP). The column generation algorithm solves the large-sized MILP by solving some LP relaxations of RMPs. For finding the added columns to an RMP, we check if there is a potential adding column with a negative reduced cost by solving an optimization subproblem, called the pricing problem. If none can be found, the current LP relaxation solution to the RMP is optimal for the original MILP. If one or more such columns are found, they will be added to RMP, and the solution process is repeated.

4.1.3. LP Relaxation Solution

Suppose that for each item i ∈ Ω, a subset of feasible schedules Ω_i' is already found and the RMP has a feasible solution θ, and let U, V be the associated dual solutions, and the unrestricted dual variables u_i(i∈Ω) are associated with the partitioning constraints and the negative dual variables v_kt (k=1,…, K; t=1, …, T) are associated with the k^th resource constraint. From linear programming duality we know that θ is the optimal solution for LP if and only if for any i∈Ω and any j∈Ω_i, the reduced costβ_ij is non-negative, i.e.,

$${\mathsf{\beta }}_{\mathrm{ij}}={b_{\mathrm{ij}}}_{}-{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{K}}\hspace{0.17em}}{v}_{kt}{r}_{ki}{x}_{ijt}}$$

(13)

Testing the optimality of θ for LP can thus be done by solving the following pricing problem

$$\mathsf{\beta }=\underset{j}{\min \arg }{\displaystyle \sum _{t=1}^{\mathrm{T}}((p_{it}-{\displaystyle \sum _{k=1}^K{v}_{kt}}{r}_{ki})x_{ijt}+s_{it}{Y}_{ijt}}+h_{it}{I}_{ijt})$$

(14)

4.1.4. Pricing problem

The pricing problem consists of N minimization subproblems.

β_i = $\underset{j}{\min \arg }$${\displaystyle \sum _{t=1}^{\mathrm{T}}((p_{it}-{\displaystyle \sum _{k=1}^K{v}_{kt}}{r}_{ki})x_{ijt}+s_{it}{Y}_{ijt}}+h_{it}{I}_{ijt})$ - u_i, for i=1,…,N. Note that for fixed i, the subscription may be eliminated. Instead of enumerating all schedules j for the negative reduced costs for item i, the pricing problem instead solves an uncapacitated single-item lot-sizing problem (ULS_i) to find the column with the most negative reduced cost:

min β

(15)

s.t.

$${\mathrm{I}}_{\mathrm{t}-1}+\mathrm{x}{\mathrm{t}}_{\hspace{0.17em}}– \mathrm{I}{\mathrm{t}}_{\hspace{0.17em}}={\mathrm{d}}_{\mathrm{t}},\forall t$$

(16)

$$x_t \text{≤} \mathrm{MY}{t}_{}{,}_{}\forall t$$

(17)

$$x_{t}I\text{≥} 0,\forall t$$

(19)

Where the meanings of the occurred notations are the same as before. This ULS_i problem can be easily solved by a dynamic programming algorithm [9]. If the cost for all i∈Ω, hold: β ≥ 0, then, the LP relaxation is solved optimal, otherwise the solution (column) (with β < 0) is added to the RMP.

4.1.5. Integer Solution

The LP relaxation solution from (3.4) is not the final integer solution we needed. To obtain integer solutions, we make use of the branch and bound technique to obtain the integer solutions with a branching scheme designed as follows [36]:

Provided that the LP relaxation solution of the SP problem, ${\displaystyle \sum _{j\in {\Omega }_i}{x}_{ijt}{\theta }_{ij}}=\alpha$, is fractional, then, we can implement a branching scheme for the MILP: on one branch we let ${\displaystyle \sum _{j\in {\Omega }_i}{x}_{ijt}{\theta }_{ij}}\le ⌊\alpha ⌋$ and on the other branch we require ${\displaystyle \sum _{j\in {\Omega }_i}{x}_{ijt}{\theta }_{ij}}\ge ⌈\alpha ⌉$. The branching scheme can be performed in the restricted main problem (RMP) by deleting columns for item i that violates the upper bound on component t on the first branch or the lower bound on the other branch. When a new column is generated for item i an upper bound of $⌊\alpha ⌋$ on component t is added to the pricing problem in the first branch and a lower bound of $⌈\alpha ⌉$ on the second branch. Then execute the traditional branch and bound procedure, until to obtain the optimal integer solution. Now, we can give the complete column generation algorithm:

5. Experimental Results

5.1. Algorithm Test

We have implemented the column generation algorithm with Visual C++. For testing the effectiveness and rightness, we have used the CG algorithm to solve two examples: No.1 from the literature [26], is a 12 item × 10-period CLSP problem, and No.2 is a group of examples randomly generated for a 10-task × 4- machine, generalized assignment problem (GAP) [34]. The original data is left out and here we only list the computational results for verifying the effectiveness, see Table 2 and Table 3.

In Table 2, The LV, DS, ABC, ABC₁₀, ABC₇₂, and LPHP2 are six heuristics for comparison, the interested reader is referred to the related literature. In Table 3, the LP value indicates the optimal objective function value of the LP relaxation of the root node of the MILP; the MILP value is the optimal objective function value of the MILP; the Nodes the number of nodes fathomed; the Dual gap = (MILP value – LP value) / LP value × 100%; the CPU time the running time of the algorithm. It can be seen that the column generation solution is very effective for most computational examples and at least we can evaluate the quality of the solution, for example, in example No.1, although the solution we obtained is not the best the maximum deviation is 2.52%.

5.2. Case Study

Now, we make use of the CG algorithm for solving the RMP in Section 2. M Steel Plant purchases its production raw materials from month to month. The number of items of raw materials is over 3000. However, in the actual decision-making process, it is not necessary to optimize the procurement of all raw materials, in fact, just choosing those raw materials with large quantities and high prices can greatly reduce the total procurement cost. These materials include iron ore and some coal. We selected 50, 100, 150, and 200 raw materials for purchasing optimization, respectively. The data is from M Steel Plant and the computational results are shown in Table 4.

The computational results show the proposed CG solution is effective. According to different procurement varieties and quantities, we give 4 raw material procurement patterns, corresponding to 50,100,150, and 200 raw materials respectively. The computational times are not more than 90 minutes, and the dual gaps are less than 0.2% for all patterns but one. This shows the efficiency and effectiveness of the algorithm. It is estimated that the application of the algorithm will save at least 80 million yuan in procurement costs for the enterprise every year.

6. Conclusion

The raw material purchasing optimization in steel plants is important because the RMP cost accounts for a significant fraction. In this paper, we proposed a MILP model and a column generation solution for the RMP for a large-scale steel plant. The computational results show that this solution is quite effective. For different varieties and quantities, we recommend several patterns for the selection of raw materials. The results show that it can save a lot of procurement costs, and it is estimated that it can save more than 80 million yuan of procurement costs every year

During the development of the solution, we note that two issues deserve more attention in the future. First, developing effective heuristics for finding feasible solutions for the lot-size problem is still a good topic, especially with tight capacity constraints. The second is the combination of column generation and neighborhood search techniques.