What if choosing inventory order sizes could be treated like arranging particles connected by springs? This paper turns that physical analogy into an exact lot-sizing solution, avoiding heuristic methods. The optimal lot size in Material Requirements Planning, or MRP, is a fundamental concept in inventory management and business logistics.

Choosing appropriate lot sizes has significant implications for operational costs, supply chain efficiency, and customer satisfaction. That is why lot sizing is considered one of the most critical and challenging problems in production planning. Classical approaches to lot sizing have been extensively studied, and one fundamental technique is the Economic Order Quantity, or EOQ, model.

The EOQ model states that the optimal lot size is achieved when total inventory holding cost equals total ordering cost. Important extensions include Fixed Order Quantity, which orders a fixed quantity when inventory reaches the reorder level, and Lot-for-Lot, which generates exactly the quantity needed to meet the plan.

Periodic Order Quantity is designed for lumpy demand that fluctuates over the planning period, with order quantity based on a period obtained by dividing EOQ by demand per period. A recent study approached the lot-sizing problem through clustering, defining a cluster as a set of elements within the same batch of orders.

Clustering analysis examines the different ways components of a system are grouped, and one-dimensional clustering defines elements with similar characteristics separated by empty spaces on both ends. The cost optimization problem here concerns ordering and storage within the lot-sizing model.

The optimization approach is based on a physical system analogy: the material supply problem is isomorphic to a one-dimensional mechanical system of point particles connected by elastic elements. That analogy enables the natural derivation of cost optimization conditions, leading to an exact solution without relying on heuristic methods.

The proposed model determines lot sizes that minimize the combined ordering and inventory holding costs while significantly reducing computational time. Its effectiveness is assessed using a real-world case study and a benchmark against classical lot-sizing algorithms.

The results demonstrate superior cost optimization and reduced execution time, underscoring the potential to improve material lot-sizing strategies in practical applications. The model is a single-item, M-period economic lot-sizing model. N denotes the number of orders over the M periods, CO is the fixed ordering cost per order, and CH is the fixed holding cost per period for the quantity required for one period.

The simplest strategy orders the required quantity for each period at the beginning of that period, so N equals M. Because each order is consumed within the same period, this strategy has no holding cost and a total cost of CT equal to MCO. Introducing inventory storage, with N less than M, opens up various lot-sizing strategies.

Each strategy begins by ordering the material required for n one periods. After n one periods, those supplies are depleted, and additional material for n two periods is ordered. This process repeats iteratively until the M periods are covered, so each strategy can be represented by a set of N integers.

Equation four corresponds, except for an additive constant, to the potential energy of a one-dimensional system shown schematically in Figure one. The system is an array of N minus one particles connected by N linear springs of elastic constant CH, with total length equal to M.

The figure makes the boundary conditions explicit: the first and last springs attach to fixed walls, while the interior springs connect neighboring blue particles. The labels n one, n two, and n sub N indicate individual spring lengths, with the dotted middle segment emphasizing that the same repeating structure extends across the array.

The optimal strategy for minimizing total cost involves placing N subscript op equally spaced orders. The resulting arrangement gives the optimal total cost. Figure two illustrates the minimum total cost as a function of the number of orders N for a planning horizon of M equal to twelve.

The curves represent gamma values of zero point one six, one, four, nine, thirty-six, and one hundred forty-four. In every case, the order cost CO is set to one, while the holding cost CH is adjusted to achieve the desired gamma values. A logarithmic scale is used on the y-axis to make the curves’ topology easier to see.

The minima move leftward as gamma increases: the blue curve bottoms near six orders, the open red curve near four, and the filled red curve near two, while the highest-gamma curve reaches its minimum at the first order. At the low-gamma end, the filled black curve continues descending across the plotted range and reaches its marked minimum only at twelve orders.

Figure two also shows intermediate scenarios: gamma equals four with N subscript op equal to six, gamma equals nine with N subscript op equal to four, and gamma equals thirty-six with N subscript op equal to two. In all these cases, the number of orders is less than the number of periods in the planning horizon.

Because the ordering cost is higher relative to the holding cost, reducing the number of orders by holding additional raw materials in inventory becomes advantageous. The real-world analysis examines a single-product purchasing and delivery process from a supplier to a customer, to assess the scope and limitations of the theoretical framework.

The data come from a concentrated pulp processing plant specializing in tomato, peach, pumpkin, and plum pulp, among others. The facility is located in the Industrial Park of San Rafael, Mendoza, Argentina. The polyethylene bag used for pulp packaging is one of the company’s most critical products because of its high turnover, and this packaging material is the focus of the study.

The company provides a planning horizon of twenty-five days, an ordering cost per order of four hundred seventy-seven point twenty-two dollars, and a holding cost per period of nine point four eight six four eight dollars. Before this analysis, the material was purchased without following formal optimization criteria.

The cost parameters for the three purchasing strategies are compiled in Table one. The total accumulated cost, CA of i, represents the sum of ordering and holding costs on day i. Figure six compares CA against i for the EOQ model, the company’s strategy, and the proposed model.

The EOQ model and the proposed model show very similar behavior, with a small difference in total cost at the end of the cycle. The company’s strategy appears highly inefficient, highlighting the importance of adopting an appropriate purchasing strategy to improve production costs.

At the right edge of the chart, the endpoint annotations show accumulated costs of two thousand three hundred twenty point forty-nine dollars and two thousand three hundred eighty point thirty-one dollars for the two circular-marker methods, while the square-marker series reaches five thousand eight hundred fifty-nine point forty-nine dollars.

The square-marker curve also exhibits larger step increases across the cycle. In the “number of orders” column, the company strategy uses thirteen orders, compared with three for the EOQ model and two for the proposed model.

The total-cost column reports five thousand eight hundred fifty-nine point forty-nine dollars, two thousand three hundred eighty point thirty-one dollars, and two thousand three hundred twenty point forty-nine dollars, respectively. Execution time is analyzed for the proposed model, the All Possible Combinations model, Fixed Order Quantity, Periodic Order Quantity, Lot-for-Lot, the Wagner-Whitin algorithm, the Silver-Meal algorithm, and Mixed Integer Linear Programming.

The execution time of the proposed model is considerably lower than that of the other models used in the comparison. The study introduces new insights into implementing optimization methods for material lot sizing using an elastic system. The preliminary results are promising, and the proposed theoretical model can be readily extended to incorporate additional complexities.

Future research will focus on variable material demands, quantity discounts, and deteriorating items. Work in this direction is currently underway. The model converts lot sizing into a one-dimensional elastic-system problem, producing equally spaced orders and much shorter execution times.

Its real-world case study also shows behavior close to EOQ and better performance than the company’s strategy.