Capacitated Single Item Problem

Capacitated restrictions enhance the complexity of lot-sizing problems. The objective of this group of lot-sizing problem is to obtain the optimal production quantity or order size for a single product that minimizes the total cost including setup/ordering, inventory, and production/purchasing costs, while meeting the known demands and satisfying capacity constraints over the planning horizon.

Lambrecht and Vanderveken (1979) developed a computationally efficient algorithm for solving a single item dynamic lot-sizing problem with capacity constraints in order to obtain an optimum production schedule that minimizes the total production and inventory costs. Reviews of the literature for this class of problem can be found in Drexl

and Kimms (1997) and Karimi et al. (2003), who offered overviews of optimal and heuristic solution procedures.

Examples of problems and related modeling approaches for dynamic capacitated lot- sizing in a single level system with discrete time representation are the ELSP (Rogers, 1958), continuous setup and lot-sizing problem (CSLP) (Karmarkar & Schrage, 1985), capacitated lot-sizing problem (CLSP) (Günther, 1987), discrete lot-sizing and scheduling problem (DLSP) (Fleischmann, 1990; Salomon et al., 1991), proportional lot-sizing and scheduling problem (PLSP) (Haase, 1994; Drexl & Haase, 1995), and capacitated lot-sizing problem with sequence-dependent setups (CLSD) (Haase, 1996).

The DLSP allows for the production of only one item in each period. The production is further assumed to be “all or nothing”, and the total capacity available per period is used for the production of the scheduled item. The CSLP is equivalent to the DLSP without the “all or nothing” requirement, which can lead to periods with some slack capacity. The PLSP goes one step further and allows for the production of a second item to avoid excessive idle time on the resource. The CLSD assumes a fixed lead time offset of one (macro) period in order to secure a feasible material flow between production stages while small bucket models usually only require a micro period as a fixed lead time offset.

Fleischmann and Meyr (1997) integrated all mentioned models (ELSP, CSLP, CLSP, DLSP, PLSP, and CLSD) within the general lot-sizing and scheduling problem (GLSP). They used a two-fold time structure, where each macro-period is divided into several micro-periods of variable length. A complete sequence of items was established. All mentioned models commonly consider that setup times can only be considered if they do not exceed the length of a period. However, Koçlar and Süral (2005), through a simple modification of the GLSP, showed that setup times exceeding the length of a

period can also be incorporated. The capacitated lot-sizing problem with linked lot sizes (CLSPL) (Suerie & Stadtler, 2003), extends the CLSP with the possibility of setup carryover. The CLSPL belongs to the class of large bucket problems, which allow many setup operations within a single period.

Florian et al. (1980) and Chen and Thizy (1990) proved that the single item CLSP is NP-hard. To deal with the intricacy of the problem and find the optimal solution in reasonable amount of time, numerous studies have applied heuristic and metaheuristic algorithms. Gavish and Johnson (1990) proposed a fully polynomial approximation scheme for solving the single item CLSP. However, their approach is more suitable for continuous models. Sandbothe and Thompson (1990) included backordering into the single item CLSP, and presented a polynomial algorithm for solving the case of constant capacities and a heuristic algorithm for solving the variable production capacity.

Kirca (1990) developed a DP-based algorithm for the single item lot-sizing problem with concave costs and arbitrary capacities. The performance of the algorithm was compared with the performance of the existing procedures in the literature for the general, the constant capacity, and the constant unit cost problems. The computational results demonstrated that proposed algorithm is at least three times faster than the other procedures for all problem types considered. Chen et al. (1994) developed a DP method for the single item capacitated dynamic lot size model with non-negative demands and no backlogging. The proposed approach produced the optimal value function in piecewise linear segments.

Chung (1994) studied a deterministic single product capacitated dynamic lot size model with linear production and holding costs where the setup costs, unit production costs, and capacities are arbitrary functions of the period, and the unit production costs

satisfy the constraint. To solve the problem, the DP algorithm was combined with branch and bound approach. Lotfi and Yoon (1994) considered a multi-period single item production scheduling problem with a deterministic time-varying demand pattern and concave cost functions. Optimal production lot sizes were determined subject to dynamic production capacity and no backlogs in addition to minimizing the total costs of production, setup, and inventory. The proposed algorithm was tested extensively by solving several randomly generated problems with varying degrees of complexity, and showed quite good performance for practical applications.

Hindi (1995a) considered a capacitated single item lot-sizing model where a startup cost is incurred for switching the production facility on, and a separate reservation cost is incurred for keeping the facility on whether it is used for production or not. A tabu search (TS) scheme was developed for solving the problem which was capable of reaching the optimal solution for a large number of varied problem instances. Hardin et al. (2007) analyzed the quality of lower and upper bounds provided by a range of fast algorithms for single item CLSP with time-varying demands.

Akbalik and Pochet (2009) provided valid inequalities for the single item CLSP with step-wise production costs. Constant-sized batch production was carried out with a limited production capacity in order to satisfy the customer demand over a finite horizon. They suggested a cutting plane algorithm for different classes of the proposed valid inequalities. Computational results showed the efficiency of the proposed algorithm compared to the existing methods. Hellion et al. (2012) examined the single item CLSP with concave production and storage costs, and minimum order quantity. They proposed a polynomial time algorithm to solve the problem optimally, and computationally tested the algorithm on various instances.