Uncapacitated Single Item Problem

The single level single item problems with no capacity constraint were at the advent of developments in the lot-sizing and scheduling arena. The EOQ model was introduced by Harris (1913), which assumes a constant demand rate for a single item, infinite planning horizon, and continuous time scale with the aim of minimizing the sum of ordering and inventory holding costs. Wagner and Whitin (WW) (1958) investigated the lot-sizing problem for a single item with unlimited capacities over a finite planning horizon divided into discrete periods. Demand and costs were accordingly time-varying.

Numerous model formulations and solution procedures have been proposed for the uncapacitated single item lot-sizing problems.

Zangwill (1969) improved the WW basic model to include backlogging of demand. Approximate solutions to the single item, single stage uncapacitated lot-sizing problem were suggested by DeMatteis (1968) and Silver and Meal (1973). The major advantage of these approaches is that they are computationally much more efficient than the exact solutions. Hax and Candea (1984) extended the EOQ model by allowing backlogging, lost sales, and quantity discounts. Fordyce and Webster (1985) modified the WW algorithm for situations in which unit cost price is not constant over the planning horizon, and included quantity discounts. Lev and Weiss (1990) and Gascon (1995) presented solutions for the finite horizon EOQ model where costs are time-dependent.

Discounts are a primary marketing mechanism for inducing customers to increase the size of their purchases. Quantity discounts from suppliers and freight discounts from shippers are commonly encountered by organizations. Tersine and Barman (1991) structured quantity and freight discounts into the order size decision in a deterministic EOQ system. Optimum lot-sizing algorithms were derived for the dual discount situations of all-units or incremental quantity discounts and all-weight or incremental freight discounts.

Gupta and Brennan (1992) introduced an easy alternative to the WW backorder algorithm. The performance of the model was compared with several of the traditional lot-sizing rules (lot for lot, EOQ, period order quantity, least unit cost, least total cost, part period algorithm, Silver-Meal algorithm, and WW algorithm) as well as the backorder versions of WW and EOQ. It was concluded that the proposed algorithm is sufficiently robust and relatively easy to apply. Most of the dynamic lot-sizing models assume that production is performed on reliable machines. Kuhn (1997) analyzed the

effects of setup recovery with machine breakdowns and corrective maintenance for the single item uncapacitated lot-sizing problem. In a first case, the assumption was made that the setup is totally lost after a breakdown. In a second case, the costs of resuming production of the same item after a breakdown was lower compared to the original setup cost.

Agra and Constantino (1999) examined the single item uncapacitated lot-sizing problem with backlogging and start-up costs where WW costs were assumed. Hernandez and Suer (1999) presented a GA approach to obtain the order quantities for a single item, single level uncapacitated lot-sizing problem. In the experimentation, different strategies were presented to evaluate the behavior of the GA under different parameters sets. The results showed that the proposed procedure generated satisfactory solutions to the considered problem. Richter and Sombrutzki (2000) studied the reverse WW dynamic production planning and inventory control model. In such reverse (product recovery) models, used products arrive to be stored and to be remanufactured at minimum cost. It was assumed that the demand can be met either from newly manufactured products or from return products which have been remanufactured.

Lee et al. (2001) discussed the single item, uncapacitated dynamic lot-sizing problem with a demand time window, where for each demand an earliest and latest delivery date is specified and the demand can be satisfied in the defined period without penalty. It was shown that there exists an optimal solution in which demand is not split, where the complete demand for a specific order can be covered by production from the same period. Loparic et al. (2001) proposed valid inequalities for solving a variant of the single item uncapacitated lot-sizing model of the WW problem involving sales instead of fixed demands and lower bounds on the stock variables. Aksen et al. (2003) introduced a profit maximization version of the WW model for the deterministic single

item uncapacitated lot-sizing problem with lost sales. It was assumed that demand cannot be backlogged, and costs and selling prices are time-variant. A forward recursive dynamic programming (DP) algorithm was developed to solve the problem optimally.

Teunter and Flapper (2003) examined a single stage single product production system, where produced units can be non-defective, reworkable defective, or non- reworkable defective. De Toledo and Shiguemoto (2005) proposed an efficient implementation of a forward DP algorithm for solving the lot-sizing problems in a single production center. Brahimi et al. (2006) reviewed various solution methods for solving the single item uncapacitated lot-sizing problem. Chiu (2008) presented a simple algebraic method to replace the use of calculus for determining the optimal lot size. Gutiérrez et al. (2008) addressed the dynamic lot-sizing problem with time-varying storage capacities with the aim of minimizing the total cost including setup, holding, and production/ordering costs.

Gaafar et al. (2009) applied the SA algorithm to find the solution of the deterministic dynamic lot-sizing problem with batch ordering and backorders, and compared the performance of the proposed SA with GA and modified Silver-Meal heuristic. Results indicated that SA algorithm had the best performance, followed by the GA, in terms of the frequency of obtaining the optimum solution and the average deviation from the optimum solution. It was also shown that SA was the most robust of the investigated heuristics as its performance was only affected by the length of the planning horizon. Hwang and van den Heuvel (2009) proposed a DP algorithm to optimally solve the classical uncapacitated single item lot-sizing problem with lost sales, upper bounds on stocks and concave costs. Vargas (2009) presented an algorithm for determining the optimal solution for the stochastic version of the WW dynamic lot-sizing problem.

Sana (2010) investigated an economic production lot size model in an imperfect production system in which the production facility may shift from an “in-control state” to an “out-of-control” state at any random time. In long-run process, the process shifts from the in-control state to the out-of-control state after certain time due to higher production rate and production run time. The proposed model was formulated assuming that a certain percent of total product is defective (imperfect), in out-of-control state, which varies with production rate and production-run time. The objective was to minimize the total cost including manufacturing cost, setup cost, holding cost, and reworking cost of imperfect quality products.

Senyiğit (2010) proposed a heuristic approach to solve the dynamic lot-sizing problem with demand and purchasing price uncertainties. Well-known least unit cost and Silver-Meal algorithms were also modified for both time-varying purchasing price and rolling horizon. The proposed heuristic was basically based on a cost-benefit evaluation at decision points. Absi et al. (2011) considered the single item uncapacitated lot-sizing problem with production time windows, lost sales, early productions, and backlogs. Several properties of the optimal solution for different variants of the problem when production time windows are non-customer specific were presented. The DP algorithm was used to solve the proposed problem.