پارامترهای MRP بهینه برای یک موجودی تکی آیتم همراه زمان فرآوری تکمیل دوباره تصادفی، سیاست POQ و محدودیت سطح خدمات

This study deals with Material Requirement Planning (MRP) software parameterization under uncertainties. The actual lead time has random deviations; so it can be considered as a random variable. MRP approach with Periodic Order Quantity (POQ) policy is considered. The aim is to find the optimal MRP time phasing corresponding to each periodicity of the POQ policy. This is a crucial issue in supply planning with MRP approach because inappropriate planned lead times under lead time uncertainties invariably lead to large and costly inventories or insufficient customer service levels. The proposed model and algorithms minimize the sum of the setup and holding costs while satisfying a constraint on the service level. Our approach does not need to employ the commonly used normal probability distributions. Instead, its originality is in finding a closed form of the objective function, valid for any probability distribution of the actual lead times.

Effective replenishment is a crucial problem in supply planning. An inadequate inventory control policy leads to overstocking or stockout situations. In the former, the generated inventories are expensive and in the latter there are shortages and penalties due to unsatisfied customer demands. Material Requirements Planning (MRP) is a commonly accepted approach for replenishment planning in major companies (Axsäter, 2006). The MRP-based software tools are accepted readily. Most industrial decision makers are familiar with their use. The practical aspect of MRP lies in the fact that this is based on comprehensible rules, and provides cognitive support, as well as a powerful information system for decision making. Some instructive presentations of this approach can be found in Baker (1993), Sipper and Bulfin (1998), Zipkin (2000), Axsäter (2006), Tempelmeier (2006), Dolgui and Proth (2010) and Graves (2011). Nevertheless, MRP is based on the supposition that both demand and lead time are deterministic. However, most production systems are stochastic. For example, a random lead time can be explained by the variability of actual supplier load (when a supplier furnishes several clients, its load depends on the timing of all client orders, if total demand outstrips production capacity, the lead time increases). There are many other external factors increasing randomness of lead times: outsourced production overseas can introduce some randomness via shipping perturbations, the orders might not arrive by the due date because of work stoppage or delays attributable to the weather (Graves, 2011). Additional random factors and unpredictable events such as machine breakdowns, absenteeism, other random variations of capacity can cause deviations in actual lead times from planning ones (Koh and Saad, 2003 and Chaharsooghi and Heydari, 2010). Therefore, as aforementioned the deterministic assumptions of MRP can be often too restrictive. As shown in Whybark and Wiliams (1976), Ho and Lau (1994), Molinder (1997) and Chaharsooghi and Heydari (2010), lead time is a principal factor foreseeing production and lead time randomness affects seriously ordering policies, inventory levels and customer service levels. Thankfully, the MRP approach can be tailored to uncertainties by searching optimal values for its parameters (Buzacott and Shanthikumar, 1994, Hegedus and Hopp, 2001, Koh and Saad, 2003, Inderfurth, 2009 and Mula and Poler, 2010). An adequate choice of these parameters increases the effectiveness of MRP techniques. Thus, one of the essential issues for companies in industrial situations is MRP parameterization. This is commonly called MRP offsetting under uncertainties. There are several MRP parameters: planned lead time, safety stock, lot-sizing rule, freezing horizon, planning horizon, etc. There exist extensive publications concerning safety stock calculation for random demand of finished products (Petruzzi and Dada, 1999 and Lee and Nahmias, 1993). In contrast, certain parameters seem not to be sufficiently examined as, for example, planned lead time (differences between due dates and release date). Optimal parameterization of most used lot-sizing rules is also an open issue. If actual lead time is random, the planned lead time can contain safety lead time, i.e. the planned lead time is calculated as the sum of the forecasted (or contracted) and safety lead times. The latter should be formulated as a trade-off between overstocking and stockout while minimizing the total cost. The search for optimal value of safety lead time, and, consequently, for planned lead time, is a crucial issue in supply planning with the MRP approach. The problem of planned lead times optimization, when safety lead times are used, has been given scant attention in the literature. In practice, often average values or percentiles of probability distributions of actual lead times are used. Nevertheless, a longer than necessary planned lead time creates excessive work in progress. Perhaps of special interest, Graves (2011) in his chapter ‘Uncertainties and Production Planning’ of the ‘Handbook of Production Planning and Inventories in the Extended Enterprise’ considers that there is “a great opportunity for developing decision support to help planners in understanding the trade-offs and in setting these parameters in a more scientific way”. This is one of motivations for this paper where we propose a decision support model for optimal MRP time phasing for each periodicity of the POQ policy. The proposed model and algorithms minimize the sum of the setup and holding costs while satisfying a constraint on the service level. Our approach does not need to employ the commonly used normal probability distributions. Instead, its originality is in finding a closed form of the objective function, valid for any probability distribution of the actual lead times.

The obtained model is useful for the optimization of the MRP parameterization with random procurement times and the POQ lot-sizing policy. The proposed model minimizes the sum of the average holding and setup costs while satisfying a constraint on a desired service level. This is a new generalization of the Newsboy model. It can be used in many industrial situations. For example, often security coefficients are introduced to calculate the planned lead time for unreliable suppliers in an MRP environment. In this case, planned lead time is equal to contractual (or forecasted) lead time multiplied by a security coefficient. This coefficient is empiric but anticipates the delay by creating safety lead time. The more unreliable a supplier is, the larger its coefficient. The model suggested in this paper can be used to better estimate these coefficients by using statistics on the procurement lead times for each supplier and taking into account the holding and setup costs. This is a multi-period model with no major restriction on the type of the lead time distribution. All discrete distributions can be used. The decision variables are integer; they represent the periodicity and planned lead time for items. Concerning the assumption of constant demand, note that this model should be used with different possible values of the demand to examine the sensitivity of the obtained parameters to the said values. If the parameter values are significantly different for the given demand levels, the approach by scenarios can be applied to choose the best parameter values. In addition, the demand variations can be decoupled from planned lead time calculation by using safety stocks. This is another promising perspective for future research.