Lot sizing problems are production planning problems of order quantity between purchasing or production lots (Brahimi, Dauzere-Peres, Najid, & Nordli, 2006). Small lot sizes lead to many orders and low inventory levels while large lot sizes lead to few orders and high inventory levels (Lee, Kramer, & Hwang, 1991). The consideration of lot sizes is, therefore, an economic problem in that the objective of inventory models is to minimize total inventory cost, which comprises unit price, ordering cost, and inventory holding cost, while satisfying demand (Brahimi et al., 2006, Hilmola and Lorentz, 2011 and Lee et al., 1991). The first inventory planning model, namely Economic Order Quantity (EOQ), was proposed by Harris (1913). It was used to find an optimal order quantity in the case of an uncapacitated single stage and single item of inventory control with a well-defined demand pattern. Wagner and Whitin (1958) proposed an inventory model with time-vary demand, namely dynamic lot sizing, and used dynamic programming techniques to find an optimal order quantity. Other subsequent inventory models have been developed, based on the above models (Askin and Goldberg, 2002 and Barancsi et al., 1990).
The USILSP is a type of inventory model with time-vary demand. Brahimi et al. (2006) stated that there are four basic formulations of the USILSP in the form of mixed integer linear programming or MILP; i.e., aggregate formulation (AGG), formulation without inventory variables (NIF), the shortest path formulation (SHP), and facility location-based formulation without inventory variables (FAL). In addition, Brahimi et al. (2006) added that USILSP modeling is popular for inventory planning for three reasons. Firstly, some industries can aggregate products to obtain a single product, for example, products that differ only in color can be treated as a single product. Secondly, the USILSP is a basic model but can be easily extended for more complex circumstances. Lastly, many complex lot sizing problems must employ USILSP as sub-problems (see Cattrysse et al., 1990 and Merle et al., 1999).
Typically, when a classical inventory model is used, the Crisp Deterministic Assumption is required. However, often the information can be uncertain such as a situation in which only qualitative information from experienced operators or personnel. As a result, the fuzzy set theory is applied to resolve this information uncertainty (Dubois and Prade, 1980, Gumus and Guneri, 2009, Ketsarapong and Punyangarm, 2010, Zadeh, 1965 and Zimmermann, 1996). In addition, Tütüncü, Akoz, Apaydin, and Petrovic (2008) suggested that some uncertainty within inventory systems should not considers the probability applications. Therefore, since 1980s, the fuzzy set theory has been widely used in modeling of inventory systems when dealing with vagueness and uncertainty (Cakir and Canbolat, 2008, Chang et al., 2006, Chen and Chang, 2008, Dutta et al., 2007, Green et al., 2011, Halim et al., 2011, Hsieh, 2002, Mandal et al., 2005, Pai, 2003 and Yao and Lee, 1999).
The rest of the paper is organized as follows: Section 2 presents the research premise; Section 3 presents the research objectives; Section 4 present the four steps of research methodology including, (1) the data collection process, (2) developing an inventory model, transforming the F-USILSP to an EC-USILSP model, developing the EC-USILSP model in the form of MILP, and an illustrative numerical example, (3) data analysis process, and (4) the decision making process. In Section 5, the case application of a petrochemical company is presented. Finally, the last Sections are the conclusions and recommendations for future studies.
The F-USILSP model was developed mainly for inventory planning. This model is suitable for inventory planning in cases where there is no information from statistical data collection, but there is verbal information from experienced operators or personnel. Such situations frequently occur for SMEs or in cases where there is information from statistical data collection which cannot be used for inventory planning due to unexpected circumstances such as natural disasters resulting in rapid changes in demand, unit price, or ordering cost.
The Possibility Approach was used to transform the F-USILSP model to the EC-USILSP model. As a result, this model can be solved by using basic software. Excel solver can be used to find an optimal solution effectively and provide useful information including total cost, ordering variables, purchasing and production quantity, and the inventory level. Furthermore, the result of Excel solver processing also showed that when the higher possible levels (α) are required the total cost will increase accordingly. Planners can therefore make decisions under the possible levels (α) that can be accepted.
This article has several important implications. The EC-USILSP model is very flexible, so other conditions can be added to this model tobe consistent with events occurring in each business. For example, in the case study of a petrochemical company, restrictions on two issues were added: transportation capacity limitations and inventory space limitations in the EC-USILSP model to better represent the actual situation.
