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Production and Operations Management: Models and Algorithms

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Production and Inventory Management with Substitutions

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 636))

Abstract

This chapter intends to give an overview of the literature on dynamic lot-sizing models and stochastic transshipment models. These two types of models are used as a basis for developing models with substitution in the following chapters. Section 2.1 contains a classification of models for dynamic lot-sizing / production planning, and selected models. In Sect. 2.2, we give a brief overview of available methods for solving deterministic dynamic lot-sizing problems modeled using mixed-integer linear programming (MILP). Section 2.3 introduces transshipment problems and presents a classification scheme for transshipment models. Section 2.4 reviews selected solution approaches that can be applied to stochastic inventory control models such as transshipment problems.

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Notes

  1. 1.

    We subsume dynamic lot-sizing problems as well as simultaneous lot-sizing and scheduling problems under the term “production planning problem”. Note that in literature, “(aggregate) production planning” may also refer to Master Planning (Rohde and Wagner, 2005) models.

  2. 2.

    Also see Fleischmann et al. (2005, p. 81f.) and Meyr (1999, p. 11ff.). regarding the classification of planning levels.

  3. 3.

    Note that the assumption of linear holding costs underlying most dynamic lot-sizing models is only a simplification: Usually, “snapshots” of the current inventory of a product are taken at the ends/beginnings of (macro-)periods to approximate the actual average inventory in each (macro-) period. For an exact calculation of holding costs based on (marginal) inventory changes within each period, it would be necessary to multiply the holding costs per quantity and time unit with the integral over the inventory level as a function of the current time. This is because the inventory of a product can change within a period if the production speed is finite. For example, considering the CLSD that assumes fixed production speeds (see Sect. 2.1.4), the inventory level as a function of time would be a piecewise linear function. However, the common approach of approximating it seems sufficiently precise for practical purposes. Beyond this, the general idea of using holding costs can be criticized because it is difficult to measure these opportunity costs due to capital lockup in practice. Also, it should be noted that lot-sizing problem data like demand forecasts are subject to uncertainty anyway, which might render the mentioned imprecision of holding costs calculations insignificant. On the accurate calculation of holding costs in dynamic lot-sizing models, also see Stammen-Hegener (2002, p. 143f.).

  4. 4.

    This constraint group is based on the same idea as the Miller–Tucker–Zemlin subtour elimination constraints for the Asymmetric Traveling Salesman Problem (ATSP) (Domschke, 1997, p. 106f.).

  5. 5.

    The constraint (2.60) is slightly more restrictive than necessary. It always ensures temporal feasibility of a solution, but excludes some feasible solutions (F. Seeanner 2009, personal communication): (2.60) is always enforced, no matter whether both the slow predecessor i and its successor k are actually produced on resources r 1 and r 2, respectively, in s − 1. If i is not produced on r 1 in s − 1, (2.60) still enforces that \({w}_{s} - {w}_{s-1} \geq {y}_{{r}_{1},s-1}^{b} + {y}_{{r}_{2},s-1}^{e}\), which unnecessarily limits the duration of split setups on r 1 and r 2 involving other products. Considering the case that k is not produced on r 2 in s − 1, (2.60) is still enforced although the solution shown in Fig. 2.12a would not violate temporal constraints. Equation (2.60) can be formulated in a less restrictive way by introducing setup splitting variables for each possible changeover from one product h to another product j, i.e., by introducing additional indices h and j for the y b and y e variables. These variables y rhjs b and y rhjs e denote the setup time consumed by a changeover from product h to j on resource r at the beginning and end of micro-period s, respectively (F. Seeanner 2009, personal communication). With these variables, one can reformulate (2.60) so that only relevant durations of changeovers involving the products i and k are included.

  6. 6.

    Just as (2.60), (2.61) is more restrictive than necessary and can be reformulated in a less restrictive way using the variables y riks b and y riks e mentioned in footnote 5 (F. Seeanner 2009, personal communication). A schedule with the property enforced by (2.61) is shown in Fig. 2.13b.

  7. 7.

    See Desrosiers and Lübbecke (2005); Huisman et al. (2005); Lübbecke and Desrosiers (2005); Ralphs et al. (2001, 2003); Vanderbeck (2003, 2005); Wilhelm (2001) for additional literature.

  8. 8.

    For additional literature see, e.g., Diaby et al. (1992a,b); Thizy and van Wassenhove (1985).

  9. 9.

    The usage of decompositions in R&F and F&O resembles the decompositions methods in SAP® APO: The Supply Network Planning (SNP) Optimizer offers decompositions by time, product, and resource (Kallrath and Maindl, 2006, pp. 84 and 89). The subproblems into which the problem is decomposed are solved sequentially. The time decomposition uses time windows as in R&F. The product decomposition optimizes the variables associated with a certain subset of products in each subproblem. It thus differs from the product decomposition in F&O that only considers one product per subproblem. So-called SNP priority profiles can be specified to provide the SNP Optimizer with information that helps decompose the problem using the product or resource decomposition in an appropriate way.

  10. 10.

    All transshipment models considered in this work are stochastic.

  11. 11.

    This criterion refers to the behavior of an algorithm, and should not be confused with the robustness criteria for solutions used in Robust Optimization.

  12. 12.

    E.g., Liao and Rittscher (2007b) consider a supplier selection problem where the flexibility provided by options for increasing or reducing supply quantities and for reducing supplier lead times is valued in the objective function. Such options that give additional flexibility can increase the (problem-specific) robustness of a solution to the problem.

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Authors and Affiliations

  1. Department of Law, Business and Economics Chair of Operations Research, Technische Universität Darmstadt, Hochschulstr. 1, 64289, Darmstadt, Germany

    Dr. J. Christian Lang

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Correspondence to J. Christian Lang .

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Lang, J.C. (2010). Production and Operations Management: Models and Algorithms. In: Production and Inventory Management with Substitutions. Lecture Notes in Economics and Mathematical Systems, vol 636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04247-8_2

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