Abstract
This chapter and Chap. 5 contain the core of this thesis, new mechanisms for collaborative supply chain planning. Recapitulating the definitions of Sect. 2.3.1, a coordination mechanism is a contractual framework for coordinating the outcomes of (self-interested) actions of the decentralized parties. Each of the mechanisms presented here comprises a scheme specifying the proposal generation and the information exchange.
For all schemes developed in this thesis, two variants will be provided that cover the different requirements for organizing the information exchange raised by the contractual frameworks presented in Chap. 5: An iterative, unilateral exchange of cost changes and a one-shot exchange by both parties. This allows us to describe and analyze the different schemes and frameworks separately.
In this chapter, different schemes are presented and customized for the Master Planning models described in Sect. 2.2. We begin with schemes for the coordination of general decentralized LP problems (Sect. 4.1) and of one buyer and several suppliers planning based on uncapacitated dynamic lot-sizing models (Sect. 4.2), and derive analytical results about their convergence behavior.
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Notes
- 1.
See also the results of our computational tests in Sect. 6.1.
- 2.
For the scheme, the initial solution might be chosen arbitrarily; when applying the mechanisms proposed in Chap. 5, the initial solution will correspond to the default solution, the solution established without coordination.
- 3.
This sequence is chosen for ease of exposition; other variants are possible, too (see Example 4.2).
- 4.
See our corresponding discussion at the end of Sect. 3.3.
- 5.
See Sect. 3.3.2.1 for basic models and input data used for the exposition of this scheme.
- 6.
Note here the different meaning of “starting solution,” which is the solution that is used as the starting point for proposal generation, compared to that of “initial solution,” which is the first solution used in the scheme.
- 7.
See also p. 34 for the statement of the model used for this evaluation.
- 8.
Note that for ease of exposition, the redundant constraint with e = st has been included here.
- 9.
See p. 57 for this model.
- 10.
Note that DPi and CS1i may have several optimal solutions including non-vertex solutions. Here we assume that CS1i is solved using an algorithm that limits its search to vertex solutions (e.g., a primal simplex).
- 11.
For ease of exposition, we skip the index i within this example.
- 12.
A proposal \({A}_{i}{x}_{i}^{n}\) can be expressed by this multiple if and only if \({A}_{i}{x}_{i}^{n}\) lies within the cone spanned by the starting proposal and the proposals found so far (here: C, D), see the left of Fig. 4.3.
- 13.
See p. 64 for a discussion of these difficulties.
- 14.
Other possible choices are, e.g., 4.00000001, 5, or even 1,000.
- 15.
This is because the proposals determined by CS-EVAL#1 and CS1 #1 have been equal here.
- 16.
A further potentially favorable rule is to choose the initial solution as the starting solution, and only if no new proposal can be found, to randomly select another solution.
- 17.
If \({Z}_{CS{1}_{i}} \leq 0\) has held in earlier steps, this obviously extends to the further steps of the scheme, where the only difference in CS1i is that \({X}_{i}^{E}\) is augmented by additional solutions.
- 18.
See Lemma 4.1 for this model.
- 19.
See p. 58.
- 20.
See Sect. 2.2.2.1.
- 21.
Note that values greater than 1 are precluded by the internal logic of the optimization model.
- 22.
Note here that MLULSP-CS does not always yield a proposal different to all previously found. If there is more than one predecessor item, a reduction of the number of setups for a buyer’s item might go along with a reduction of the number of setups for several items of the supplier and, hence, cause an overfulfillment of the setup target [i.e., some slack in (4.30)]. A proposal that has been generated under such an overfulfillment will be repeated at least once since further increases of the setup target by 1 will not enforce an additional reduction of the number of orders.
- 23.
Note that we consider end item demand only, i.e., we assume zero demand for intermediate items. A further (implicit) assumption is the time independence of holding costs h j .
- 24.
See, e.g., Simpson (2007, p. 136) and our computational results for the MLULSP (p. 172).
- 25.
In fact, other costs like those for shelf space use are often minor compared to the costs of capital bound in inventory, e.g., Schneeweiß (1981, p. 69).
- 26.
See p. 27 for the modeling of the supplier’s implementation of a supply target.
- 27.
- 28.
Especially with level demand, there may be several equally optimal outcomes of MLULSP-CS; which of them are determined by the solver running this model, depends on external factors like the model structure and the solver characteristics and cannot be determined a priori.
- 29.
Note that a corollary to this theorem is stated in Albrecht (2008) (called Theorem 2 there), where convergence is shown under the additional requirements on N B ∕ N S and | T | ∕ N B specified in Lemma 4.3.
- 30.
Since the buyer’s costs for a dominating frequency are lower, proposals with dominated frequencies – which are not identified by our procedure – are obviously inferior.
- 31.
See also equation (4.39).
- 32.
E.g., ILOG (2008) and Dash (2008).
- 33.
See Baker and Lasdon (1985, p. 264).
- 34.
See, e.g., Kallrath and Wilson (1997, p. 376).
- 35.
- 36.
See, e.g., Gupta and Ravindran (1985, p. 1534).
- 37.
See Geoffrion (1972, p. 237).
- 38.
See Duran and Grossmann (1986, p. 307).
- 39.
See, e.g., Floudas (1995, p. 114 and p. 144).
- 40.
See Chap. 6.
- 41.
A thorough investigation of the effectiveness of different MINLP solution procedures for CS1i , in turn, does not seem necessary for that purpose.
- 42.
See, e.g., Williams (1993, p. 152).
- 43.
- 44.
In Sect. 4.3.2, we present a model where these functions are explicitly linearized for GM.
- 45.
See Beale and Tomlin (1970, p. 447).
- 46.
See Sharpe (1971, p. 1269).
- 47.
In the preceding example, lb = x O and ub = x C hold.
- 48.
See Sect. 2.2.1 for the mathematical formulation of this model.
- 49.
For a reversed information flow, GM-CS2S is applied instead of GM-CS2B. Further note that for adapting the models stated below to the MLCLSP and MLCLSP-C presented in Sect. 2.1.2, these models simply have to be augmented by the additional features for lot-sizing and campaign planning.
- 50.
The modeling of these constraints instead of (2.43) and (2.44) – which have been introduced for clearness of exposition in Sect. 2.3.2 – is more direct and allows to reduce the number of variables used.
- 51.
This insight has also been used in our computational tests of Chap. 6.
- 52.
In this context, note that inventory holding of supplied items at the buyer’s site will become more expensive than at the supplier’s if the holding costs comprise capital costs only with identical interest rates for parties and if the purchase prices for these items exceed the supplier’s production costs.
- 53.
The introduction of these additional variables helps to avoid potential errors in the approximation leading to negative values for actually positive \({K}_{jt}^{+}\), \({K}_{jt}^{-}\). Such effects are natural for the approximation applied and would cause some degradation in the solution quality in case of an alternative modeling without these variables.
- 54.
- 55.
For a further illustration of this idea, see Example 2.6 on p. 31.
- 56.
For the determination of b jt , see p. 13 of this work.
- 57.
See, e.g., Kersten (2003, p. 332).
- 58.
For ease of exposition, we will limit to the formulations with nonlinear objective functions. Their linearizations can be formulated analogously to \({\text{ GM-CS1}}_{\text{ B}}^{\text{ L}}\).
- 59.
- 60.
See Chap. 6. A further difficulty experienced in computational tests is that the optimizer running models GM-CS1B and GM-CS1S might declare barely feasible problems as infeasible. This can be caused by a large downscaling of the matrix elements in case of huge differences between the upper and lower bounds used for the linearization.
- 61.
See Dudek and Stadtler (2005, p. 677). The difference to our approach is that their penalty costs are not fixed, but updated by exponential smoothing in each iteration. We did not follow their approach because it did not prove significantly better in computational tests, while augmenting the complexity of the scheme.
- 62.
See also p. 139.
- 63.
A similar idea has been used by the scheme of Dudek and Stadtler (2005, p. 677).
- 64.
See p. 139.
- 65.
See the sensitivity analyses in Sect. 6.1.3.
- 66.
See also the results of our computational study in Sects. 6.1 and 6.3.
- 67.
Note that the use of all levels of modifications proposed is not always advisable. E.g., for GM investigated in our computational tests (Sect. 6.1), the omittance of GM-\({\text{ CS1}}_{\text{ i}}^{\text{ agg-P}}\) turns out to be superior.
- 68.
Since in a two-party setting only one cost-reporting party exists, we can omit the index i here (as for Π E).
- 69.
Note that we use the MLCLSP as the base model here since CS-LOT only applies for lot-sizing models and, thus, not for GM.
- 70.
See p. 31.
- 71.
Further modifications that help to somewhat sharpen the linearization are the introduction of additional variables for the costs for lost sales (analogous to \({C}_{jt}^{p,+}\),\({C}_{jt}^{p,-}\)) and of constraints assuring that, in case of shortages for a single item, the penalty costs for lost sales are directly imputed to this item. These modifications have been considered in our computational study for the real-world planning problems in Sect. 6.5, but not in the models stated above for ease of exposition.
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Albrecht, M. (2010). New Coordination Schemes. In: Supply Chain Coordination Mechanisms. Lecture Notes in Economics and Mathematical Systems, vol 628. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02833-5_4
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