K-convexity in Rn
K-convexity in Rn is a mathematical concept.
Formula
Let [math]\displaystyle{ \Kappa }[/math]= (K0,K1,...,Kn) to be a vector of (n+1) nonnegative constants and define a function [math]\displaystyle{ \Kappa }[/math](.): [math]\displaystyle{ \Re_+^n }[/math] → [math]\displaystyle{ \Re_+^1 }[/math] as follows:
[math]\displaystyle{ \Kappa }[/math]([math]\displaystyle{ x }[/math]) = K0[math]\displaystyle{ \delta }[/math](e[math]\displaystyle{ x }[/math]) + [math]\displaystyle{ \sum_{i=1}^n }[/math]Ki[math]\displaystyle{ \delta }[/math]([math]\displaystyle{ x_i }[/math]),
where e = (1,1,...,1) ∈ [math]\displaystyle{ \Re^n }[/math], [math]\displaystyle{ \Re_+^n = \{x \in \Re^n | x \geq 0 \} }[/math], [math]\displaystyle{ \delta }[/math](0) = 0 and [math]\displaystyle{ \delta(z) }[/math]= 1 for all [math]\displaystyle{ z }[/math] > 0.[1]
The concept of K-convexity generalizes K-convexity introduced by Scarf (1960)[2] to higher dimensional spaces and is useful in multiproduct inventory problems with fixed setup costs. Scarf used K-convexity to prove the optimality of the (s, S) policy in the single product case. Several papers are devoted to obtaining optimal policies for multiple product problems with fixed ordering costs.[1][3][4][5][6][7][8][9][10]
This definition introduced by Gallego and Sethi (2005)[1] is motivated by the joint replenishment problem when we incur a setup cost K0, whenever we order an item or items and an individual setup cost Ki for each item [math]\displaystyle{ i }[/math] we order. There are some important special cases:
(i) The simplest is the case of one product or n = 1, where K0 + K1 can be considered to be the setup cost.
(ii) The joint setup cost arises when Ki = 0, [math]\displaystyle{ i }[/math] = 1, 2, . . . , n, and a setup cost of [math]\displaystyle{ \Kappa_0 }[/math] is incurred whenever any one or more of the items are ordered. In this case, [math]\displaystyle{ \Kappa }[/math] = (K0, 0, 0, . . . , 0) and [math]\displaystyle{ \Kappa }[/math]([math]\displaystyle{ x }[/math]) = K0[math]\displaystyle{ \delta }[/math](e[math]\displaystyle{ x }[/math]).
(iii) When there is no joint setup cost, i.e., K0 = 0, and there are only individual setups, we have [math]\displaystyle{ \Kappa }[/math]([math]\displaystyle{ x }[/math]) = [math]\displaystyle{ \sum_{i=1}^n }[/math]Ki[math]\displaystyle{ \delta }[/math]([math]\displaystyle{ x_i }[/math]).
References
- ↑ 1.0 1.1 1.2 Gallego, G.; Sethi, S. P. (2005). "[math]\displaystyle{ \mathcal{K} }[/math]-convexity in [math]\displaystyle{ \mathfrak{R}^n }[/math]". Journal of Optimization Theory & Applications 127 (1): 71–88. doi:10.1007/s10957-005-6393-4. https://www.researchgate.net/publication/257604732_K-Convexity_in_Rn.
- ↑ Scarf, Herbert (1960). The Optimality of (S, s) Policies in the Dynamic Inventory Problem (Report). Stanford, CA: Stanford University Press. https://purl.stanford.edu/bt748ps8245.
- ↑ Johnson, Ellis L. (1967). "Optimality and computation of (σ, s) policies in the multi-item infinite horizon inventory problem". Management Science 13 (7): 475–491. doi:10.1287/mnsc.13.7.475.
- ↑ Kalin, Dieter (1980). "On the optimality of (σ, S) policies". Mathematics of Operations Research 5 (2): 293–307. doi:10.1287/moor.5.2.293.
- ↑ Sulem, Agnès (1986). "Explicit solution of a two-dimensional deterministic inventory problem". Mathematics of Operations Research 11 (1): 134–146. doi:10.1287/moor.11.1.134.
- ↑ Liu, Baoding; Esogbue, Augustine O. (1999). Decision Criteria and Optimal Inventory Processes. Springer Science & Business Media. doi:10.1007/978-1-4615-5151-5.
- ↑ Ohno, K.; Ishigaki, T. (2001). "A multi-item continuous review inventory system with compound Poisson demands". Mathematical Methods of Operational Research 53 (1): 147–165. doi:10.1007/s001860000101.
- ↑ Li, Yu; Sethi, Suresh (August 24, 2022). Optimal Ordering Policies for Two-Product Inventory Models with Fixed Ordering Costs (Report). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4199040.
- ↑ Perera, Sandun C.; Sethi, Suresh P. (2023). "A Survey of Stochastic Inventory Models with Fixed Costs: Optimality of (s, S) and (s, S)-type Policies—Discrete-time case". Production and Operations Management 32 (1): 131–153. doi:10.1111/poms.13820.
- ↑ Perera, Sandun C.; Sethi, Suresh P. (2023). "A Survey of Stochastic Inventory Models with Fixed Costs: Optimality of (s, S) and (s, S)-type Policies—Continuous-time Case". Production and Operations Management 32 (1): 154–169. doi:10.1111/poms.13819.
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