K-convexity in Rn

Let $\backslash Kappa$= (K₀,K₁,...,K_n) to be a vector of (n+1) nonnegative constants and define a function $\backslash Kappa$(^.): $\backslash Re\_+^n$ → $\backslash Re\_+^1$ as follows:

$\backslash Kappa$($x$) = K₀$\backslash delta$(e$x$) + $\backslash sum\_\{i=1\}^n$K_i$\backslash delta$($x\_i$),

where e = (1,1,...,1) ∈ $\backslash Re^n$, $\backslash Re\_+^n\; =\; \backslash \{x\; \backslash in\; \backslash Re^n\; |\; x\; \backslash geq\; 0\; \backslash \}$, $\backslash delta$(0) = 0 and $\backslash delta(z)$= 1 for all $z$ > 0.{{cite journal

| last1=Gallego | first1=G.

| last2=Sethi | first2=S. P.

| date=2005

| url=https://www.researchgate.net/publication/257604732_K-Convexity_in_Rn

| title=$\backslash mathcal\{K\}$-convexity in $\backslash mathfrak\{R\}^n$

| journal=Journal of Optimization Theory & Applications

| volume=127

| issue=1

| pages=71–88

| doi=10.1007/s10957-005-6393-4}}

The concept of K-convexity generalizes K-convexity introduced by Scarf (1960){{cite report

| last1=Scarf | first1=Herbert

| date=1960

| title=The Optimality of (S, s) Policies in the Dynamic Inventory Problem

| publisher=Stanford, CA: Stanford University Press

| url=https://purl.stanford.edu/bt748ps8245}} 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.{{cite journal

| last1=Johnson | first1=Ellis L.

| date=1967

| title=Optimality and computation of (σ, s) policies in the multi-item infinite horizon inventory problem

| journal=Management Science

| volume=13

| issue=7

| pages=475–491

| doi=10.1287/mnsc.13.7.475}}{{cite journal

| last1=Kalin | first1=Dieter

| date=1980

| title=On the optimality of (σ, S) policies

| journal=Mathematics of Operations Research

| volume=5

| issue=2

| pages=293–307

| doi=10.1287/moor.5.2.293}}{{cite journal

| last1=Sulem | first1=Agnès

| date=1986

| title=Explicit solution of a two-dimensional deterministic inventory problem

| journal=Mathematics of Operations Research

| volume=11

| issue=1

| pages=134–146

| doi=10.1287/moor.11.1.134}}{{cite book

| last1=Liu | first1=Baoding

| last2=Esogbue | first2=Augustine O.

| date=1999

| title=Decision Criteria and Optimal Inventory Processes

| publisher=Springer Science & Business Media

| doi=10.1007/978-1-4615-5151-5}}{{cite journal

| last1=Ohno | first1=K.

| last2=Ishigaki | first2=T.

| date=2001

| title=A multi-item continuous review inventory system with compound Poisson demands

| journal=Mathematical Methods of Operational Research

| volume=53

| issue=1

| pages=147–165

| doi=10.1007/s001860000101}}{{cite report

| last1=Li | first1=Yu

| last2=Sethi | first2=Suresh

| title=Optimal Ordering Policies for Two-Product Inventory Models with Fixed Ordering Costs

| date=August 24, 2022

| url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4199040}}{{cite journal

| last1=Perera | first1=Sandun C.

| last2=Sethi | first2=Suresh P.

| title=A Survey of Stochastic Inventory Models with Fixed Costs: Optimality of (s, S) and (s, S)-type Policies—Discrete-time case

| journal=Production and Operations Management

| date=2023

| volume=32

| issue=1

| pages=131–153

| doi=10.1111/poms.13820}}{{cite journal

| last1=Perera | first1=Sandun C.

| last2=Sethi | first2=Suresh P.

| title=A Survey of Stochastic Inventory Models with Fixed Costs: Optimality of (s, S) and (s, S)-type Policies—Continuous-time Case

| journal=Production and Operations Management

| date=2023

| volume=32

| issue=1

| pages=154–169

| doi=10.1111/poms.13819}}

This definition introduced by Gallego and Sethi (2005) 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 K_i for each item $i$ 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 K_i = 0, $i$ = 1, 2, . . . , n, and a setup cost of $\backslash Kappa\_0$ is incurred whenever any one or more of the items are ordered. In this case, $\backslash Kappa$ = (K₀, 0, 0, . . . , 0) and $\backslash Kappa$($x$) = K₀$\backslash delta$(e$x$).

(iii) When there is no joint setup cost, i.e., K₀ = 0, and there are only individual setups, we have $\backslash Kappa$($x$) = $\backslash sum\_\{i=1\}^n$K_i$\backslash delta$($x\_i$).

{{Math-stub}}

Category:Convex analysis