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Centralized inventory in a farming community

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Abstract

A centralized inventory problem is a situation in which several agents face individual inventory problems and make an agreement to coordinate their orders with the objective of reducing costs. In this paper we identify a centralized inventory problem arising in a farming community in northwestern Spain, model the problem using two alternative approaches, find the optimal inventory policies for both models, and propose allocation rules for sharing the optimal costs in this context.

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Notes

  1. In this paper we assume that the waiting time since an order is posed until it is received is deterministic, in which case it can be assumed to be zero. The deterministic nature of waiting times in this problem is realistic. In principle, the supplier guarantees to serve the orders the next day after they are posed.

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Acknowledgments

Authors acknowledge the financial support of Ministerio de Ciencia e Innovación through projects MTM2011-23205 and MTM2011-27731-C03, and Generalitat Valenciana through project ACOMP/2014.

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Correspondence to I. García-Jurado.

Appendix

Appendix

Here, the reader can find the proofs of the main theorems stated in this paper.

1.1 Proof of Theorem 4.1

Proof

Assume first that (Nc) is subadditive and take a pair of non-empty coalitions \(S,T\subset N\) with S T = ∅ and \(\max_{i\in S}\{a_i\}\leq \max_{i\in T}\{a_i\}.\) It holds that

$$c({S\cup T})=\max_{i\in S\cup T}\{a+a_i\}\max_{i\in S\cup T} \left \{\frac{d_i}{K_i}\right \} =\max_{i\in T}\{a+a_i\}\max_{i\in S\cup T} \left \{\frac{d_i}{K_i}\right \}.$$

Then, the subadditivity condition implies that

$$\max_{i\in T}\{a+a_i\}\left (\max_{i\in S\cup T} \left \{\frac{d_i}{K_i}\right \}-\max_{i\in T}\left \{\frac{d_i}{K_i}\right\}\right) \leq \max_{i\in S}\{a+a_i\}\max_{i\in S} \left \{\frac{d_i}{K_i}\right \}.$$

If \(\max_{i\in S}\{\frac{d_i}{K_i}\}> \max_{i\in T}\{\frac{d_i}{K_i}\},\) then \(\max_{i\in S\cup T} \{\frac{d_i}{K_i}\}=\max_{i\in S} \{\frac{d_i}{K_i}\}.\) Thus, dividing both sides by \(\max_{i\in T}\{a+a_i\}\max_{i\in S} \{\frac{d_i}{K_i}\},\) the inequality above becomes

$$1-\frac{\max_{i\in T}\{\frac{d_i}{K_i}\}}{\max_{i\in S} \{\frac{d_i}{K_i}\}}\leq \frac{\max_{i\in S}\{a+a_i\}}{\max_{i\in T}\{a+a_i\}}$$

which is equivalent to

$$\frac{\max_{i\in T}\{a_i\}-\max_{i\in S}\{a_i\}}{a+\max_{i\in T}\{a_i\}}\leq \frac{\max_{i\in T}\{\frac{d_i}{K_i}\}}{\max_{i\in S} \{\frac{d_i}{K_i}\}}.$$

Conversely, take a pair of non-empty coalitions \(S,T\subset N\) with ST = ∅ and \(\max_{i\in S}\{a_i\}\leq \max_{i\in T}\{a_i\} .\) If condition 1 in the statement holds, then

$$c({S\cup T})=\max_{i\in S\cup T}\{a+a_i\}\max_{i\in S\cup T} \left \{\frac{d_i}{K_i}\right \}=\max_{i\in T}\{a+a_i\}\max_{i\in T}\left \{\frac{d_i}{K_i}\right \}=c(T)\leq c(S)+c(T).$$

If condition 2 in the statement holds, then

$$\left (\max_{i\in T}\{a_i\}-\max_{i\in S}\{a_i\}\right)\max_{i\in S}\left \{\frac{d_i}{K_i}\right \}\leq \max_{i\in T}\{a+a_i\}\max_{i\in T}\left \{\frac{d_i}{K_i}\right \}$$

and

$$\begin{aligned} c(S)+c(T)&=\max_{i\in S}\{a+a_i\}\max_{i\in S}\left\{\frac{d_i}{K_i}\right\}+\max_{i\in T}\{a+a_i\}\max_{i\in T}\left\{\frac{d_i}{K_i}\right\}\\ &\geq \max_{i\in S}\{a+a_i\}\max_{i\in S}\left\{\frac{d_i}{K_i}\right\}+(\max_{i\in T}\{a_i\}-\max_{i\in S}\{a_i\})\max_{i\in S}\left\{\frac{d_i}{K_i}\right\}\\ &=\max_{i\in T}\{a+a_i\}\max_{i\in S}\left\{\frac{d_i}{K_i}\right\}=\max_{i\in S\cup T}\{a+a_i\}\max_{i\in S\cup T}\left\{\frac{d_i}{K_i}\right\}=c({S\cup T}).\\ \end{aligned}$$

1.2 Proof of Theorem 4.2

Proof

Take \(\sigma\in \Uppi(N)\) satisfying that \(\sigma^{-1}(1)\in N\) is an agent whose distance to the supplier is maximal. We prove that m σ(Nc) belongs to the core of (Nc). It suffices to show that for every non-empty coalition \(S\subset N,\) it holds that \(\sum_{i\in S}m^\sigma_{i}(N,c)\leq c(S).\) We distinguish two cases.

  1. (a)

    S contains the agent σ−1(1). Then,

    $$\begin{aligned} \sum_{i\in S}m^{\sigma}_{i}(N,c)&=c(\sigma^{-1}(1))+\sum_{j\in S \setminus\{\sigma^{-1}(1)\}}\left(c({P}^{\sigma}_{j}\cup\{j\})-c({P}^{\sigma}_{j})\right) \\ &= c(\sigma^{-1}(1))+\sum_{j\in S \setminus\{\sigma^{-1}(1)\}}\left((a+\max_{i\in N}\{ a_i\})\max_{i\in {P}^{\sigma}_{j}\cup\{j\}}\left \{{{d_i}\over {K_i}}\right \} - (a+\max_{i\in N}\{ a_i\})\max_{i\in {P}^{\sigma}_{j}}\left \{{{d_i}\over {K_i}}\right \}\right) \\ &= \sum_{j\in S}(a+\max_{i\in N}\{ a_i\})\left(\max_{i\in {P}^{\sigma}_{j}\cup\{j\}}\left \{{{d_i}\over {K_i}}\right \} - \max_{i\in {P}^{\sigma}_{j}}\left \{{{d_i}\over {K_i}}\right \}\right) \\ & \leq \sum_{j\in S}(a+\max_{i\in N}\{ a_i\})\left(\max_{i\in ({P}^{\sigma}_{j}\cup\{j\})\cap S}\left \{{{d_i}\over {K_i}}\right \} - \max_{i\in {P}^{\sigma}_{j}\cap S}\left \{{{d_i}\over {K_i}}\right \}\right) \\ &=(a+\max_{i\in N}\{ a_i\}) \sum_{j\in S}\left(\max_{i\in ({P}^{\sigma}_{j}\cup\{j\})\cap S}\left \{{{d_i}\over {K_i}}\right \} - \max_{i\in {P}^{\sigma}_{j}\cap S}\left \{{{d_i}\over {K_i}}\right \}\right) \\ &=(a+\max_{i\in S}\{ a_i\}) \max_{i\in S}\left \{{{d_i}\over {K_i}}\right \} \\ &=c(S), \\ \end{aligned}$$

    where the inequality follows from the fact that the function f(x) = max{xy} − x is non-increasing in x for all \(y\in [0,\infty )\) (and taking \(y=\frac{d_j}{K_j}\)).

  2. (b)

    S does not contain the agent σ −1(1). In this case denote \(\bar{S}=S\cup \{\sigma^{-1}(1)\}.\) Using the same proof above we conclude that

    $$\sum_{i\in \bar{S}}m^{\sigma}_{i}(N,c)\leq c(\bar{S}).$$

    Now, taking into account that \(m^{\sigma}_{\sigma^{-1}(1)}(N,c)=c(\sigma^{-1}(1))\) and that (Nc) is subadditive, it holds that

    $$c(\sigma^{-1}(1))+\sum_{i\in {S}}m^{\sigma}_{i}(N,c)=\sum_{i\in \bar{S}}m^{\sigma}_{i}(N,c)\leq c(\bar{S})\leq c(\sigma^{-1}(1))+c(S),$$

    which implies that \(\sum_{i\in S}m^\sigma_{i}(N,c)\leq c(S).\)

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Fiestras-Janeiro, M.G., García-Jurado, I., Meca, A. et al. Centralized inventory in a farming community. J Bus Econ 84, 983–997 (2014). https://doi.org/10.1007/s11573-014-0710-z

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