Safety stock management in single vendor–single buyer problem under VMI with consignment stock agreement

doi:10.1016/j.ijpe.2014.04.007

International Journal of Production Economics

Volume 154, August 2014, Pages 16-31

https://doi.org/10.1016/j.ijpe.2014.04.007 Get rights and content

Highlights

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We model the VMI policy with consignment agreement considering safety stock.
•
We consider the single vendor–single buyer problem, with equal shipments size.
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The safety stock is expressed using an approximation of the standard normal c.d.f.
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The service level takes into account the number of admissible stockouts per year.
•
A minimum cost solution is obtained.

Abstract

In this paper a new analytical approach to safety stock management, within single buyer–single vendor framework under VMI with consignment agreement, is presented. In particular, the cost of the safety stock is evaluated adopting a logistic approximation of the standard normal cumulative distribution. The service level is put in relation to the dimension of the single shipment, to the average demand on the buyer and to the number of admissible stockouts. Once the mean joint total cost function is defined, on the basis of the model of Braglia and Zavanella, (2003. Int. J. Prod. Res. 41, 3793–3808) modified according to Braglia et al. (2014. Int. J. Logist. Syst. Manage. forthcoming 2014), a minimum cost solution (within a predefined domain) is derived. Moreover, a numerical study is carried out. First, a sensitivity analysis is shown. Then, the model of Braglia and Zavanella, (2003. Int. J. Prod. Res. 41, 3793–3808), in its original expression, is compared with both Hill (1997. Eur. J. Oper. Res. 97, 493–499), Hill (1999. Int. J. Prod. Res. 37, 2463–2475) and Hill and Omar (2006. Int. J. Prod. Res. 44, 791–800). In particular, the comparison considers the equal-sized shipments case without delayed deliveries taking into account different reciprocal values for the stockholding costs.

Introduction

To improve their competitive capacity, firms tend to become integral part of a supply chain, rather than being single entities. According to this point of view, the development of Joint Economic Lot Size (JELS) models still represents one of the main research topic in the Supply Chain Management (SCM) field.

Among the others, the Consignment Stock (CS) policy stimulated great interest. According to this strategy, the supplier autonomously manages the stock of its own items at the customer warehouse and both decides the dimension of the batches and the time of delivery. In the vendor–buyer relation only the former manages operatively, in an integrated and optimized fashion, the whole stock level of the considered product within the supply chain. Ultimately, the consignment stock concept means that the supplier holds the stock ownership until the customer actually uses it. A comprehensive analysis of the CS policy is provided, e.g., by Valentini and Zavanella (2003) and Gümüs et al. (2008).

An early analytical formulation of the CS was proposed by Braglia and Zavanella (2003), who proved the better performances of CS in a stochastic environment (with particular reference to the equal-sized shipments with delayed deliveries case) than the standard JELS model proposed by Hill, 1997, Hill, 1999. Their model has later been extended taking into account various other aspects and/or new methods of solution. Indeed, Tang et al. (2007) considered the impact of variability within a decentralized decision system, Zavanella and Zanoni (2009) studied the one-vendor multi-buyer case, Battini et al. (2010) took into account obsolescence, safety stock, and limited space availability, Wang et al. (2012) analyzed the CS model for a deteriorating item and capacity constrained buyer warehouse. Finally, Bylka (2012) faced the CS modeling under a game theoretic perspective. An exhaustive literature review on the whole JELS modeling is provided by Glock (2012).

In this paper a new analytical approach to the safety stock management, within the single buyer–single vendor framework under VMI with consignment agreement, is presented. In particular, the safety stock at the buyer side is evaluated exploiting a logistic approximation of the standard normal cumulative distribution. Moreover, the service level is put in relation to the average demand on the buyer, to the dimension of the single shipment and to the number of admissible stockouts per year. Taking into account these aspects a mean joint total cost formulation is first derived (which is essentially based on the model of Braglia and Zavanella (2003) modified according to Braglia et al. (2014)), and a minimum cost solution (within a predefined domain) is then provided. Finally, the model developed is studied numerically. First, a sensitivity analysis is carried out. Then, the model of Braglia and Zavanella (2003), in its original expression, is compared with both Hill, 1997, Hill, 1999 and Hill and Omar (2006). In particular, the comparison considers the equal-sized shipments case without delayed deliveries taking into account different reciprocal values for the stockholding costs.

The paper is thus organized: Section 2 introduces the analytical model, Section 3 presents the numerical study, and, finally, Section 4 deals with the conclusions.

Section snippets

Notation and assumptions

TC(q,n)=

mean joint total cost per year ($/year);

average demand rate on the buyer (units per year);

σ_D=

standard deviation of demand (units per year);

vendor production rate, constant and continuous (units per year);

number of shipments per batch production run;

size of the production batch (units);

q_i=

size of the ith shipment in a batch production run (units), with $\sum_{i = 1}^{n} q_{i} = n \times q = Q$ ;

A_V=

fixed production setup cost ($/setup);

A_B=

fixed order/shipment cost ($/order);

h_{B}^{p h} (h_{V}^{p h})

physical storage

Sensitivity analysis

Let us consider the following fixed values took from Braglia and Zavanella (2003):

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A_V=400 $/setup;
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A_B=25 $/order;
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$h_{V, V}^{f i n}$ =2 $/item/year;
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$h_{V}^{p h}$ =2 $/item/year;
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$h_{V, B}^{f i n}$ =3 $/item/year;
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$h_{B}^{p h}$ =2 $/item/year;
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P=3200 units/year;
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D=1000 units/year.

We have reasonably assumed that $h_{B}^{p h} = h_{V}^{p h}$ (Valentini and Zavanella, 2003). It is worth highlighting that, in our example, $h_{1}^{C S} = 4 < h_{2}^{C S} = 5 .$ If $h_{1}^{C S} > h_{2}^{C S},$ the main results do not change. In addition, let us fix α^⁎ equal to 20%.

To ease the analysis, let us consider different

Conclusions and future works

The present paper introduces new aspects into the single vendor–single buyer problem under Vendor Managed Inventory with consignment agreement. Starting from the early model proposed by Braglia and Zavanella (2003), and involving the observations given by Braglia et al. (2014) concerning the evaluation of the stockholding cost at the buyer, an innovative formulation concerning the safety stock cost is provided. As a matter of fact, the safety stock is introduced adopting a logistic

Acknowledgments

The authors gratefully acknowledge the reviewers, whose precious suggestions and comments permitted to improve considerably the quality and the clarity of the manuscript.

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The single-vendor single-buyer integrated production–inventory model with a generalized policy
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(1997)
G. Valentini et al.
The consignment stock of inventories: industrial case and performance analysis
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A one-vendor multi-buyer integrated production–inventory model: the ‘Consignment Stock’ case
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Consignment stock inventory policy: methodological framework and model
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S.R. Bowling et al.
A logistic approximation to the cumulative normal distribution
J. Ind. Eng. Manage.
(2009)

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Safety stock management in single vendor–single buyer problem under VMI with consignment stock agreement

Highlights

Abstract

Introduction

Section snippets

Notation and assumptions

Sensitivity analysis

Conclusions and future works

Acknowledgments

Int. J. Prod. Econ.

Int. J. Prod. Econ.

Eur. J. Oper. Res.

Int. J. Prod. Econ.

Int. J. Prod. Econ.

Int. J. Prod. Econ.

Consignment stock inventory policy: methodological framework and model

Int. J. Prod. Res.

A logistic approximation to the cumulative normal distribution

J. Ind. Eng. Manage.