# Large scale stochastic inventory routing problems with split delivery and service level constraints

## Abstract

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.

## Abbreviations

i,j=0,1,…,N :

Index of customer or depot, where i,j=1,…,N are customer indexes, and 0 is the depot index,

t=1,…,T :

Period index,

C :

Vehicle capacity in volume,

c ij :

Variable shipping cost per unit of product along arc (i,j) where c ij =c ji and triangle inequality holds (c ij +c jk c ik ),

$$c_{i0}^{b}$$ :

Traveling cost of an empty vehicle from customer i back directly to the depot,

f t :

Fixed vehicle cost per tour in period t,

h it :

Holding cost per unit product for customer i in period t,

I i0 :

Initial inventory level at the beginning of period 1,

I it :

Inventory level of customer i at the end of period t,

$$I_{it}^{+}=\max\mathrm{max}\,(0,I_{it})$$ :

On-hand inventory of customer i at the end of period t, which excludes the stockout of I it <0,

V i :

The inventory capacity for customer i’s warehouse,

α it :

Service level for customer i’s demand in period t (probability that customer i’s demand is satisfied in period t),

β it :

The service level of customer i’s warehouse in period t (probability that customer i’s warehouse is not overfilled in period t),

ζ it :

Stochastic demand of customer i in period t,

ζ i,(1,t) :

$$=\sum_{s=1}^{t}\zeta_{is}$$, cumulative stochastic demand from period 1 to t,

F i,(1,t)(.):

Accumulative probability distribution function of stochastic demand ζ i,(1,t),

d it :

Delivery volume to customer i in period t,

q ijt :

Demand quantity transported on directed arc (i,j) in period t,

x ijt :

Number of the times that customer j is visited directly after customer i in period t.

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Yu, Y., Chu, C., Chen, H. et al. Large scale stochastic inventory routing problems with split delivery and service level constraints. Ann Oper Res 197, 135–158 (2012). https://doi.org/10.1007/s10479-010-0772-4