The problem of the distribution of products by vehicle and keeping the inventory levels within specified bounds at warehouse in a logistic system is called the inventory routing problem (IRP). The IRP assumes application of vendor management inventory (VMI) concept where suppliers determine an order quantity and the time of its delivery. The IRP is usually formulated and solved under the assumptions of deterministic consumption and delivery times. However, in real-life scenario demand is stochastic in its nature. This paper presents a comprehensive literature review related to stochastic inventory routing problem (SIRP). Under this approach, the order delivery is in congruence with stochastic demand rates. The objective of SIRP is to resolve a flow strategy that reduce operational and inventory holding costs without having a stock-out. In this study, an extensive literature survey focusing on inventory policy, fleet characteristics, types and problem structure and solving approach is summarized and analyzed. The gaps in the field are identified. Besides, future trends, opportunities, and suggestions are presented.

Keywords: - Inventory Routing Problem, Stochastic IRP, Vendor Managed Inventory, attributes

### Introduction

The inventory routing problem (IRP) is one of the challenging optimization problems in logistics and supply network. It focuses on optimally integrate inventory control and vehicle routing operations in a supply network as shown in figure 1. Its main goal is to determine an optimal replenishment policy, which includes vehicle routes, quantity delivered and lead time helps to minimize inventory holding and distribution costs. Implementing policies such as vendor managed inventory (VMI) has proven to considerably improve the overall performance of a supply network. However, this does not guarantee that implementing VMI elements such as visibility of demand, inventory location, and replenishment decision will impact or improve VMI performance (service and cost) Radzuan, Rahim [1].

More specifically, previous studies focused on single-warehouse, multiple-retailer vendor managed inventory, in cases where all retailers face a stationary demand. However, in real-world issues, demand rates are not usually constant rather it’s stochastic. This review focus on stochastic inventory routing problem (SIRP), where retailers consume the product at a stochastic demand rate.

Moreover the current study seeks unearth articles, published from 1984 to 2018 in the refereed journals. This study borrows a leave from Andersson, Hoff [2], Roldán, Basagoiti [3] and studies with a different classification technique and study period. However it differs from Andersson, Hoff [2] as its focused on stochastic IRP from 1984 to 2018. The paper is to provide and elucidate a comprehensive review of stochastic IRP and the associated methodologies used.

Figure1. An illustration of IRP

Stochastic IRP (SIRP)

In this section, we present the literature review of several variants of the SIRP. This paper also includes the review of relevant literature on the problem arising in inventory management, as long as it is connected and relevant to IRP. Some methods that have been used in vehicle routing are also described, which could also be applied in the IRP. Logistics is a very challenging area. From a cost center, logistics is now seen as a value adding center, controlling the delivery process and inventory, and managing the demand. IRP is a viable alternative that can be employed. A classification of IRP models by Andersson, Hoff [2], is shown in Table 1 below.

X_ijkt={█(1 if vehicle k uses node( i ",j) in period t; @0",otherwise)┤

The objective of classical IRP can be mathematically formulated as in equation 1.

Minimized=∑_(t∈T)▒∑_(i∈N)▒〖ICr〗_it +∑_(i∈N)▒∑_(t∈T)▒〖ICw〗_it +∑_(k∈K)▒∑_(t∈T)▒〖VC〗_kt (1)

Where

〖ICr〗_it Inventory cost at retailer i on time t

〖ICw〗_it Inventory cost at the warehouse i on time t

〖VC〗_kt Distribution cost of vehicle k on time t

Subject to:-

Constraints ((2)–(4)) represent routing constraints are describe as follows

∑_(j∈N)▒∑_(k∈K)▒X_ijkt ≤1",∀i∈N∖{0}",∀ t∈T

(2)

∑_(i≠j, i∈N)▒X_ijkt -∑_(p≠j, p∈N∪M)▒X_jkt =0",∀j∈N ∕{0}",∀ t∈T",∀ k∈K

(3)

∑_( i∈M)▒X_i0kt -∑_(p∈N∕{0})▒X_0pkt =0",∀ t∈T",∀ k∈K (4)

Constraint (2) confirms that each customer visited at once. Constraint (3) shows that if vehicle k arrives at retailer j on time unit t, then it must depart from retailer j . The same constraints of supplier are given by (4)

∑_(i∈N∪M)▒∑_(j∈N∪M)▒〖〖θ 〗_ij X〗_ijkt ≤τ_t",∀ k∈K

(5)

∑_(k∈K)▒∑_(i∈N∪M)▒β_ijk -∑_(k∈K)▒∑_(i∈N∪M)▒β_jk =q_it",∀j∈N ∪M",∀ t∈T

(6)

Constraints (5) ensure that trucks complete their routes within one travel period, so the total travel time of a truck should not exceed the total working hours. Constraints (6) determine the quantity delivered to a customer.

U_i≥ q_it+I_it≥ d_it",∀ i∈N⁄{0} ",∀ t∈T

(7)

U_i ∑_j▒∑_k▒X_ijtk -I_it ≤ q_it",∀ i∈N⁄{0} ",∀ t∈T

(8)

I_it≥0",〖 X〗_ijtk ∈{0",1} q_it≥0",∀ i",j∈N",∀ t∈T

Constraints (7) and (8) guarantee that stocking out does not occur and up-to-level policy is implemented respectively.

3. Methodology

In order to investigate the sate of knowledge in stochastic IRP, this paper adopts meta-analysis as a systematic research method on SIRP. The research review covers basically covers publications within the period under review. After browsing a set of academic studies like Google and Google scholar, science Direct, Springer, DBLP database and others to search SIRP literatures.

Finally based on meta-analysis, articles were categorized based on classification methods so as to identify their distribution as shown in table 1.

Table 1. SIRP classification methods

1 Inventory policy 4 Type of problem

1.1. Maximum level 4.1. Direct

1.2. Deliver up to level 4.2. Pick up & delivery

1.3. Back-order 4.3. Time window

1.4. Non-negative 4.4. Multiple

2 Fleet composition 5 Structure

2.1. Homogeneous 5.1. One-to one

2.2. Heterogeneous 5.2. One-to-many

3 Fleet size 5.3. Many-to many

3.1. Single 6 Solution methods

3.2. Multiple

3.3. Unconstrained

Adopted from Andersson, Hoff [2]

Table 1 depicts classification methods for SIRP. For the first method, inventory policy is defined as rules to replenish user. Through results found in the literature, there are maximum, deliver up to level, back-order and non-negative policies attributes. If the inventory cannot supply the demand, then back-ordering would occur, which will restock the lack of inventory through newly delivered shipments. Managing the inventory shows how the model can be determined. If there is no back-order, then the extra demand is considered lost sales and stock-out may occur. In logistics, the number of supplier and retailers may change, where it could entail one-to-one, one-to-many, and sometimes many-to-many relationships. Meanwhile, fleet composition and size can be homogeneous or heterogeneous. Homogeneous is when the vehicle delivers the inventory using fixed capacity in a route. Heterogeneous refers to when the delivery use and the number of vehicles available may be fixed at single, multiple, or unconstrained.