Estimating freight rates in inventory replenishment and supplier selection decisions

Abstract

Given the impact of transportation costs on both supplier selection and inventory replenishment decisions in today’s enterprises, this article addresses both problems simultaneously by proposing a mixed integer nonlinear programming model to properly allocate order quantities to the selected set of suppliers while taking into account the purchasing, holding and transportation costs under suppliers’ capacity and quality constraints. It is shown that incorporating transportation costs in the process of selecting suppliers and establishing an inventory policy, not only affects the order quantity shipped from selected suppliers, but also the actual selection of suppliers. Because of the difficulty that arises when working with actual transportation freight rates in large-scale problems, two continuous functions that estimate actual freight rates are analyzed. It is shown that the use of these functions is very practical and easy to implement.

Appendix 1: Mendoza and Ventura’s model [13]

The mathematical model for supplier selection and order quantity allocation using actual transportation freight rates from Mendoza and Ventura [13] is presented in this appendix. With the exception of λ_i,k, b_i,k, g_i,k, which are introduced and explained below, the notation is the same as that used in the present article:

$$ \begin{aligned} ({\text{AO}})\,{\text{minimize}}\quad Z_{A} & = {\frac{d}{M}}\left[ {\frac{1}{Q}}\sum\limits_{i = 1}^{r} J_{i} k_{i} + \sum\limits_{i = 1}^{r} J_{i} p_{i}\right.\\ &\left. \quad+ \,{\frac{1}{Q}}\sum\limits_{i = 1}^{r} J_{i} \cdot TC_{i} (Qw) + {\frac{h}{Y}} \cdot \sum\limits_{i = 1}^{r} J_{i} l_{i} \right] + {\frac{hQ}{2}} \\ {\text{subject}}\,{\text{to}}\quad J_{i} d & \le c_{i} M,\quad i = 1, \ldots ,r, \\ \end{aligned} $$

(18)

$$ \sum\limits_{i = 1}^{r} J_{i} q_{i} \ge q_{a} M, $$

(19)

$$ \sum\limits_{i = 1}^{r} J_{i} = M, $$

(20)

$$ Q \cdot w = \sum\limits_{k = 1}^{{u_{i} + 1}} b_{i,k} \cdot \lambda_{i,k} ,\quad i = 1, \ldots ,r, $$

(21)

$$ TC_{i} (Qw) = \sum\limits_{k = 1}^{{u_{i} + 1}} g_{i,k} \cdot \lambda_{i,k} ,\quad i = 1, \ldots ,r, $$

(22)

$$ \lambda_{i,k} \le Z_{i,k} ,\quad i = 1, \ldots ,r; \quad k = 1, $$

(23)

$$ \lambda_{i,k} \le Z_{i,k - 1} + Z_{i,k} ,\quad i = 1, \ldots ,r;\quad k = 2, \ldots ,u_{i} , $$

(24)

$$ \lambda_{i,k} \le Z_{i,k - 1} ,\quad i = 1, \ldots ,r;\quad k = u_{i} + 1, $$

(25)

$$ \sum\limits_{k = 1}^{{u_{i} + 1}} \lambda_{i,k} = 1,\quad i = 1,\ldots,r, $$

(26)

$$ \sum\limits_{k = 1}^{{u_{i} + 1}} Z_{i,k} = 1,\quad i = 1, \ldots ,r, $$

(27)

$$ Z_{i,k} \in \{ 0,1\} ,\quad i = 1, \ldots ,r;\quad k = 1, \ldots ,u_{i} , $$

(28)

$$ \lambda_{i,k} \ge 0,\quad i = 1, \ldots ,r;\quad k = 1, \ldots ,u_{i} + 1, $$

(29)

$$ Q \ge 0, $$

(30)

$$ J_{i} \ge 0,\quad {\text{integer}},\quad i = 1, \ldots ,r, $$

(31)

$$ M \ge 1,\quad {\text{integer}}. $$

(32)

The total weight shipped from supplier ‘i’ (in one order) is defined in Eq. 21, where b_i,k (i=1,…,r; k =1,…,u _i +1) represents a break point (lb) that can be obtained from the actual LTL rate structure, and u _i is the total number of break points in the actual LTL rate structure. Furthermore, b_i,1 = 0 and $b_{{i,u_{i} + 1}}$ equals the capacity of a TL. The total transportation cost charged to supplier i for the weight (Qw) shipped is defined in Eq. 22, where g_i,k (i = 1,…,r; k =1,…,u _i +1) is the total transportation cost obtained by evaluating the corresponding break point b_i,k into the actual LTL rate structure. As per the definition of $b_{{i,u_{i} + 1}} ,$$g_{{i,u_{i} + 1}}$ is the cost of a full TL. Moreover, each binary variable, Z_i,k, represents one linear segment of the freight rate function (see Fig. 3). By constraint 27, only one Z_i,k per supplier can get a value of ‘1’. Then, the specific segment chosen contains the weight shipped (Qw) and its corresponding total transportation cost TC _i (Qw) is expressed as the linear combination of λ_i,k and λ_i,k+1 (0 ≤ λ_i,k ≤ 1, i = 1,…,r; k = 1,…,u_i + 1). This clearly explains the presence of constraints 23–27.