Coordinated Location, Distribution, and Inventory Decisions in Supply Chain Network Design

We consider a firm that owns a manufacturing plant that is capable of producing multiple products. The products are then delivered to different distribution centres (warehouses/wholesalers) in order to satisfy their associated dynamic demands. The network is illustrated in Figure 1.

Assume that in the company the logistics centre seeks to determine the right transportation plan to allocate multiple commodities from the source (factory) to J warehouses (DCs). Each destination has a forecast demand of each commodity to be received from the plant. The forecast demand of each warehouse depends on uncertain demands of allocated customers to this warehouse. This work focuses on developing a MOSNLP method to optimise distribution decisions such as location/allocation and inventory policy in a food industry company.

This problem in fact integrates three decision sub-problems: (1) selecting the optimum numbers, locations, and capacity levels of the warehouses to deliver products to retailer/customer at the least cost while satisfying desired service level to retailers; (2) allocation of these retailers to the open warehouses; and (3) inventory decisions for the supply chain.

Decision-making in such a complex supply chain network requires the consideration of conflicting objectives and of different constraints imposed by the manufacturer and distributors. Moreover, in practical situations, due to the variability and/or uncertainty of required data over the strategic and mid-term horizon, most of the parameters embedded in a DPD problem are frequently stochastic in nature, and can be obtained through probabilities or subjectively in a fuzzy environment. For example, in a real decision problem, market demands, cost/time coefficients, and the amounts of available resources are usually imprecise over the planning horizon, and therefore assigning a set of crisp values for such ambiguous parameters is not appropriate. We rely on probability theory to model this uncertainty. This theory uses statistical distributions to handle this inherently ambiguous phenomenon in the problem parameters.

Problem description, assumptions, and notation

The stochastic multi-objective DPD problem examined here can be described as follows:

• In the analysed case study, the plant location is known and fixed. The network considered encompasses a set of retailers with known locations, and a discrete set of possible location zones/sites where the plant and warehouses are located.
• The final products have stochastic retailer demand over the given finite planning horizon with mean dil and variance vil (note that our customers are mostly retailers, not end consumers).

• There are multiple values for storage capacity at each warehouse (five level storage capacities).
• The distribution costs and delivery time on the given route are directly proportional to the shipped units.
• Products are independent of each other, related to marketing and sales price.
• The number of potential DCs and their maximum capacities are known.
• Retailers receive each product only from a single DC.
• No inventory is held in the plant.
• Decisions are made within a single period.

The indices, parameters, and variables used to formulate the mathematical problem are described as follows:

Indices:

$I$            index set of customers/customer zones (i=1,....., I)

$J$            index set of potential warehouse sites (j=1,....,J)

$L$            index set of products (l=1,...,L)

$H$            index set of capacity levels available to the potential warehouses (h=1,_,H)

$G$            index for objectives for all g=1,2,3

$η$             investment cost performance index [0,1]

$γ$             delivery time performance index [0,1]

$K$            customer service performance index [0,1]

$Z_{l-a}$        normal distribution value for system service level

Parameters:

$TC_{jjt}$       unit cost of supplying product l to customer zone i from warehouse on site j

$\overline{TC} _{jl}$      unit cost of supplying product l to warehouse on site j from the plant

$t_{ijl}$          delivery time per unit delivered from warehouse j to customer zone i for each product l

$t_{ij}$           delivery time per unit delivered from the plant to warehouse j for each product l

$LT_{j^{'}j}$        the elapsed time between two consecutive orders of product l for site j

$Ρ$             fixed cost for opening and operating warehouse with capacity level h on

$F_{jh}$          site j per time unit

$d_{il}$           mean demand of product l from customer zone i per time unit

$v^{'}_{il}$          variance demand of product l from customer zone i per time unit

$HC_{jl}$         holding cost of product l in warehouse on site j per time unit

$OC_{jt}$        ordering cost of product l from warehouse on site j to the plant

$cap_{jh}$       capacity of warehouse on site j with capacity level h

$s_t$            space requirement of product l at any warehouse

$PH$            planning horizon

Decisions variables:

$X_{jh}$           it takes value 1, if a warehouse with capacity level h is installed on potential site j, and 0 otherwise;

$D_{ij}$            it takes value 1, if the warehouse on site j serves product l of customer i, and 0 otherwise;

$D_{ji}$           mean demand of product l to be assigned to warehouse on site j per time unit

$V_{jl}'$          variance demand of product l to be assigned to warehouse on site j per time unit

Stochastic multi-objective non-linear programming model

Objective functions

We have selected the multi-objective functions for solving the DPD problem by reviewing the literature and considering practical situations. In particular, these objective functions are normally stochastic or fuzzy in nature owing to incomplete and/or uncertain information over the planning horizon. Accordingly, three objective functions are simultaneously considered in formulating the original stochastic DPD problem, as follows:

• Minimise total investment (INV) in opening DCs/warehouses:
  
  $\operatorname{Min} Z_{1}=\sum_{j=1}^{J} \sum_{h=1}^{H} F_{j h} \cdot X_{j h}$
  
  
• Minimise total costs (TCOST)

This objective function contains (see Appendix):

• Transportation cost of products from the plant to the warehouses and from the warehouses to the retailers
• Holding cost for mean inventory and safety stocks
  
  $\begin{aligned}&\operatorname{Min} Z_{2}=\sum_{i=1}^{I} \sum_{j=1}^{J} \sum_{l=1}^{L}\left(T C_{j l}+T C_{i j l}\right) \cdot d_{i l} \cdot Y_{i j l} \\&+\sum_{j=1}^{J} \sum_{l=1}^{L} \sqrt{2 \cdot H C_{j l} \cdot O C_{j l}} \cdot \sqrt{D_{j l}}+\sum_{j=1}^{J} \sum_{l=1}^{L} H C_{j l} \cdot Z_{1-\alpha} \cdot \sqrt{L T_{j l}} \cdot \sqrt{V_{j l}}\end{aligned}$
  
  
• Minimise total delivery time (TDELT)
  
  $\operatorname{Min} Z_{3}=\sum_{i=1}^{I} \sum_{j=1}^{J} \sum_{l=1}^{L} (t_{j l} + t_{i j l}) \cdot d_{i l} \cdot Y_{i j l }$
  
  

Constraints

• Constraints that ensure that each retailer is served exactly for each product by one warehouse (single source):
  
  $\sum_{j=1}^{j} Y_{i j l} = 1 \quad \forall i=1, \ldots, I, \forall l=1, \ldots, L$
  
  
• Constraints of the warehouse capacity:
  
  $\sum_{i=1}^{I} \sum_{l=1}^{I} d_{i l } \cdot S_{l} \cdot Y_{i j l} \leq \sum_{h=1}^{H} cap \; _{j h} \cdot X_{j h} \quad \forall l=1, \ldots, J$
  
  
• Constraints that compute the served average demand by each warehouse:
  
  $\sum_{i=1}^{I} d_{i l} \cdot Y_{i j l}=D_{j l} \quad \forall j=1, \ldots, J, \forall l=1, \ldots, L$
  
  
• Constraints that indicate the total variance of the served demand by each warehouse:
  
  $\sum_{i=1}^{I} v_{i j l} \cdot Y_{i j l}=V_{j l} \quad \forall j=1, \ldots, J, \forall l=1, \ldots, L$

Implicitly, we assume that the demands are independently distributed across the retailers, and thus that all the covariance terms are zero.

• Constraints that ensure that each warehouse can be opened at least at one capacity level
  
  $\sum_{h=1}^{H} X_{j h} \leq 1 \quad \forall j=1, \ldots, J$
  
  
• Binary constraints of decision variables:
  $X_{j h}, Y_{i j l} \in\{0,1\} \quad \forall i=1, \ldots, I, \forall j=1, \ldots, J, \forall l=1, \ldots, L, \forall h=1, \ldots, H$