## Location, Routing, and Inventory

Read this article. In it, a model is presented to help determine the number of distribution centers, their locations, and capacity among other factors. Among the 15 assumptions presented, which do you feel are most important and least important?

### Problem Formulation

#### Problem description

The trade-off
between cost and time creates a bi-objective problem. One criterion
tries to minimize the fixed cost of locating the opened distribution
centers, the safety stock costs of distribution center by considering
uncertainty in customer's demand, inventory ordering and holding costs,
the transportation costs from a plant to its allocated distribution
centers, and also vehicle routing costs beginning from a distribution
center (DC) with the aim of replying to and covering the devoted
customer's demands to the DC by considering risk-pooling. The other
criterion looks for the reduction of the time to transport the product
along the supply chain. It is desired to minimize the transportation
time from a plant to customers. The important assumptions in this paper
are as follows:

- One kind of product is involved.
- Each distribution center j is assumed to follow the (, ) inventory policy.
- The inventory control is to be conducted only at DCs in this paper.
- A single-sourcing strategy is considered in the whole supply chain.
- It is considered that the customers' demands after reaching the retailer are independent and follows a normal distribution.
- Each plant has a limited capacity.
- We consider different capacity levels for each distribution center, and finally, one capacity for each of them is selected.
- Each DC with the limited capacity carries on-hand inventory to satisfy demands from customer demand zones as well as safety stock to deal with the mutability of the customer demands at customer demand zones to attain risk-pooling profits.
- All customers should be served.
- The number of available vehicles for each type and the number of allowed routes for each DC are limited.
- There are several modes of transportation between two consecutive levels.
- Between two nodes on an echelon, only one type of vehicle is used.
- A faster transportation mode is the more expensive one.
- The amount of products is transported from each plant to each distribution center that is associated with it, and an equal amount of products has been ordered from the desired distribution center to that plant.
- To determine all feasible routes, the following factors are taken into account:

- Each customer should be visited by only one vehicle.
- Each route begins at a DC and ends at the same DC.
- The sum of the demands of the customers served in each route must not exceed the capacity of the associated vehicle.
- Each of the distribution center and the vehicle have various limited, and determined capacity.

#### Model formulation

Following are the notations introduced for the mathematical description of the proposed model.

1. Indices

- , set of plants indexed by
- , set of candidate DC locations indexed by
- , set of customer demand zones indexed by
- , set of capacity levels available to
- , set of all feasible routes using a vehicle of type from
- , set of vehicles between nodes and
- , set of vehicles between nodes and

2. Parameters

- , yearly fixed cost for opening and operating distribution center with capacity level
- , cost of transporting one unit of product from plant to distribution center using vehicle
- , cost of sending one unit of product in route using vehicle (These costs include the fixed cost of vehicle plus the transportation cost of each demand unit in route r. The mentioned transportation cost for each demand unit is not related to customer demand zone, and it is considered fixed for all locations in each route ).
- , time for transporting any quantity of a product from plant to using vehicle
- , time for transporting any quantity of a product from on route using vehicle
- , safety stock factor of
- , unit inventory holding cast at , (annually)
- , mean demand at customer demand zone
- , variance of demand at customer demand zone
- , fixed inventory ordering cost at
- , capacity with level for
- , capacity of plant
- , number of available vehicles of each type
- , number of routes associated with each distribution center

3. Binary coefficients

4. Decision variables

- if distribution center is opened with capacity level , and 0 otherwise
- \(A_{ijl1}, binary variable equal to 1 if vehicle connecting plant and is used, and equal to 0 otherwise
- , binary variable equal to 1 if vehicle connecting and customer is used, and equal to 0 otherwise
- , 1 if and only if route is selected, and 0 otherwise
- , quantity transported from plant to using vehicle

5. Mathematical model

(a) The problem formulation is as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(21)

In
this model, the first objective function minimizes the total expected
costs consisting of the fixed cost for opening distribution centers with
a certain capacity level, transportation costs from plants to
distribution centers, annual routing costs between distribution centers
and customer demand zones, and expected annual inventory costs. The
second objective function looks for the minimum time to transport the
product along any path from the plants to the customers.

Constraint
(1) ensures that each distribution center can be assigned to only one
capacity level. Constraints (2) and (3) are the capacity constraints
associated with the distribution centers, and also, constraint (4) is
the capacity constraints associated with the plants. Constraint (5)
states that if distribution center with capacity is opened, it is
serviced by a plant. Constraint (6) represents the single-sourcing
constraints for each customer demand zone. Constraints (7) and (8)
ensure that if two nodes on an echelon are related to each other, one
type of vehicle transports products between them. Constraint (9) makes
sure that if the distribution center j gives the service to the customer
k, that center must get services at least from a plant. Constraint (10)
ensures that if the distribution center j is allocated to customer k by
vehicle , that center should certainly be established with a
determined capacity level. Constraint (11) is the standard set covering
constraints, modeling assumption 9. Constraints (12) and (13) impose
limits on the maximum number of available vehicles of each type and the
maximum number of allowed routes for each DC, modeling assumption 10.
Constraint (14) makes sure that if plant i gives the service to the DC j
, the amount of transported products from that plant to the desired
distribution center would be more than one. Constraint (15) implies that
customers' demands of zone are more than 1. Constraint (16) is the
capacity constraint associated with plant i. Constraints (17) to (20)
enforce the integrality restrictions on the binary variables. Finally,
constraint (21) enforces the non-negativity restrictions on the other
decision variables.