Location, Routing, and Inventory

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:

1. One kind of product is involved.
2. Each distribution center j is assumed to follow the ($Q_i$, $R_j$) inventory policy.
3. The inventory control is to be conducted only at DCs in this paper.
4. A single-sourcing strategy is considered in the whole supply chain.
5. It is considered that the customers' demands after reaching the retailer are independent and follows a normal distribution.
6. Each plant has a limited capacity.  
   
7. We consider different capacity levels for each distribution center, and finally, one capacity for each of them is selected.
8. 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.
9. All customers should be served.
10.  The number of available vehicles for each type and the number of allowed routes for each DC are limited.
11. There are several modes of transportation between two consecutive levels.
12. Between two nodes on an echelon, only one type of vehicle is used.
13. A faster transportation mode is the more expensive one.
14. 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.
15. To determine all feasible routes, the following factors are taken into account:

Model formulation

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

1. Indices

1.  $I$, set of plants indexed by $i$
2. $J$, set of candidate DC locations indexed by $j$
3. $K$, set of customer demand zones indexed by $k$
4. $N_j$, set of capacity levels available to $DC_j (j ∈ J)$
5. $Ω_{jl2}$, set of all feasible routes using a vehicle of type $l_2$ from $DC_+{j} (j∈J)$
6. $LP_{ij}$, set of vehicles $l_1$ between nodes $i$ and $j$
7. $LW_{jk}$, set of vehicles $l_2$ between nodes $j$ and $k$

2. Parameters

1. $F_{j}^{n}$, yearly fixed cost for opening and operating distribution center $j$ with capacity level $n (∀ n ∈ N_j , ∀ j ∈ J)$
2. $CP_{ijl_1}$, cost of transporting one unit of product from plant $i$ to distribution center $j$ using vehicle $l_1$
3. $CW_{rl_2}$ , cost of sending one unit of product in route $r$ using vehicle $l_2$ (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 $r$).
4. $TP_{ijl_1}$, time for transporting any quantity of a product from plant $i$ to $DC_j$ using vehicle $l_1$
5. $TW_{jl_2r}$, time for transporting any quantity of a product from $DC_j$ on route $r$ using vehicle $l_2$
6. $λ_j$, safety stock factor of $DC_j (j ∈ J)$
7. $h_j$, unit inventory holding cast at $DC_j (j ∈ J)$, (annually)
8. $μ_k$, mean demand at customer demand zone $k$
9. $δ^{2}_{k}$, variance of demand at customer demand zone $k$
10. $E_j$, fixed inventory ordering cost at $DC_j$
11. $b_{j}^{n}$, capacity with level $n$ for $DC_j$
12. $MP_i$ , capacity of plant $i$
13. $ω_{l_2}$, number of available vehicles of each type $l_2$
14. $g j$ , number of routes associated with each distribution center $j$

3. Binary coefficients

1. $P_{kr}, 1$ if and only if customer $k$ is visited by route $r$, and 0 otherwise

4. Decision variables

1. $U^{n}_{j} , 1$ if distribution center $j$ is opened with capacity level $n$, and 0 otherwise
2. \(A_{ijl1}, binary variable equal to 1 if vehicle $l_1$ connecting plant $i$ and $DC_j$ is used, and equal to 0 otherwise
3. $B_jkl_2$, binary variable equal to 1 if vehicle $l_2$ connecting $DC_j$ and customer $k$ is used, and equal to 0 otherwise
4. $X_r$ , 1 if and only if route $r$ is selected, and 0 otherwise
5. $X_{ijl_1}$, quantity transported from plant $i$ to $DC_j$ using vehicle $l_1$

5. Mathematical model

(a) The problem formulation is as follows:

$minf_1 = ∑_{n ∈ N_j} ∑_{j ∈ J} F_{j}^{n} U_{j}^{n} + ∑_{i ∈ I } ∑_{j ∈ J} ∑_{l_1 ∈ LP_{ij}} CP_{ijl_1} A_{ijl_1} X_{ijl_1}$

$+ ∑_{j ∈ J} ∑_{k ∈ K} ∑_{l_2 ∈ LW_{jk}} ∑_{r ∈ Ω_{jl_2}} CW_{rl_2} μ_k P_{kr} X_r$

$+∑_{i ∈ I} ∑_{j ∈  J} ∑_{k ∈ K} ∑_{l_1  ∈ LP_{ij}}  ∑_{l_2}  ∈ LW_{jk} \dfrac{E_j B_{jkl_2 μk}} {X_{ijl_1} A_{ijl_1}}$

$+∑_{i ∈ I} ∑_{j ∈ J} ∑_{l_1  ∈ LP_{ij}} \dfrac{A_{ijl1} X_{ijl_1} h_j} {2}$

$+∑_{j ∈ J} λ_j h_j \sqrt{∑_{i ∈  I} ∑_{k ∈ K} ∑_{l_1 ∈ LP_{ij}} ∑_{l_2 ∈ LW_{jk}} δ_{k}^{2} B_{jkl_2} L_{jil_1} A_{ijl_1}}$

$min f_2 = max_j (max_{i,l_1} (TP_{ijl_1} A_{ijl_1})+max_{r,l_2} (TW_{jl_2r}X_r))$

$s.t.$

$∑_{n ∈ N_j} U_{j}^{n} ≤ 1 ∀j ∈ J$

(1)

$∑_{k ∈ K} ∑_{l_2 ∈ LW_{jk}} μ_k B_{jkl_2} ≤ ∑_{n ∈ N_j} b_{j}^{n} U_{j}^{n} ∀j ∈ J$

(2)

$∑_{i ∈ I} ∑_{l_1 ∈ LP_{ij}} X_{ijl_1} + λ_{j} \sqrt{∑_{i ∈ I} ∑_{k  ∈ K} ∑_{l_1 ∈ LP_{ij}} ∑_{l_2 ∈ LW_{jk}} δ_{k}^{2} B_{ikl_2} L_{ji l_1} A_{ijl_1}} ≤ ∑_{n ∈ N_j} b_{j}^{n} U_{j}^{n}∀j ∈ J$

(3)

$∑_{j ∈ J} ∑_{l_1 ∈ LP){ij}} X_{ijl_1} ≤ MP_i ∀i ∈ I$

(4)

$∑_{i ∈ I} ∑_{l_1 ∈ LP_{ij}} A_{ijl_1} ≥ ∑_{n ∈N_j} U_{j}^{n} ∀j ∈ J$

(5)

$∑_{j ∈ J} ∑_{l_2 ∈ LW_{jk}} B_{jkl_2} =1∀k ∈ K$

(6)

$∑_{l_2 ∈ LW_{jk}} B_{jkl_2} ≤ 1∀j ∈ J,∀k ∈ K$

(7)

$∑_{l_1 ∈ LP_{ij}} A_{ijl_1} ≤ 1∀i ∈ I,∀j ∈ J$

(8)

$∑_{l_1 ∈ LP_{ij}} ∑_{i ∈ I} A_{ijl_1} ≥ ∑_{l_2 ∈ LW_{jk}} B_{jkl_2} ∀j ∈ J,∀k ∈ K$

(9)

$∑_{n ∈ N_j} U_{j}^{n} ≥ ∑_{l_2 ∈ LW_{jk}} B_{jkl_2}∀j ∈ J,∀k ∈ K$

(10)

$∑_{j ∈ J } ∑_{l_2 ∈ LW_{jk}} ∑_{r ∈ Ω_{jl_2}} P_{kr} X_{r} ≥1∀k ∈ K$

(11)

$∑_{j ∈ J} ∑_{r ∈ Ω_jl_2} X_{r} ≤ W_{l_2}∀l_2 ∈ LW_{jk}$

(12)

$∑_{l_2 ∈ LW_{jk}} ∑_{r∈Ω_{jl_2}} X_r ≤ g_{j} ∀j ∈ J$
(13)

$X_{ijl_1} −A_{ijl_1} ≥ 0∀i ∈ I,∀j ∈ J,∀l_1 ∈ LP_{ij}$
(14)

$μk ≥ ∑_{l_2∈LW-_{jk}} ∑_{n ∈ N_j} ∑_{j ∈ J } B_{jk l_2} U_{j}^{n}∀k ∈ K$
(15)

$MP_i−A_{ijl_1} X_{ijl_1} ≥ 0∀i ∈ I,∀j ∈ J,∀l_1 ∈ LP_{ij}$
(16)

$U_{j}^{n} ∈ {0,1} ∀j ∈ J,∀n ∈ N_j$
(17)

$X_r ∈ {0,1} ∀r ∈ ∪ _{j∈J , l_2 ∈LW_{jk}} Ω_{jl_2}$
(18)

$A_{ijl_1} ∈ {0,1} ∀i ∈ I,∀j ∈ J, ∀l_1 ∈ LP_{ij}$
(19)

$B_{jkl_2} ∈ {0,1} ∀j ∈ J,∀k ∈ K,∀l_2 ∈ LW_{jk}$
(20)

$X_{ijl_1} ≻ 0∀i ∈ I,∀j ∈ J,∀l_1∈LP_{ij}$
(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 $j$ with $n$ 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 $l_2$, 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 $k$ 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.

Optimization is a mathematical procedure to determine devoting the optimal allocation to scarce resources, and it helps to get the best result from the model solution. In this paper, we consider five examples, and then they are solved by the Lingo 9.0 software to show that this model works well. This software is a comprehensive tool designed to make building and solving linear, nonlinear, and integer optimization models faster, easier, and more efficient. It provides a completely integrated package that includes a powerful language for expressing optimization models, a full featured environment for building and editing problems, and a set of fast built-in solvers. Objective functions $(f_i)$ have been normalized between zero and one. In other words, they have been without any dimension (i.e., scaleless). By using the following formula, these objectives are converted to a single objective function, where $f_{1}′$ and $f_{2}′$ are the normalized forms of $f_1$ and $f_2$ objective functions.

$min f = αf_{1}′+(1−α)f_2′$

To minimize deviations from the ideal, this function is reduced. As the first objective function ($f_1$) is more important than the second objective function ($f_2$) in the given problem, the coefficients of the above formula are considered in the form of $α = 0.7$ and $1 − α = 0.3$.

This problem is implemented by this software, and a global optimal solution is obtained. The computational results are shown in Tables 1, 2, 3, 4, and 5.

Table 1 $U_{j}^{n}$ is 1 if distribution center $j$ is opened with capacity level $n$. and 0 otherwise