Optimization of network redundancy and contingency planning in sustainable and resilient supply chain resource management under conditions of structural dynamics

Abstract

One of the key issues in supply chain sustainability is the efficient usage of the available resources. At the same time, proactive supply chain design with disruption risk considerations frequently leads to a network redundancy which implies some resource reservations in anticipation of possible disruptions. Even if resilient supply chain design has received much attention in literature, there is a research gap in designing both resilient and sustainable supply chains. This study contributes to closing the given gap by proposing a novel methodological approach to modelling network redundancy optimization. This allows for simultaneous computation of both optimal network redundancy and proactive contingency plans, considering both supply dynamics and structural disruption risks. The novelties of this study are the integration of sustainable resource utilization and SC resilience based on coordination of structure- and flow-oriented optimization. The model uncovers a practical approach to analyze and optimize supply chain redundancy by varying processing intensities of resource consumption in the network according to supply and structural dynamics. This makes it possible to explicitly include the dynamics of resource consumption for contingency plan realization in disruption scenarios.

Appendix 1: Input data for computational experiments

Notation	Meaning	Experimental value
Decision-maker preferences
α1	Priority coefficient for outbound shipment of the trans-shipped cargo to the customers	0.6
α2	Priority coefficient for the storage of the cargo flow ρ-гo at the node $i$	0.3
α3	Priority coefficient for the efficient resource consumption	0.1
The priorities of the sending of cargo flow
$\gamma_{1}$	Priority coefficient of the $\rho_{1}$ flow type 1 (low-tonnage) delivered at the customer	1
$\gamma_{2}$	Priority coefficient of the $\rho_{2}$ flow type 2 (wagon or carload) delivered at the customer	1
$\gamma_{3}$	Priority coefficient of the $\rho_{3}$ flow type 4 (group) delivered at the customer	1
Node characteristics
$Y_{i}$	Storage capacity at the nodes	16,000 tons of cargo
$E_{1}$	Budget at the node 1	3600 $
$E_{2}$	Budget at the node 2	3200 $
$E_{3}$	Budget at the node 3	3000 $
$E_{4}$	Budget at the node 4	3000 $
$E_{5}$	Budget at the node 5	3200 $
$E_{6}$	Budget at the node 6	3400 $
$E_{7}$	Budget at the node 7	3400 $
Structural dynamics of seaport operations
$T_{1}$	Duration of the 1st interval of structural constancy	60 h
$T_{2}$	Duration of the 2nd interval of structural constancy	120 h
$T_{3}$	Duration of the 3rd interval of structural constancy	180 h
$T_{4}$	Duration of the 4th interval of structural constancy	60 h
$T_{5}$	Duration of the 5th interval of structural constancy	90 h
$I_{121}$	Incoming flow of the cargo flow type 2 (i.e., carload) at the node 1 at k = 1	12,000 cargo units
$I_{611}$	Incoming flow of the cargo flow type 1 (i.e., low-tonnage) at the node 6 at k = 1	8000 cargo units
$I_{711}$	Incoming flow of the cargo flow type 1 (i.e., low-tonnage) at the node 7 at k = 1	5000 cargo units
$I_{522}$	Incoming flow of the cargo flow type 2 (i.e., carload) at the node 5 at k = 2	4000 cargo units
$I_{322}$	Incoming flow of the cargo flow type 2 (i.e., carload) at the node 3 at k = 2	10,000 cargo units
$I_{312}$	Incoming flow of the cargo flow type 1 at the node 3 at k = 2	6000 cargo units
$I_{412}$	Incoming flow of the cargo flow type 2 at the node 4 at k = 2	6000 cargo units
$I_{123}$	Incoming flow of the cargo flow type 2 at the node 1 at k = 3	15,000 cargo units
$I_{613}$	Incoming flow of the cargo flow type 1 at the node 6 at k = 3	6000 cargo units
$I_{413}$	Incoming flow of the cargo flow type 1 at the node 4 at k = 3	7000 cargo units
$I_{213}$	Incoming flow of the cargo flow type 1 at the node 2 at k = 3	3000 cargo units
$I_{114}$	Incoming flow of the cargo flow type 1 at the node 1 at k = 4	5000 cargo units
$I_{214}$	Incoming flow of the cargo flow type 1 at the node 2 at k = 4	7000 cargo units
$I_{225}$	Incoming flow of the cargo flow type 2 at the node 2 at k = 5	9000 cargo units
Operation parameters «LOW»
$\omega_{i11}$	Processing intensity for cargo flow type 1 trans-shipment	16, 7 cargo units/h
$\omega_{i21}$	Processing intensity for cargo flow type 2 trans-shipment	50 cargo units/h
$\omega_{ij\rho 1}$	Processing intensity for forwarding the cargo flow between the nodes i and j	16, 7 cargo units/h
$\omega_{{i\rho_{0} 1}}$	Trans-shipment costs	1 $/h
$\omega_{i41}$	Processing intensity for building the cargo flow of type 4 (group flow)	66, 7 cargo units/h
$\omega_{{ij\rho_{0} 1}}$	Cargo processing and forwarding costs between the nodes	3, 3 $/h
Operation parameters «HIGH»
$\omega_{i12}$	Processing intensity for cargo flow type 1 trans-shipment	50 cargo units/h
$\omega_{i21}$	Processing intensity for cargo flow type 2 trans-shipment	150 cargo units/h
$\omega_{ij\rho 2}$	Processing intensity for forwarding the cargo flow between the nodes i and j	200 cargo units/h
$\omega_{{i\rho_{0} 2}}$	Trans-shipment costs	15 $/h
$\omega_{i42}$	Processing intensity for building the cargo flow of type 4 (group flow)	50 cargo units/h
$\omega_{{ij\rho_{0} 2}}$	Cargo processing and forwarding costs between the nodes	50 $/h
Operation parameters «FLEX»
$\omega_{i13}$	Processing intensity for cargo flow type 1 trans-shipment	[16, 7, 50] cargo units/h
$\omega_{i23}$	Processing intensity for cargo flow type 2 trans-shipment	$\left[ {50,150} \right]$ cargo units/h
$\omega_{{i\rho_{0} 3}}$	Trans-shipment costs	$\left[ {1,15} \right]$ $/h
$\omega_{i43}$	Processing intensity for forwarding the cargo flow between the nodes i and j	[66, 7, 200] cargo units/h
$\omega_{ij\rho 3}$	Processing intensity for building the cargo flow of type 4 (group flow)	[16, 7, 50] cargo units/h
$\omega_{{ij\rho_{0} 3}}$	Cargo processing and forwarding costs between the nodes	[3, 50] $/h

1.1 Appendix 2: Decomposition procedure for large scale linear optimization problems with block-diagonal constraints

With regards to the large-scale optimization problems using the model (2)–(10), the block programming method has been used in this study (Dantzig and Wolfe 1960) based on the problem decomposition. An example of the block-diagonal view of the constraint matrix is shown in Fig. 9.

In Fig. 9, an example of block-diagonal view of the constraint matrix for the constraint system (2)–(10) is presented. The peculiarity of the block programming method is the use of a coordinating problem that comprises a lower number of rows and columns as compared to the initial problem. An explicit definition of all the columns is not required for coordinating problem solution. They are generated in the progress of simplex method computation. The initial solution is being iteratively improved using the method of iterative plan improvement with two-side constraints until the optimal solution $U_{{}}^{*}$ is found.

In general, the transformation of the constraint matrix from a general to block-diagonal structure with top edging is a task of very high combinatorial complexity. This complexity results first from the row permutation (i.e., top edging formation), and second from the column permutation (i.e., separation of non-zero blocks without common members). However, for the considered planning problem, it became possible to apply quite a simple approach to form the required blocks in the constraint matrix. The constraints which are related to different structural constancy intervals are blocked independently. The constraints which connect different structural constancy intervals (i.e., the matrix top edging) are grouped as a set of flow balance constraints (Eq. 3) that are using the variables from all the structural constancy intervals. The blocks can be aggregated. The analysis of aggregation impacts on the computational productivity is beyond the scope of this paper.