Flexibility evaluation of multiechelon supply chains.

Multiechelon supply chains are complex logistics systems that require flexibility and coordination at a tactical level to cope with environmental uncertainties in an efficient and effective manner. To cope with these challenges, mathematical programming models are developed to evaluate supply chain flexibility. However, under uncertainty, supply chain models become complex and the scope of flexibility analysis is generally reduced. This paper presents a unified approach that can evaluate the flexibility of a four-echelon supply chain via a robust stochastic programming model. The model simultaneously considers the plans of multiple business divisions such as marketing, logistics, manufacturing, and procurement, whose goals are often conflicting. A numerical example with deterministic parameters is presented to introduce the analysis, and then, the model stochastic parameters are considered to evaluate flexibility. The results of the analysis on supply, manufacturing, and distribution flexibility are presented. Tradeoff analysis of demand variability and service levels is also carried out. The proposed approach facilitates the adoption of different management styles, thus improving supply chain resilience. The model can be extended to contexts pertaining to supply chain disruptions; for example, the model can be used to explore operation strategies when subtle events disrupt supply, manufacturing, or distribution.

Introduction

Supply chains (SCs) are required to handle environmental uncertainties. Therefore, it is important to develop strategies to improve the flexibility and resilience of SCs without compromising their operation efficiency and effectiveness [1]. In a generic system, suppliers transport raw materials in multiple lots to factories. Manufacturing processes convert these materials into finished products. Periodically, these products need to be transported to other factories, to distribution centers, or directly to customers. SCs that decide to maintain a high level of finished goods inventories combined with fast shipping fleets increase their chances of meeting customer demand on time, thus improving their service level and revenue. However, SCs also need to maintain low inventory levels and choose economic modes of transport to minimize production, inventory, and logistics costs. The allocation of production capacity, inventories, and transportation simultaneously to satisfy demands can be very challenging. Nevertheless, to develop a tactical SC plan, these interactions must be considered.

In uncertain environments, an important requirement for SC planning models is the ability to cope with flexibility. However, in many situations, the parameters of deterministic models are not known completely. In such cases, sensitivity analysis combined with parametric optimization is commonly adopted. However, this strategy, known as parametric linear programming, is hardly relevant to optimization under uncertainty. This strategy makes predictions only when a model faces certainty scenarios [2], [3]. Therefore, during optimization, the proper approach to handle uncertainty is to use stochastic models. However, when the number of scenarios increases, stochastic SC planning models become complex and difficult to solve. Flexibility analysis often considers only one or two echelons of SCs.

The present study is aimed at investigating a four-echelon SC tactical planning model by robust stochastic programming. To the best of our knowledge, the adopted approach has not yet been explored in previous studies. This study intends to bridge this gap. The proposed model combines robust optimization and stochastic programming features to evaluate SC flexibility and resilience, considering scenarios with stochastic parameters and adjustable levels of demand variability.

The rest of the paper is organized as follows: In the next section, the relevant literature on SC planning under uncertainty is revisited. Then, the SC problem is presented in terms of a numerical example with deterministic parameters, and the proposed formulation for the stochastic problem is explained. In the section following this, a computational study is presented. SC flexibility was evaluated in three dimensions: supply, production, and distribution. The model also enabled the analysis that considers the tradeoff between demand variability and the service level. Finally, the conclusions and directions for future research are presented.

Literature review

The assumption that a system will operate in a stable environment is not realistic. Customers can change their needs, suppliers can offer discounts on products, and regulations can impact transportation costs. To achieve SC flexibility, uncertainty must be considered in systems and models [4]. The idea of incorporating uncertainty in mathematical programming models was pioneered by Dantzig [5]. Since then, the understanding of uncertainty via stochastic programming has progressed [6], [7]. Particularly, SC planning models have been reviewed by some researchers, e.g., Birge [8], Mula et al. [9], and Sodhi and Tang [10]. Uncertainty can also be modeled by robust optimization.

Through robust optimization, a set of computationally tractable uncertainty scenarios can be selected and evaluated simultaneously; this approach was initially proposed by Soyster [11] and Falk [12]. Recent advances in robust optimization have been presented by Gabriel et al. [13]. The approach is an alternative to the use of sensitivity analysis because the latter is a reactive post-optimality study that cannot assess the impact of data uncertainty. In robust optimization, the knowledge of probability distribution parameters is not previously assumed. Its use is motivated by the tradeoff between the value of the objective function and the risk of infeasibility in cases where sufficient reliable data are not available to elaborate decision scenarios [14], [15], [16]. It was developed to ensure problem feasibility given a set of realizations of uncertainty. However, the model can become very conservative, in which case, one can previously set the uncertainty budgets [17].

For SC planning under uncertainty, stochastic programming and robust optimization have been successfully adopted. Stochastic programming formulations are frequently used because they usually yield good expected performance estimates [18]. On the other hand, robust optimization can favor SC planning systems that need to handle parameter uncertainties [19], [20]. Some applications include reverse logistics [21] and manufacturing under the build-to-order strategy [22].

Recent studies on SC flexibility include: the use of decomposition approaches for solving large-scale models to improve SC resilience [23]; the study of cycle and delivery times for SC systems considering failure in rework [24]; multiobjective simulation-based optimization to evaluate supplier flexibility and safety stock levels [25]; the development of closed-loop SC models for evaluating uncertainty on demand and returns for tire remanufacturing [26], and the evaluation of corporate social responsibility [27].

A recent review on SC flexibility identified the use of quantitative models as a new research direction for achieving tradeoff between performance measures [28]. To the best of our knowledge, no work has evaluated four-echelon SC flexibility by simultaneously considering demand variability and service levels for supply, production, and distribution scenarios. To address this challenge, we develop a hybrid model that combines stochastic programming and robust optimization for tactical policies. The model also includes a bill of materials for a generic product structure and one sublevel. The supply–production–distribution model is capacitated, multiplant, multiproduct, multiperiod, and multimodal. Table 1 presents the recent and relevant works that contribute to the proposed integrated approach of evaluating the flexibility of the tactical supply chain planning problem using two-stage stochastic programming and robust optimization.

Table 1

Recent and relevant works that contribute to the proposed approach for supply chain planning.

Author(s)	Year	Tactical level	Uncertainty	2SSP1	RO2	4ESC3 Flexibility
[29]	2000			✓
[18]	2003		✓	✓
[16]	2004	✓	✓	✓
[20]	2005	✓	✓		✓
[9]	2006	✓	✓	✓
[10]	2009	✓	✓
[21]	2011				✓	✓
[22]	2013	✓	✓		✓
[17]	2013		✓	✓	✓
[13]	2014				✓
[23]	2014			✓		✓
[1]	2015		✓			✓
[28]	2016	✓	✓			✓
[24]	2016	✓	✓
[26]	2017		✓			✓
[25]	2017	✓	✓

1Two-Stage Stochastic Programming.

2Robust Optimization.

3Four-Echelon SC.

Problem definition and modeling

The proposed multiechelon supply chain model considers a two-stage stochastic programming formulation [6] with robust optimization elements for the objective function and constraints [14]. For readability we consider the following nomenclature: MESC-2SSP-RO. Probability distributions represent the price ranges and demand values for uncertainty modeling. The goal is to find an optimal policy that maximizes the expected profit. The stochastic model is risk neutral and insensitive to results that are far from the expected solution. The objective function assumes the generic nonlinear form according to Eq (1): where f is the variance of second-stage costs and α is a scalar used by the decision maker to determine nonnegative risk tolerance. Larger values of α produce solutions that reduce the variance, whereas smaller values of α increase the expected profit. The exploitation of this formulation gives quadratic terms. We linearize the mean absolute deviation of the objective function [19]. The new formulation on Eq (2) and constraints (3) and (4) reduces the computational effort by using half the variables used in the generic Eq (1): where θ represents the average deviation violation of scenario s from the X scenarios. λ is a weight that measures the tradeoff between risk and the expected value. In Eq (2) and constraints (3), if ξ − ∑ ρ ξ ≥ 0, then θ = 0 under the optimal plan and Ψ = ∑ ρ ξ + λ∑ ρ(ξ − ∑ ρ ξ). Else, if ξ − ∑ ρ ξ ≤ 0, then θ = ∑ ρ ξ − ξ under the optimal plan and Ψ = ∑ ρ ξ + λ∑ ρ(∑ ρ ξ − ξ). The second and third terms of Eq (2) refer to the solution and model robustness, respectively. When solution robustness is ensured, the solution is close to optimality under any scenario, whereas when model robustness is ensured, the solution satisfies the demand under any scenario. δ represents a negative deviation from the demand, that is, a situation in which customers face a lack of products, whereas ω represents a weight of penalty over δ for the tradeoff between model robustness and solution robustness.

Notation and model formulation

Consider a SC where is the set of products consisting of raw materials and finished products (i.e. ). Let be a location in a network consisting of suppliers, industrial plants, distribution hubs, and customers; thus, . In this four-echelon SC, suppliers provide raw materials to industrial plants. These plants process raw materials on resources, thus producing finished products over periods to meet the demands of customers. The sets of the products, locations, resources, and periods are indexed by p, l, r and t, respectively. Fig 1 shows a schematic of the SC.

Fig 1

Four-echelon SC.

The supply–production–distribution model is capacitated, multiplant, multiproduct, multiperiod, and multimodal.

We list the deterministic and stochastic parameters in Table 2 and the first and second-stage variables in Table 3. The elements of the objective function (13) are described in Table 4.

Table 2

MESC-2SSP-RO model parameters.

Deterministic parameters
TlrpR	∈{0, 1}: Technical route of product p on resource r at location l
Bp′p	Bill of materials p′ required to produce a unit of product p
MlrpC	Unit time required to produce product p on resource r at location l
ElrtF	Resource efficiency r at location l in period t
AtH	Available hours in each period t
AlrtX	Extra hours available on resource r at location l in period t
LlpM	Lot size of product p at location l
PlrtM	Hours of preventive maintenance required for resource r at location l in period t
SlptS	Safety stock of product p at location l in period t
SlptX	Stock capacity of product p at location l in period t
AlptR	Availability of raw materials for product p at location l in period t
Slp0	Initial inventory of product p at location l
Ylr	Raw material yield on resource r at location l
NlrM	Number of resources of type r at location l
Tmll^′CX	Transportation capacity of raw-material on modal m from location l to l′
Tmll^′CY	Transportation capacity of finished product on modal m from location l to l′
ChtI	Inbound handling capacity at location l in period t
ChtO	Outbound handling capacity on location l and time period t
AlrtV	(AtHNlrM-PlrtM)(ElrtFY_lr) Available hours on resource r at location l in period t
Stochastic parameters
Dcpts	Demand of customer c for product p in period t and scenario s
Rps	Sales revenue of finished product p in scenario s
Nps	Fictitious cost penalty for not meeting demand p in scenario s
ClrsF	Fixed cost of resource r at location l in scenario s
ClpsV	Variable cost of production of p at location l in scenario s
ClrsX	Extra capacity cost on resource r at location l in scenario s
ClpsS	Unit inventory cost of product p at location l in scenario s
Cmll^′sL	Unit transport cost on modal m from location l to l′ in scenario s
ClpsP	Unit procurement cost of raw material x at location l in scenario s
TlpsX	Tax over finished product y sold to customer c in scenario s
ρs	Probability of each scenario s, ∑_s∈Sρ_s=1
λ	Weight for measuring tradeoff between risk and expected value
ω	Penalty for measuring tradeoff between solution and model robustness

Table 3

MESC-2SSP-RO model decision variables.

First-stage decision variables
αlp	∈Z^+: Production of product p at location l in first period
alrp	Production of product p on resource r at location l in first period
blp	Consumption of raw material x at location l in first period
slpt	Stock of product p at location l at the end of first period
dlp	Met demand of product p at location l in first period
nlp1s	Nonsatisfied demand of product p at location l in first period in scenario s
rlp	∈Z^+: Procurement of lots of raw material x at location l in first period
tmll′p	Quantity of product p transported on modal m from l to l′
clr	Consumption of resource r at location l in first period
clr′	Overtime percentage for resource r at location l
ylr	∈ {0, 1}: Decision of activate (or not activate) resource r at location l in first period
Second-stage decision variables
αlpts	∈Z^+: Production of product p at location l in period t and scenario s
alrpts	Production of product p on resource r at location l in period t and scenario s
blpts	Consumption of raw material x at location l in period t and scenario s
slpts	Stock of product p at location l at end of period t in scenario s
dlpts	Met demand of product p at location l in period t and scenario s
nlpts	Nonsatisfied demand of product p at location l in period t and scenario s
rlpts	∈Z^+: Procurement of lots of raw material x at location l in period t and scenario s
tmll′pts	Quantity of product p transported on modal m from l to l′ in period t and scenario s
clrts	Consumption of resource r at location l in period t and scenario s
clrts′	Overtime percentage for resource r at location l in period t and scenario s
ylrts	∈ {0, 1}: Decision to activate (or not activate) resource r at location l in period t and scenario s
θ1s	Deviation of mean violation in first-stage scenario s
θ2s	Deviation of mean violation in second-stage scenario s

Table 4

Objective function elements of four-echelon SC planning model.

Elementsa	Description
R1P	Revenue after taxes from sales in first stage
C1L	Logistics cost from origin to destination on different transport modes in first stage
C1F	Fixed cost for machine activation in each plant in first stage
C1V	Finished product cost in each plant in first stage
C1P	Procurement cost of raw material in first stage
C1S	Storage cost in each plant in first stage
C1X	Capacity expansion cost of resources in each plant in first stage
C1N	Nondelivery cost for each customer in first stage
R2P	Revenue after taxes from sales in second stage
C2L	Logistics cost from origin to destination on different transport modes in second stage
C2F	Fixed cost for machine activation in each plant in second stage
C2V	Finished product cost in each plant in second stage
C2P	Procurement cost of raw material in second stage
C2S	Storage cost in each plant in second stage
C2X	Capacity expansion cost of resources in each plant in second stage
C2N	Nondelivery cost for each customer in second stage

aElements subscript 1 and 2 represents first stage[1] and second stage[2].

Where: , , , , , , , , , , , , , , and .

The robust stochastic programming model is described as follows:

Objective function of MESC-2SSP-RO on Eqs (5)–(10) maximizes the expected profit. It follows the robust two-stage stochastic programming formulation presented in Eq (2), where ρ is the occurrence probability of each scenario s; , λ is a weight used to measure the tradeoff between risk and the expected value; and ω is a penalty used to measure the tradeoff between the solution and model robustness. The objective function is obtained as the difference between the after-tax revenue and the costs of procurement, production, inventory, and transportation. The costs consist of fixed and variable parts. Fixed costs are incurred on resource activation during certain periods. Variable costs are incurred on different levels of procurement, production, extra capacity use, inventory, logistics, and delivery.

The objective function of MESC-2SSP-RO is subjected to constraints in the first (t = 1) and second () stages. Constraint (11) expresses the initial stocks of raw materials and finished products that are considered to exist in industrial plants and distribution centers. Constraints (12) and (13) describe the stored volumes of raw materials and finished products that must consider the inventory safety levels and must not exceed the storage capacity limits of these locations. Constraints (14) and (15) indicate that the purchase of lots of raw materials or finished products must consider their availability with suppliers in each period.

Flow balance constraints (16)–(28) integrate the different parts of the problem. The end of each period is connected by the sum of the input and output flows; therefore, the transportation of products is not allowed if the product does not reach the destination within the planned horizon. The input and output flows are considered for each location, product, and period. The input flow is represented by the transportation of raw materials or finished products from the previous SC echelon, the production of finished products, the inventory level, and the procurement of raw materials or finished products at the end of the previous period. The output flow is the result of the balance of transportation of items to the next SC echelon, the met demand, the inventory level, and the consumption of raw materials in the production process at the end of period.

Constraints (29)–(32) represent the inbound and outbound handling capacities at the distribution centers for each period. The production in each process depends on the route and production time of each item. Eqs (33) and (34) describe the production capacity use. The maintenance of a process activated during low demand times may incur unnecessary fixed costs of operation. Temporary process stoppages are tactical decisions that can lead to a reduction in operation costs because teams would then be reallocated to alternate company sites for training and supporting activities.

Constraints (35) and (36) express the capacity of a process, which is measured by the total available production time. In this time period, a process may or may not be activated. If activated, its capacity can be reduced by performing scheduled preventive maintenance, controlling operational efficiency, or/and controlling raw material yield.

Constraints (37) and (38) indicate that the decision to use overtime can be a profitable alternative. The use of extra capacity results in extra costs, which are included in the objective function. However, the value of the extra costs is limited by the company. These constraints also guarantee that extra capacity can be activated only if there is a need for production in the period.

Constraints (39) and (40) ensure that a finished product is produced in the latest machine of the technical route in each manufacturing plant. Constraints (41) and (42) represent the bill of materials for a generic product structure [30]; therefore, a finished product is a result of the combination of raw materials in different proportions. Constraints (43)–(46) guarantee that the product flow does not exceed the transportation capacity for each transport mode. Constraints (47) and (48) indicate that eventually, some part of the original demand is not met. Constraints (49)–(60) characterize the domain of the variables.

As shown in constraint (3), the difference between the total average cost and the total cost of scenarios should be nonnegative and equivalent to the deviation for violating the average. These conditions are maintained by constraints (61)–(64). Constraints (61) and (63) determine these conditions for the first stage, whereas constraints (62) and (64) determine these conditions for the second stage of the robust two-stage stochastic program. These constraints ensure that distinct and nonnegative objective functions are generated by different scenarios.

Numerical example

A numerical example is designed to demonstrate the scope of the proposed formulation. For didactic purposes, without loss of generality, we consider one scenario wherein parameters lambda and omega (penalty) equal zero, leading to a deterministic program. Parameters are kept simple so that the resulting plans, i.e., the purchasing, production, material use, storage, and transport levels, in each period can be intuitive. Although the model is simplified, it defines the optimal flow that maximizes the operating profit considering both production and logistics constraints throughout the four-echelon SC. The elements of this chain are listed in Table 5.

Table 5

Sets of numerical example.

Elements of numerical example
Suppliers	F1, F2
Industrial Plants	I1, I2
Hubs of Distribution	H1, H2
Modal of Transport	M1, M2
Raw-materials	X1, X2
Finished products	Y1, Y2
Machines of Plant 1	MA, MB
Machines of Plant 2	MC, MD

Fig 2 illustrates a small SC that plans its operations for the following two months. Two industrial plants sell two types of finished products to two customers. Each customer demands 10 units of each product in each month, i.e., D = 10, leading to a total demand of 80 units throughout the period. Products can be sent by two types of transport modes from industrial plants to two distribution centers and from there to the customers. The distribution hubs can handle the two products for dispatch or for stock, considering the safety stock, handling, and storage capacity limits.

Fig 2

Schematic of numerical example SC.

Supply, production, and distribution plan for two periods.

In the industrial plants, machines process the raw materials into finished products following a technology roadmap. Raw materials are processed by machines M and then M in plant I1 and by machines M and M in plant I2. The bill of materials considers a generic product structure and sets the raw materials used by each finished product, as illustrated in Fig 3. The industrial plants buy raw materials in multiple lots of 10 units from two suppliers. Finished products can be purchased only from supplier F1. Either of the two transport modes can be used to transport these items. Table 6 lists the availability of raw materials and finished products with suppliers.

Fig 3

Generic product structure of the bill of materials.

A generic product structure is adopted to set the raw materials used by finished products Y1 and Y2.

Table 6

Monthly availability of raw materials and finished products.

Quantity	X1	X2	Y1	Y2
F1	50	10	10	10
F2	10	50	–	–

The initial stock in each plant is 100 units of each raw material and 5 units of each finished product. In both the distribution hubs, the initial stock is 5 units of each finished product. The safety stock of raw materials and finished products is 10 units for both the industrial plants and distribution hubs. Each machine has a capacity of 50 h and an extra capacity of 10 h, that is, = 10. The production of finished product Y1 must occur in multiple lots of 5 units. There are no restrictions on the multiple lots for the production of finished product Y2. The transportation capacity per month is 10 units on transport mode M1 and 20 units on transport mode M2. The input and output material handling capacity is 50 units per month. Each finished product is sold at $100.00 to customers. A 5% tax is deducted from the sales revenue. The fixed cost of production for each machine is $500.00. Table 7 lists the variable production costs, whereas Table 8 lists the purchase costs of the raw materials or the finished product. The overtime cost for each machine is $100.00. The shipping cost using either of the transport modes is $10.00/(km⋅unit). For didactic purposes, without loss of generality, parameters , , , , Y and take the value of 1 for any set combination. No preventive maintenance is planned for the considered period, i.e., .

Table 7

Production variable cost on each industrial plant.

Raw material/Product	Y1	Y2
I1	20	20
I2	10	20

Table 8

Procurement costs of raw material or finished products.

Procurement	X1	X2	Y1	Y2
F1	1	1	40	50
F2	1	1	–	–

The SC plan is presented in the following tables. The financial report and model statistics are given in Table 9. The entire demand is met, as listed in Table 10. From the gross sales revenue of $8,000.00, 5% tax is discounted, i.e., $400.00. In the production process, overtime is not necessary. In the optimal SC plan, production processes are partially activated. Table 11 indicates that production is complemented by the decision of buying finished products from a supplier.

Table 9

Finance performance and statistics of numerical example.

Finance Report	Value ($)
Gross revenues	8000.00
Net revenues	7600.00
Logistic cost	2000.00
Opportunity cost	0.00
Production fixed cost	2000.00
Production variable cost	940.00
Procurement cost	1002.00
Overtime cost	0.00
Inventory cost	80.00
Operating cost	1578.00
Model Statistics	Value (unit.)
Equations	687
Variables	606
Integer variables	128
Binary variables	8
Computational run time	0.1 s

Table 10

Demand meeting plan.

Month	Customer	Product	Quantity
1	C1	Y1	10
		Y2	10
	C2	Y1	10
		Y2	10
2	C1	Y1	10
		Y2	10
	C2	Y1	10
		Y2	10

Table 11

Procurement plan.

Month	Supplier	Item	Quantity
1	F1	X1	20
		Y2	10
2	F1	Y2	10

The purchasing plan determines the optimal flow of raw materials and finished products from suppliers to industrial plants considering the transportation capacity of each transport mode. Table 12 indicates that the optimal plan is to purchase 10 units of finished product Y2 in each month. However, this plan can be realized only with supplier F1, given the unavailability of these products with supplier F2. Furthermore, these products are transported only by transport mode M2 because the capacity of transport mode M1 is completely exhausted for the transportation of raw materials.

Table 12

Capacitated supply chain transportation plan.

Month	Modal	Origin	Destiny	Item	Quantity	Modal utilization
1	M1	F1	I1	X1	10	100%
			I2	X1	10	100%
		I1	H1	Y2	10	100%
			H2	Y1	5	50%
		I2	H2	Y1	10	100%
		H1	C1	Y1	10	100%
		H2	C2	Y1	10	100%
	M2	F1	I1	Y2	5	25%
			I2	Y2	5	25%
		I1	H1	Y2	15	75%
			H2	Y2	15	75%
		I2	H1	Y1	15	75%
			H2	Y1	20	100%
		H1	C1	Y2	10	50%
			C2	Y2	10	50%
2	M1	I2	H1	Y2	10	100%
		H2	C1	Y1	10	100%
			C2	Y1	10	100%
	M2	F1	I2	Y2	10	50%
		H1	C1	Y2	10	50%
		H2	C2	Y2	10	50%

Table 13 indicates that the inventories of the raw materials and the finished products meet the storage capacity and safety stock limits in the industrial plants and the distribution hubs. Raw material X2 has a stock of 50 units because according to the bill of materials, its use is half that of raw material; therefore, in the production process, less units of X2 are consumed.

Table 13

Planned inventory level on supply chain echelons.

Month	Location	Item	Quantity
1	I1	X1	10
		X2	50
		Y1	10
		Y2	10
	I2	X1	10
		X2	50
		Y1	10
		Y2	10
	H1	Y1	10
		Y2	10
	H2	Y1	30
		Y2	20
2	I1	X1	10
		X2	50
		Y1	10
		Y2	10
	I2	X1	10
		X2	50
		Y1	10
		Y2	10
	H1	Y1	10
		Y2	10
	H2	Y1	10
		Y2	10

Table 12 presents the optimal transportation plan for each transport mode along the SC, considering the initial stocks, safety stocks, storage, and handling capacity of the SC echelons. Each transport mode can carry more than one type of product in each period. In this optimal plan, both the transport modes are used. Transport mode M1 is often engaged to its limit. Its capacity is half that of transport mode M2.

The optimal product delivery plan for each customer is listed in Table 10. However, information such as (i) the finished product flow from the distribution hubs and (ii) the quantity of products transported by each transport mode is complementary (Table 12). The inbound and outbound logistics in each hub are limited by the handling capacity of 50 units per month for each hub. Table 14 presents the dynamic use of handling resources.

Table 14

Input and output amount on hubs.

Month	Hub	Quantity	Utilization
1	H1	40	80%
	H2	50	100%
2	H1	10	20%
Input amount on distribution hubs
Month	Hub	Quantity	Utilization
1	H1	30	60%
	H2	10	20%
2	H1	10	20%
	H2	30	60%
Output amount on distribution hubs

Table 15 presents the optimal use of raw materials in an industrial plant and the quantity of the finished product obtained. Recalling the bill of materials of Fig 3, in plant I1, the production of 10 units of Y1 uses 20 units of X1 and 10 units of X2, whereas the production of 40 units of Y2 uses 80 units of X1 and 40 units of X2. Thus, the number of units of X1 used in plant I1 is 100 (20 + 80), whereas the number of units of X2 used in the same plant is 50 (10 + 40). In plant I2, the production of 50 units of Y1 uses 100 units of X1 and 50 units of X2.

Table 15

Raw materials consumption and production on plants.

Month	Plant	Raw material	Quantity
1	I1	X1	100
		X2	50
	I2	X1	100
		X2	50
Raw material consumption and production
Month	Plant	Product	Quantity
1	I1	Y1	10
		Y2	40
	I2	Y1	50
Production on industrial plants

Table 16 presents the production plan for each machine. To obtain the optimal solution, production is not activated in the second month. The high fixed costs of production contribute to the decision to force resource activation in the first month only. In addition, inventory accumulation in this period compensates for the resource deactivation in the second month. Each industrial plant makes 50 finished products, so, a total of 100 units are produced. Of these, 80 units are used to meet the demand and 20 units are allocated to the safety stock supply: 10 units for each plant (5 of Y1 and 5 of Y2).

Table 16

Production plan on industrial plants.

Month	Plant	Resource	Activated?	Production
1	I1	MA	Yes	50
		MB	Yes	50
	I2	MC	Yes	50
		MD	Yes	50
2	I1	MA	No	0
		MB	No	0
	I2	MC	No	0
		MD	No	0
Production plan by resource
Month	Plant	Resource	Product	Quantity
1	I1	MA	Y1	10
		MA	Y2	40
		MB	Y1	10
		MB	Y2	40
	I2	MC	Y1	50
		MD	Y1	50
Detailed production plan by resource

Although the procurement cost of Y1 is less than that of Y2 (Table 8), the variable production cost is lower in plant I2 (Table 7). Consequently, the entire available capacity in plant I2 is used to produce Y1. We realize that the activation costs of production resources are considerably higher than storage costs, so production is enabled only during the first month and is carried out at maximum capacity. Then, the finished products are transported to the distribution hubs where they remain in stock until the demand is met in the following month.

In comparison to traditional approaches, the proposed model present advantages on demonstrating quantitatively the impact of changes on each element on all echelon of SC, providing a holistic view to the managers responsible for SC tactical planning. The SC elements are, for example, the purchasing activity; the material used on production; the limited production or transportation capacity; or the demand in each period. For the robust-stochastic version, multiple scenarios can be simultaneously evaluated empowering decision-makers to select the tactical SC plan that best fits the global SC metrics, as presented in the results and discussion’s section.

This section proposed a numerical example to demonstrate the scope of the MESC-2SSP-RO formulation. The SC problem was based on two elements for each component for didactic purposes, however, solving the complete MILP problem with numerous elements is a challenge. The problem, namely dynamic lot-sizing, has been recognized to be NP-hard [31] since multiple product lots have to be planned within a capacity that varies over time.

Results and discussion

The flexibility evaluation of the four-echelon SC is based on a three-dimensional framework, presented by Esmaeilikia et al. [28], and the tradeoff analysis of demand variability and service levels. The evaluation facilitates the investigation of the flexibility criteria for supply, manufacturing, and logistics. Supply flexibility includes make-or-buy and sourcing decisions. Manufacturing flexibility includes the production of multiple product types in each plant on machines and the expansion of production capacity. Logistics flexibility includes multimodal, transportation, handling, and storage strategies throughout the network. The literature on stochastic SC models that incorporate three-dimensional flexibility options is limited [28].

We considered a baseline stochastic example to evaluate three equiprobable scenarios [29] over 12 periods. The problem involves determining the optimal tactical plan for an SC with 3 suppliers, 2 industrial plants, 2 distribution hubs, and 10 customers. Four types of raw materials are processed in 5 machines in each plant, producing 10 types of finished products. Although this is a small problem, the stochastic four-echelon SC problem yields a big model; therefore, we did not describe the results in detail as we did in the previous deterministic numerical example. The model has 39,103 variables and 13,873 constraints with 9,652 continuous variables and 892 integer variables, of which 69 are binary. For evaluating flexibility by the comparison of scenarios, we present a report with only the aggregated values of financial and operational results (Table 17). Demand and price are the stochastic parameters. Demand is uniformly distributed with a minimum of 6 units and a maximum of 10 units, whereas the price is normally distributed with a mean of 100 units and a standard deviation of 10 units. The procurement cost of the raw material is $85.00/unit; 20 units are available with the suppliers. The fixed and variable costs for each plant are $500.00/(machine⋅period) and $20.00/unit, respectively. The overtime costs are $875.00/(machine⋅period), and the transportation costs are $2.50/(km⋅unit) for both the transport modes. For each period and plant, the safety stocks are set to 10 and 5 units for the raw materials and the finished products, respectively, and the safety stocks of finished products are set to 2 units in the distribution hubs. All the other parameters are kept constant in the following experiments.

Table 17

Performance of the baseline test-problem for flexibility evaluation.

Financial report	Value ($)	Operational report	Value (unit.)
Sales revenues	797,240.95	Raw material procurement	180,768
Logistics cost	190,400.00	Finished product procurement	0
Production fixed-cost	60,000.00	Inventory on supply chain	9,901
Production variable-cost	152,440.00	Production on plant-[1]	11,433
Procurement cost	30,128.00	Production on plant-[2]	11,433
Overtime cost	0.00	Transport on modal-[1]	172,746
Inventory cost	3,300.33	Transport on modal-[2]	55,734
Expected overall profit	360,972.62
Profit scenario-[1]	361,959.75	Total demand	30,703
Profit scenario-[2]	374,603.85	Met demand	24,246
Profit scenario-[3]	346,354.25	Nonsatisfied demand	6,457

Under the above settings, the total demand is partially met. The industrial plants buy only raw materials from suppliers. Although 600 (20 × 10 × 3) finished products are available with suppliers in every period, the procurement cost of these products is not compensatory. The maximum capacity of manufacturing is used in the industrial plants, although extra capacity is not used. The items transported are assigned to both modal-[1] and modal-[2]. At the same costs, there is no preferable shipping mode between the two transport modes.

Stock-out situations may be affected by production lead times leading to non-satisfied demand. Therefore we conducted an additional experiment for the baseline test-problem changing the unit time required to produce product p on resource r on industrial plants, represented by the parameter , to evaluate these joint effects. The reduction of up to 20% of production lead time reduced stock-out units form 6,457 to 3,718, while the increase in production lead time have unfavorable impacts on stock-out as presented in Table 18.

Table 18

Joint effect of stock-out and production lead time for baseline test-problem.

Leadtime	Delivery	Stock-out
-50%	26,985.00	3,718.00
-40%	26,985.00	3,718.00
-30%	26,985.00	3,718.00
-20%	26,985.00	3,718.00
-10%	26,476.60	4,226.40
Base	24,246.00	6,457.00
10%	22,076.30	8,626.70
20%	20,392.80	10,310.20
30%	18,870.20	11,832.80
40%	17,360.00	13,343.00
50%	16,042.00	14,661.00

Supply flexibility assesses strategies to choose suppliers based on the price of raw materials or their availability. It also evaluates the strategy to choose between making the finished product and buying it directly from a supplier. In this experiment, supply flexibility occurs when the company achieves a reduction in the procurement price of a finished product from $85.00/unit to $45.00/unit from one supplier. The results listed in Table 19 show that at this cost, the company can increase its profit by 16.74% by purchasing all the 600 finished products over 12 periods i.e. 7,200 units from suppliers, thereby increasing the overall service level and revenue. The costs of production and raw material procurement are reduced. Buying instead of producing finished products increases the overall profit by reducing both the consumption of manufacturing resources and the need to purchase raw materials.

Table 19

Performance of the test-problem with flexible supply.

Financial report	Value ($)	Operational report	Value (unit.)
Sales revenues	964,588.52	Raw material procurement	172,568
Logistics cost	199,858.33	Finished product procurement	7,200
Production fixed-cost	57,323.33	Inventory on supply chain	10,944
Production variable-cost	145,606.67	Production on plant-[1]	10,974
Procurement cost	136,761.33	Production on plant-[2]	10,867
Overtime cost	0.00	Transport on modal-[1]	186,799
Inventory cost	3,648.00	Transport on modal-[2]	53,031
Expected overall profit	421,390.85
Profit scenario-[1]	418,951.60	Total demand	30,703
Profit scenario-[2]	442,929.84	Met demand	30,421
Profit scenario-[3]	402,291.12	Nonsatisfied demand	282

Manufacturing flexibility assesses volume flexibility and operational decisions. In this case, the tradeoffs between extra production costs and nonsatisfied demand costs are evaluated. We evaluate the manufacturing flexibility by reducing plant-[2]’s fixed costs of production by $100.00/(machine⋅period), reducing overtime costs by $250.00/(machine⋅period), and increasing the safety stock of raw materials from 10 units to 20 units in each industrial plant. Table 20 presents the financial and operational results. In this experiment, the average profit increases by 3.96% over the baseline scenario because of the reduction in operation costs and the increment on sales. The latter is attributed to the increase in production, which exceeds the available capacity. Thus, overtime is used mainly in plant-[2], which has a lower fixed cost of operation compared to plant-[1]. We also note that the variable cost of production increases and overtime is activated. On overtime activation, the consumption of raw materials increases. This change is operational, so there is no need to purchase finished goods from the supplier. The production in factories increases, so the quantity of inventory and goods transported throughout the chain also increases.

Table 20

Performance of the test-problem with flexible production volume.

Financial report	Value ($)	Operational report	Value (unit.)
Sales revenues	877,510.25	Raw material procurement	203,476
Logistics cost	214,004.17	Finished product procurement	0
Production fixed-cost	54,000.00	Inventory on supply chain	17,306
Production variable-cost	171,163.33	Production on plant-[1]	12,618
Procurement cost	33,912.67	Production on plant-[2]	13,056
Overtime cost	23,404.17	Transport on modal-[1]	178,953
Inventory cost	5,768.83	Transport on modal-[2]	77,852
Expected overall profit	375,257.08
Profit scenario-[1]	376,319.75	Total demand	30,703
Profit scenario-[2]	392,915.10	Met demand	27,054
Profit scenario-[3]	356,536.40	Nonsatisfied demand	3,648

Logistics flexibility assesses the ability to adopt alternative strategies for transport and storage to meet the customer demand on time. In this experiment, the logistics flexibility is evaluated by reducing the distribution cost for modal-[2] from $2.50/(km⋅unit) to $2.40/(km⋅unit). Additionally, we change the inventory safety level of finished products from 2 units to 5 units in distribution centers. The financial and operational results are presented in Table 21. In previous experiments, products are allocated to a transport mode by considering only its capacity because there is no difference in the transportation costs. In this case, there is neither purchase of products nor overtime activation. Revenue, production costs, and purchases are not altered compared to the baseline scenario. Nevertheless, the transportation of units is concentrated on modal-[2] because of its lower cost. The increase in the total expected profit is a consequence of the reduced distribution costs. However, a part of this profit is consumed by the cost of maintaining higher safety stocks of finished products in distribution centers.

Table 21

Performance of the test-problem with flexible logistics.

Financial report	Value ($)	Operational report	Value (unit.)
Sales revenues	792,132.80	Raw material procurement	180,768
Logistics cost	184,345.60	Finished product procurement	0
Production fixed-cost	60,000.00	Inventory on supply chain	11,727
Production variable-cost	152,440.00	Production on plant-[1]	11,433
Procurement cost	30,128.00	Production on plant-[2]	11,433
Overtime cost	0.00	Transport on modal-[1]	51,168
Inventory cost	3,909.00	Transport on modal-[2]	177,132
Expected overall profit	361,310.20
Profit scenario-[1]	362,284.15	Total demand	30,703
Profit scenario-[2]	374,685.80	Met demand	24,066
Profit scenario-[3]	346,960.65	Nonsatisfied demand	6,637

We perform additional experiments to evaluate the solution robustness and the model robustness. The quality of flexibility and tradeoff analysis is enhanced by setting the parameters of demand variability, λ, and the penalty for nonsatisfied demand, ω. Figs 4 and 5 shows that as the penalty for nonsatisfied demand increases, the total expected profit is reduced and the unmet demand is also reduced. Figs 6 and 7 shows that under great demand variability, the nonsatisfied demand increases; however, the global expected profit of the SC can be enhanced by selecting profitable orders and increasing the inventory level over SC echelons. Such a strategy increases the availability of products to customers. The rise in the inventory costs over SC echelons is compensated by more sales.

Fig 4

Service level with the increase of penalty ω.

Nonsatisfied demand reduces due to the increase of penalty ω: Model robustness.

Fig 5

Overall profit by increasing of penalty ω.

Monotonic reduction of the total profit due to the increase of penalty ω: Solution robustness.

Fig 6

Overall profit of scenarios with variability λ.

Increase in total profit due to the increase on demand variability λ): Solution robustness.

Fig 7

Service level of scenarios with variability λ.

Increase of nonsatisfied demand due to the increase of variability λ): Model robustness.

Decision makers aim to evaluate supply, manufacturing, and distribution scenarios and select the option that provides the best contribution to the global SC metrics. For managers, it is important to know the quantity of materials and products that should be bought, produced, stocked, transported, and sold over different periods. The results show that demand variability and the service level are in conflict with each other. Our proposed robust stochastic programming model helps decision makers to evaluate supply, manufacturing, and distribution strategies, decide as to which service level should be selected from the perspective of demand variability, and understand the impact of a decision on the overall expected profit. Moreover, our proposed tactical model provides strategic decisions in case of SC disruptions. Strikes, accidents, or supply shortage can occur; in such cases, managers can, for example, plan optimal changes in the transport mode, material flow on machines, or supplier used.

Conclusion

This study developed a robust stochastic programming model to evaluate the flexibility of a four-echelon SC at a tactical level. The proposed model simultaneously considered demand variability and service levels for scenarios of flexibility in supply, production, and distribution. Although such a strategy increases the model size and its complexity, the approach fills the gaps in the literature pertaining to the flexibility analysis of an integrated SC and is aligned to current research trends.

The supply flexibility evaluated the strategy to buy the finished product directly from a supplier. In this case, the company acts as a link, activating only its logistics infrastructure to meet demand. This situation occurs when the company’s fixed costs of operational activation do not compensate for the level of demand. However, the company runs the risk of having its image damaged by providing a competitor’s product. The manufacturing flexibility evaluated the strategy of reducing fixed costs of machines and overtime costs in plant-[1]. From this analysis, the idea of increasing production in plant-[2] emerged because the product families can be manufactured in both the industrial plants to prevent stock disruption. The logistics flexibility evaluated a reduction in the distribution cost for modal-[2] and an increase in the inventory safety level. The reduction in the distribution costs increased the total expected profit, but a part of it was consumed by the cost of maintaining higher safety stocks in distribution centers. Finally, additional experiments were performed to assess the flexibility and the robustness simultaneously. The proposed approach facilitates the adoption of different management styles and improves SC resilience. The model can be extended to contexts pertaining to SC disruptions by exploring strategies when unexpected events disrupt the supply, manufacturing, or distribution network of a company.

In summary, the main features of this paper are as follows: (1) introduction of a robust stochastic programming formulation to a four-echelon SC model; (2) consideration of parameter uncertainty for the evaluation of flexibility in three dimensions: supply, production, and distribution; and (3) increase in SC resilience based on flexibility analysis that considers the tradeoff between demand variability and the service level.

Although we have demonstrated the advantages of our model, our study has some limitations. Backlogging is not considered, and each product has one technical route. In future research, the stochastic parameters of the transportation lead time and production can be considered. The study of a risk-averse robust formulation is also a future research avenue. Moreover, as the amount of data grows, this problem, which is nondeterministic polynomial-time hard, becomes difficult to solve. Future developments can include the development of an efficient method of decomposition for solving the large-scale multiechelon SC planning problem.