Mathematical Programming Approaches

Abstract

Mathematical programming is one of the most important techniques available for quantitative decision-making. The general purpose of mathematical programming is finding an optimal solution for allocation of limited resources to perform competing activities. The optimality is defined with respect to important performance evaluation criteria, such as cost, time, and profit. Mathematical programming uses a compact mathematical model for describing the problem of concern. The solution is searched among all feasible alternatives. The search is executed in an intelligent manner, allowing the evaluation of problems with a large number of feasible solutions.

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1 Introduction

Mathematical programming is one of the most important techniques available for quantitative decision-making. The general purpose of mathematical programming is finding an optimal solution for allocation of limited resources to perform competing activities. The optimality is defined with respect to important performance evaluation criteria, such as cost, time, and profit. Mathematical programming uses a compact mathematical model for describing the problem of concern. The solution is searched among all feasible alternatives. The search is executed in an intelligent manner, allowing the evaluation of problems with a large number of feasible solutions.

Mathematical programming finds many applications in supply chain management, at all decision-making levels. It is also widely used for supply chain configuration purposes. Out of several classes of mathematical programming models, mixed-integer programming models are used most frequently. Other types of models, such as stochastic and multi-objective programming models, are also emerging to handle more complex supply chain configuration problems. Although these models are often more appropriate, computational complexity remains an important issue in the application of mathematical programming models for supply chain configuration.

This chapter describes application of mathematical programming for supply chain configuration. The general overview is given in Sect. 8.2. It is followed by a description of generic supply chain configuration mixed-integer programming model in Sect. 8.3. This model is based on the data model presented in Chap. 7. Computational approaches for solving problems of large size are also discussed along with typical modifications of the generic model, especially, concerning global factors. Section 8.4 outlines the application of other classes of mathematical programming models. In Sect. 8.5, the generic optimization model is used to optimize the SCC Bike ’s supply chain configuration. Section 8.6 details a model integration procedure, whereby optimization models for supply chain configuration problems can be built on the basis of pertinent information models.

2 Purpose

Mathematical programming models are used to optimize decisions concerning execution of certain activities subject to resource constraints. Mathematical programming models have a well-defined structure. They consist of mathematical expressions representing objective function and constraints. The expressions involve parameters and decision variables. The parameters are input data, while the decision variables represent the optimization outcome. The objective function represents modeling objectives and makes some decisions more preferable than others. The constraints limit the values that decision variables can assume.

The main advantages of mathematical programming models are that they provide a relatively simple and compact approximation of complex decision-making problems, an ability to efficiently find an optimal set of decisions among a large number of alternatives, and supporting analysis of decisions made. Specifically, in the supply chain configuration problem context, mathematical programming models are excellent for modeling its spatial aspects.

There are also some important limitations. Mathematical programming models have a lower level of validity compared to some other types of models—particularly, simulation. In the supply chain configuration context, mathematical programming models have difficulties representing the dynamic and stochastic aspects of the problem. Additionally, solving of many supply chain configuration problems is computationally challenging.

Following the supply chain configuration scope , mathematical programming models are suited to answer the following supply chain configuration questions:

• 1. Which partners to choose?

• 2. Where to locate supply chain facilities?

• 3. How to allocate production and capacity?

• 4. Which transportation mode to choose?

• 5. How do specific parameters influence supply chain performance?

The most common type of mathematical programming models is linear programming models. These models have all constraints and the objective function expressed as a linear function in variables. However, many real-life problems cannot be represented as linear functions. A typical example is representation of decisions concerning the opening of supply chain facilities. These decisions assume values equal either to 0 or 1. Integer programming models are used to model such problems. Their computational tractability is lower than that of linear programming models. Nonlinear expressions are often required to represent inventory and transportation-related issues of supply chain. That results in nonlinear programming models, which have high computational complexity.

Given the heterogeneous nature of supply chains, optimization often cannot be performed with respect to a single objective. Multi-objective programming models seek an optimal solution with regard to multiple objectives. These models rely on judgmental assessment of the relative importance of each objective.

Generally, as one moves from linear programming to more complex mathematical programming models, the validity of representing real-world problems is improved at the expense of model development and solving simplicity. Specialized model-solving algorithms are often required to solve complex problems.

Mathematical programming modeling systems (Greenberg 1993) have been developed for elaboration, solving, and analysis of mathematical programming models. These include GAMS, ILOG, and LINGO, to mention a few. These systems provide the means for data handling, model composition using special-purpose mathematical programming languages, and model solving. From the perspective of integrated decision modeling frameworks, these systems can be easily integrated into the decision support system to provide optimization functionality. The integration is achieved by using some types of application programming interfaces. Data structures used, generally, are system specific. Therefore, these need to be mapped to data sources using information modeling.

The role of mathematical programming systems in the overall strategic decision-making system has been described by Shapiro (2006). The described optimization modeling system includes links from the mathematical programming system to a decision-making database and other data sources, as well as advanced tools for conducting analysis. Generation of optimization models from data stored in the decision-making database is considered.

3 Mixed-Integer Programming Models

Traditional supply chain configuration models are mixed integer programming models. This section starts with presenting a generic model formulation which includes only the most frequently used decision variables , parameters, and constraints, as identified during construction of the generic supply chain configuration data model. The presentation of the generic model is followed by an overview of most frequently used modifications.

3.1 Generic Formulation

The following subsections define notation used to specify the generic supply chain configuration optimization model, and present the object function and constraints of this model.

Notations

Notation	Definition
Indices
i j k s m n t	Products, i = 1,…, I Materials, j = 1,…, J Plants, k = 1,…, K Suppliers, s = 1,…, S Distribution centers, m = 1,…, M Customer zones, n = 1,…, N Time period, t = 1,…, T
Parameters
d int	Demand
π i	Revenues per product
h k	Plant capacity
γ i	Capacity requirements for product
δ ij	Material consumption per product
ρ js	${\rho}_{js}=1$ if the supplier offers a material and ${\rho}_{js}=0$ otherwise
ω js	Material purchasing cost from supplier per unit
${\lambda}_{ik}$	Production cost at plant per unit
β im	Inventory storage cost at the distribution center over the planning horizon per product
${r}_{im}$	Handling cost at distribution center per unit
${c}_{1jsk}$	Transportation cost from supplier to plant per material unit
${c}_{2ikm}$	Transportation cost from plant to distribution center per product unit
${c}_{3imn}$	Transportation cost from distribution center to customer per product unit
f 1k	Plant fixed opening/operating cost per time period
f 2m	Distribution center fixed opening/operating cost per time period
P	A large constant number
Decision variables
X imnt	Quantity of products sold from distribution center to customer
Q ikt	Quantity of products produced at plant
Y ikmt	Quantity of products shipped from plant to distribution center
B imt	Inventory size at distribution center per product
V jskt	Quantity of materials purchased and shipped from supplier to plant
W k	Plant open indicator equals 1 if plant is open and 0 otherwise
U m	Distribution center open indicator equals 1 if distribution center is open and 0 otherwise
A mn	Customer zone to distribution center allocation indicator equals 1 if customer is served by distribution center and 0 otherwise

Objective Function

The objective function (Eq. 8.1) maximizes profit E determined as a difference between revenues Φ and total cost TC. As indicated in the previous chapter, profit maximization increasingly is considered as one of the main supply chain configuration performance measures. The total cost consists of multiple cost components including production cost (TC ₁), materials purchasing and transportation cost (TC ₂), products transportation cost from plants to distribution centers, product handling and transportation cost from distribution centers to customers (TC ₃), fixed costs for opening and operating plants and distribution centers (TC ₄), and inventory holding cost (TC ₅). Revenues, total cost, and its components collectively are referred as measures used to evaluate supply chain configuration performance.

$$ E=\Phi -TC\to \max $$

(8.1)

Measures

$$ \Phi ={\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{n=1}^N{\displaystyle {\sum}_{t=1}^T{\pi}_i{X}_{imnt}}}}} $$

(8.2)

$$ TC={\displaystyle {\sum}_{l=1}^5T{C}_l} $$

(8.3)

$$ T{C}_1={\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{t=1}^T{\lambda}_{ik}{Q}_{ikt}}}} $$

(8.4)

$$ T{C}_2={\displaystyle {\sum}_{j=1}^J{\displaystyle {\sum}_{s=1}^S{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{t=1}^T\left({\omega}_{js}+{c}_{1jsk}\right){V}_{jskt}}}}} $$

(8.5)

$$ T{C}_3={\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{t=1}^T{c}_{2ikm}{Y}_{ikmt}}}}}+{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{n=1}^N{\displaystyle {\sum}_{t=1}^T\left({r}_{im}+{c}_{3imn}\right){X}_{imnt}}}}} $$

(8.6)

$$ T{C}_4={\displaystyle {\sum}_{k=1}^K{f}_{1k}{W}_k}+{\displaystyle {\sum}_{m=1}^M{f}_{2m}{U}_m} $$

(8.7)

$$ T{C}_5={T}^{-1}{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{t=1}^T{\beta}_{im}{B}_{imt}}}} $$

(8.8)

Constraints

$$ {\displaystyle {\sum}_{m=1}^M{X}_{imnt}}\le {d}_{int},\forall i,n,t $$

(8.9)

$$ {\displaystyle {\sum}_{n=1}^N{X}_{inmt}}\le {B}_{imt}+{\displaystyle {\sum}_{k=1}^K{Y}_{ikmt}},\forall i,m,t $$

(8.10)

$$ {B}_{imt}={B}_{imt-1}+{\displaystyle {\sum}_{k=1}^K{Y}_{ikmt}}-{\displaystyle {\sum}_{n=1}^N{X}_{inmt}},\forall i,m,t $$

(8.11)

$$ {\displaystyle {\sum}_{m=1}^M{Y}_{ikmt}}\le {Q}_{ikt},\forall i,k,t $$

(8.12)

$$ {\displaystyle {\sum}_{i=1}^I{\gamma}_i{Q}_{ikt}}\le {h}_k{W}_k,\forall k,t $$

(8.13)

$$ {\displaystyle {\sum}_{i=1}^I{\delta}_{ij}{Q}_{ikt}}\le {\displaystyle {\sum}_{s=1}^S{\rho}_{js}{V}_{jskt}},\forall j,k,t $$

(8.14)

$$ {\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{n=1}^N{X}_{imnt}}}<P{U}_m,\forall m,t $$

(8.15)

$$ {{\displaystyle {\sum}_{n=1}^NA}}_{mn}=1,\forall n $$

(8.16)

$$ {B}_{im0}=0,\forall i,m $$

(8.17)

$$ {W}_k,{U}_m\in \left\{0,1\right\},\forall k,m $$

(8.18)

$$ {A}_{mn}\in \left\{0,1\right\},\forall m,n $$

(8.19)

Equation (8.9) enforces the balance between products sold and demand. The balance between incoming and outgoing flows at distribution centers is defined by Eqs. (8.10) and (8.11). This balance is achieved by satisfying the customer demand with newly arrived shipments from plants or from the inventory. If some of the newly arrived shipments are not sold to customers, they are retained in inventory at distribution centers. The balance between products produced and products shipped to the distribution centers is enforced by Eq. (8.12). Equation (8.13) restricts capacity availability. Availability of materials to produce products is checked by Eq. (8.14). Equation (8.15) states that product flows are allowed only through open distribution centers. Equation (8.16) control allocation of customer zones to distribution centers by requiring that each customer zone is served by only one distribution center. The initial inventory of products at distribution centers is set to zero Eq. 8.17. Variables W _k , U _m and A _mn are binary Eqs. (8.18) and (8.19).

Comments

The model does not explicitly include parameters characterizing a spatial location of supply chain units. Alternative locations for a particular supply chain unit are evaluated by allowing for several units with equal characteristics but different transportation costs, which characterize the location of the unit.

There are two factors affecting the model composition: (1) the broker and power structure ; and (2) the initial state of the network. Depending upon the organizational and power structure of the supply chain and a decision maker’s point of view (i.e., interests of the whole supply chain vs. interests of the dominant member), some of the cost parameters are set to zero because the total cost the broker is concerned about is not affected by these cost parameters, even if these are relevant to the overall supply chain modeling (e.g., a final assembler pays only purchasing costs for components and is not concerned about processing costs at the supply level). The initial state of the network determines whether some of the decision variables already do not have a fixed value. For instance, the location of several assembly plants is already fixed and cannot be changed. Similarly, long-term purchasing contracts with some suppliers can set definite limits on purchasing volume from these suppliers.

Reconfiguration

The model implicitly assumes that greenfield supply chain configuration is performed and there are no fixed supply chain units or links. In the case of supply chain reconfiguration, additional constraints are imposed to represent the reconfiguration options. If a unit or link is indicated as design time selection, then constraints (8.13) and (8.15)–(8.16) are not changed. If a unit or link is indicated as fixed, then the corresponding constraints are set equal to one (i.e., the decision variable becomes a parameter).

If configuration decision variables n are made at execution time, then the selection variables can assume values either 1 or 0. In this case, it is suggested that design time evaluation of the impact of execution time decisions should be performed by means of robust optimization.

3.2 Modifications

The generic formulation obviously needs to be adjusted to include factors relevant to a particular decision-making problem. The literature analysis suggests that the most frequently considered factors are international factors, inventory, capacity treatment, transportation, and supply chain management policies. We discuss these below.

International Factors

Given that many supply chains involve partners from different countries, international factors need to be addressed in supply chain configuration. This problem is of particular importance for large multinational companies manufacturing and selling their products worldwide. Mathematical programming models consider quantitative factors, while there are also numerous qualitative factors influencing international decision-making.

Table 8.1 lists selected decision variables, parameters, and constraints used in some international supply chain configuration models. Goetschalckx et al. (2002) provide a summary table on works considering international factors. This summary indicates that taxes and duties are the most often considered international factors. In a similar work by Meixell and Gargeya (2005), the most frequently considered international factors besides tariffs and duties are currency exchange rates and corporate income taxes. However, many of the models surveyed use already fixed supply chain configuration. Kouvelis et al. (2004) present an extensive sensitivity analysis of the impact of international factors on supply chain configuration. The transfer pricing to optimize overall global supply chain profitability is analyzed in a recent contribution by De Matta and Miller (2015).

Table 8.1 Selected international factors considered in literature

Inventory

There has been a significant increase of supply chain configuration models including inventory management related issues as supply chain configuration problem solving. The literature review shows that 26 out of the 68 survey mathematical programming models are multi-period models including inventory management decisions. This trend is driven by an increasing need to analyze supply responsiveness. The inventory management decisions are represented not only at the tactical level but also at the operational decision including safety stock and ordering quantity.

Capacity Treatment

A majority of models have some sort of flow intensity and transformation capacity limits as a parameter. A parameter characterizing capacity consumption per unit processed or handled is also widely used (e.g., Pirkul and Jayaraman 1998; Sabri and Beamon 2000). Sabri and Beamon (2000) and Yan et al. (2003) use product specific capacity, while Pirkul and Jayaraman (1998) the flexible capacity. Bhutta et al. (2003) is one of the few papers using capacity as a decision variable. This paper allows either increasing or decreasing capacity at the facility.

In order to account for environmental factors, it is also important to consider a kind of capacity or resources used in supply chain processes. For instance, Chaabane et al. (2012) set capacity limits for specific production technologies and the most appropriate production technology is used to minimize emissions associated with production as one of the supply chain configuration objectives.

Transportation

The most common way of representing transportation is considering just one mode and including variable costs per unit shipped between supply chain units. However, transportation-related issues generally are much more complex and several models attempt to account for this complexity. Nonlinear dependence of transportation costs according to quantity shipped is modeled by Tsiakis et al. (2001). This dependence is represented by a piece-wise linear function. Transportation costs are not calculated for individual products but for families of similar products, thus reducing the model complexity. Syam (2002) and Viswanadham and Gaonkar (2003) include a fixed charge per unit using a particular link to transfer products between units. Arntzen et al. (1995), Dogan and Goetschalckx (1999), and Viswanadham and Gaonkar (2003) also include the transportation time parameter. Prakash et al. (2012) consider different transportation modes that allows for multi-objective evaluation of the supply chain configuration in order to minimize costs and maximize demand fill rate.

Ross et al. (1998) have transportation as one of the key specific problems of supply chain configuration decision-making and the model represents individual vehicles with their characteristics. Farahani et al. (2015) combine distribution network design with vehicle routing by assigning retailers to a specific delivery route. This approach is particularly useful in the case of agile supply chain, where distribution network design decision are revised relatively often. Vidal and Goetschalckx (2001) split transportation costs between supplier and manufacturer to take advantage of lower taxes.

Capacity limits are also frequently used for links between units. Arntzen et al. (1995) and Syam (2002) represent transportation capacity by limiting the total weight of products shipped. The shipment weight-based representation of shipments costs and transportation capacity is often used in applied studies.

Detailed representation of transportation is a feature of many commercial supply chain network design models. These are based on detailed databases of distance and freight rates. These data as well as transportation cost structure and shipment planning are described by Bowersox et al. (2002).

Supply Chain Management Policies

Configuration decisions concerning use of particular supply chain facilities are often tightly interrelated with strategic-level decisions in relation to the particular managerial policies used. Two cases of representing management policies are distinguished:

• Policies are represented structurally;

• Policies are represented through values of parameters.

An example of structurally represented policies is a decision between using direct shipments and using a centralized warehouse. Evaluation of such alternatives effectively implies development of two separate models, which share common features. However, it is also possible to construct a single model with binary variables used for switching between different structures.

An example of policies represented through values of parameters is a decision between using Electronic Data Interchange (EDI) or the Internet as a communication mode among supply chain units. In this case, a binary variable can be used to represent the decision between policies, and values of parameters representing fixed costs for establishing links among units and variable costs for transferring products are specified for each of the two policies.

A combined example, where policies are represented both structurally and through values of parameters, is a decision variable between using flexible manufacturing facilities or specialized manufacturing facilities. Structurally different product-to-facility assignments are given as inputs (i.e., multiple flexibility scenarios are evaluated). At the same time, representing flexible manufacturing facilities influences the value of the fixed cost parameter.

The literature on including policy-related variables in the quantitative supply chain configuration models is scarce. Truong and Azadivar (2005) include a decision variable representing a choice between using push and pull manufacturing policies.

Analyzing many different policies might lead to explosive growth of the computational time needed to solve the model. Therefore, many policy related decisions are already made at earlier steps of the supply chain configuration.

3.3 Computational Issues

Model solving is an important part of supply chain configuration problem solving because the direct use of commercially available solvers might not be sufficient. Geoffrion and Powers (1995), in their discussion of developments in design of integrated production–distribution networks, indicate that corresponding large-scale models are difficult to solve in reasonable time because it is an NP-hard problem. Small to medium problems can be solved using standard software on personal computers (Kouvelis et al. 2004). However, that depends on the structure of a particular model and values of parameters. Specialized model-solving algorithms are generally required to solve large-scale problems.

There are two major approaches to elaboration of computationally efficient algorithms. These are based on Lagrangian relaxation and Bender’s decomposition. A short overview of these methods is provided here. Readers are referred to Avriel and Golany (1996) for a detailed coverage of mathematical programming.

Lagrangian Relaxation

The Lagrangian relaxation schema assumes that problem solving is complicated by a few difficult constraints. It attempts to simplify the problem by dualizing the difficult constraints (i.e., constraints are introduced into the objective function with a penalty function). As a result, a relaxed problem of the original problem is obtained. The relaxed problem is solved to obtain an upper bound (for maximization problems) of the original problem. Any feasible solution of the original problem provides a lower bound. Iterative heuristic algorithms are used in searching for the optimal solution of the original problem in this narrowed range. The upper and lower bounds are continuously updated. A good overview of the general theory on the Lagrangian relaxation is provided by Magee and Glover (1996).

Pirkul and Jayaraman (1998) successfully applied the Lagrangian relaxation problem for the supply chain configuration problem. Similar results have been obtained by Jang et al. (2002) and Amiri (2006). The supply chain configuration model by Pirkul and Jayaraman (1998) locates a specified number of manufacturing facilities and warehouses to minimize fixed and transformation costs subject to customer demand satisfaction and capacity constraints.

The mathematical representation of their model is as follows. Parameters of the model are:

• C _ijl —the variable cost to distribute a unit of product l from warehouse j to customer zone i;

• T _jkl —a unit cost to transport product l from plant k to warehouse j;

• f _k and g _j —fixed cost to open and operate plant k and warehouse j, respectively;

• a _il —demand for product l at customer zone i;

• D _k —capacity of plant k;

• W _j —throughput limit at warehouse l;

• q _l —plant capacity consumption by product l;

• s _l —is warehouse throughput capacity consumption by product l;

• W and P—upper limit on the number of warehouses and plants that can be opened, respectively.

Variables X _ijl and Y _jkl denote the total number of units of product l distributed through warehouse j to customer zone i and the total number of units of product l shipped from plant k to warehouse j, respectively. P _k and Z _j are binary variables denoting whether plant k is open and whether warehouse j is open, respectively.

The objective function and constraints are given below.

$$ \min Z={\displaystyle \sum_i{\displaystyle \sum_j{\displaystyle \sum_l{C}_{ijl}{X}_{ijl}}}}+{\displaystyle \sum_j{\displaystyle \sum_k{\displaystyle \sum_l{T}_{jkl}{X}_{jkl}}}}+{\displaystyle \sum_k{f}_k{P}_k}+{\displaystyle \sum_j{g}_j{Z}_j} $$

(8.20)

subject to

$$ {\displaystyle \sum_j{X}_{ijl}}={a}_{il},\forall i,l $$

(8.21)

$$ {\displaystyle \sum_i{\displaystyle \sum_l{s}_l{X}_{ijl}\le {Z}_j{W}_j}},\forall j $$

(8.22)

$$ {\displaystyle \sum_j{Z}_j}\le W $$

(8.23)

$$ {\displaystyle \sum_i{X}_{ijl}}\le {\displaystyle \sum_k{Y}_{jkl}},\forall j,l $$

(8.24)

$$ {\displaystyle \sum_i{\displaystyle \sum {q}_l}{Y}_{jkl}}\le {D}_k{P}_k,\forall k $$

(8.25)

$$ {\displaystyle \sum_k{P}_k}\le P,\forall k $$

(8.26)

After relaxing constraints Eqs. (8.21) and (8.24), the Lagrangian relaxation of the problem is

$$ \begin{array}{c} \min {Z}_{LR}={\displaystyle \sum_i{\displaystyle \sum_j{\displaystyle \sum_l{C}_{ijl}{X}_{ijl}}}}+{\displaystyle \sum_j{\displaystyle \sum_k{\displaystyle \sum_l{T}_{jkl}{X}_{jkl}}}}+{\displaystyle \sum_k{f}_k{P}_k}+{\displaystyle \sum_j{g}_j{Z}_j}\\ {}+{\displaystyle \sum_i{\displaystyle \sum_l{\gamma}_{il}\left({\displaystyle \sum_j{X}_{ijl}-{a}_{il}}\right)}}+{\displaystyle \sum_j{\displaystyle \sum_l{\beta}_{jl}\left({\displaystyle \sum_i{X}_{ijl}}-{\displaystyle \sum_k{Y}_{jkl}}\right)}}\end{array} $$

(8.27)

where γ _il and β _jl are Lagrangian multipliers (dual prices). The relaxed problem is further decomposed into a subproblem representing manufacturing plants and a subproblem representing warehouses. An iterative model-solving procedure is used to solve the configuration problem. The Lagrangian subproblems are used to narrow the gap between lower and upper bounds until the difference is less than one percent or 500 iterations have been executed. Computational efficiency of the procedure has been tested for different numbers of products, potential plants and warehouses, and customer zones, as well as for different levels of capacity load. For instance, the problem-solving time for a problem with 100 customer zones, 20 warehouses, 10 plants and 3 products is about 60 s.

Bender’s Decomposition

The main idea behind the Bender’s decomposition approach is partitioning the original mixed-integer problem into its linear and integer parts (Salkin 1975). The steps of the problem-solving algorithm are as follows:

• 1. Fix values of integer variables and determine upper and lower bounds.

• 2. Solve a dual problem of the linear programming model obtained by fixing the integer variables and update the upper bound (for minimization problems).

• 3. Solve an integer problem obtained from the original problem by fixing the continuous part of the problem and update lower bound.

• 4. Iterate until the gap between the upper and lower bound is sufficiently small.

• 5. Upon convergence, compute optimal values of continuous decision variables.

At the first step, not only integer variables can be fixed but also any variables deemed as complicated. The Benders decomposition for solving supply chain configuration problems has been used by Geoffrion and Graves (1974), and Dogan and Goetschalckx (1999). In both cases, it has allowed solving large industrial-scale problems within a reasonable time. The former authors additionally develop a specialized acceleration technique, which has been shown to decrease computational time substantially.

4 Other Mathematical Programming Models

Multi-objective, stochastic, and nonlinear mathematical programming models are other models that find application in supply chain configuration.

4.1 Multi-objective Programming Models

A multi-objective evaluation is needed to represent various aspects of supply chain performance and customers’ requirements satisfaction, as well as to balance the performance of individual supply chain units. Two main technical approaches to representing multi-objective situations are: (1) assigning weights to each objective, characterizing relative importance; and (2) preemptive optimization starting with the most important objective. Choice of appropriate weights and prioritization of objective relies on the decision maker’s judgment and substantially affects modeling results.

The generic formulation can be extended to multi-objective setting in various ways. Objectives associated with environment al issues, responsiveness, and reliability including customer service are considered most frequently. The generic supply chain configuration optimization model is extended to incorporate these additional objectives and the objective function (8.1) is reformulated as

$$ Z= \min \left({Z}_1,{Z}_2,{Z}_3,{Z}_4\right), $$

(8.28)

where ${Z}_1=TC$ represents total costs, Z ₂ represents environmental impact, Z ₃ represents responsiveness and Z ₄ represents reliability (other notation is used as in Sect. 8.3.1 but omitting time period index t). The environmental impact is evaluated as quantity of carbon emissions due to transportation

$$ \begin{array}{l}{Z}_2={\displaystyle {\sum}_{j=1}^J{\displaystyle {\sum}_{s=1}^S{\displaystyle {\sum}_{k=1}^K{e}_j{\tau}_{1jsk}{V}_{jsk}}}}\\ {}+{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{m=1}^M{e}_i{\tau}_{2ikm}{Y}_{ikm}}}}+{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{n=1}^N{e}_i{\tau}_{3imn}{X}_{imn}}}}\to \min, \end{array} $$

(8.29)

where e _j and e _i are carbon emissions associated with transportation of materials and products, respectively, and ${\tau}_{1jsk}$ is transportation time from supplier to plant per material unit, ${\tau}_{2ikm}$ is transportation time from plant to distribution center per product unit, and ${\tau}_{3mn}$ is transportation time from distribution center to customer per product unit.

The responsiveness is evaluated as a time spent during transportation of materials and products along the supply chain links

$$ \begin{array}{l}{Z}_3={\displaystyle {\sum}_{j=1}^J{\displaystyle {\sum}_{s=1}^S{\displaystyle {\sum}_{k=1}^K{t}_{1jsk}{V}_{jsk}}}}\\ {}+{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{k=1}^K{\displaystyle {\sum}_{m=1}^M{t}_{2ikm}{Y}_{ikm}}}}+{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{n=1}^N{t}_{3imn}{X}_{imn}}}}\to \min \end{array} $$

(8.30)

The supply chain reliability is evaluated by the fill rate

$$ {Z}_4={D}^{-1}{\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{m=1}^M{\displaystyle {\sum}_{n=1}^N{X}_{imn}}}}\to \max $$

(8.31)

where $D={\displaystyle {\sum}_{i=1}^I{\displaystyle {\sum}_{n=1}^N{d}_{in}}}$ is the total demand for all products in the supply chain.

Other multi-objective supply chain configuration models have been developed by Li and O’Brien (1999), Sabri and Beamon (2000), Talluri and Baker (2002), Brandenburg (2015) and Das and Rao Posinasetti (2015) (see Chap. 3).

4.2 Stochastic Programming Models

The models discussed above assume that all parameters are known with certainty, which is not the case in real-life situations. To obtain robust results, the impact of uncertainty needs to be assessed. Stochastic programming is one of the techniques allowing accounting for stochastic parameters.

Many of the stochastic programming models developed for supply chain configuration have demand as a stochastic parameter. Demand uncertainty usually is represented by multiple demand scenarios (Mirhassani et al. 2000; Tsiakis et al. 2001). In this case, a prototype objective function can be expressed as

$$ Z=\underset{\mathbf{Q},\mathbf{Y}}{ \max }E\left[F\left(\mathbf{c},\mathbf{D},\mathbf{Q},\mathbf{Y}\right)\right]=\underset{\mathbf{Q},\mathbf{Y}}{ \max }{\displaystyle {\sum}_{s=1}^SF\left(\mathbf{c},{\mathbf{D}}_s,\mathbf{Q},\mathbf{Y}\right)}, $$

(8.32)

where c represents all parameters of the supply chain configuration problem, D represents demand, Q represents continuous decision variables and Y represents binary decision variables (e.g., inclusion of units in the supply chain). F is an abstract function, E is the expected profit, and s = 1,…, S are evaluated demand scenarios.

Other stochastic parameters can also be represented by evaluation of multiple scenarios (e.g., Gutiérrez et al. 1996). The obvious limitation of this approach is a limited number of considered scenarios and there is little assurance that the coverage of uncertainty has been adequate.

Kim et al. (2002) develop a model for determining ordering quantities from suppliers for a fixed supply chain network subject to demand uncertainty. The demand uncertainty is represented using demand probability density function and an iterative model-solving procedure is developed without relying on using scenarios.

Santoso et al. (2005) develop a stochastic programming model for a typical supply chain configuration problem. The model minimizes total investment and operating costs by deciding which facilities to build and routing products from suppliers to customers. It allows for uncertainty in processing/transportation costs, demand, supplies, and capacities and for limited, but a very large number of scenarios representing uncertainty in demand, as well as in other parameters. The main constraints enforce capacity limits, flow conversion limits, and facility opening requirements (i.e., facility is operational only if open). The model is a two-stage stochastic program that minimizes the current investment cost and expected operational costs.

A specialized model-solving algorithm is developed. It uses an accelerated Benders decomposition to solve the facility opening problem and the sample average approximation scheme to solve the stochastic part of the model. The model is tested by its application in designing a supply chain in the packaging industry. The authors show that the developed model-solving algorithm allows solving large scale problems (13 products and 142 facilities) in less than two hours for one scenario and, more importantly, growth of computational time as the number of scenarios increases is slow. The stochastic approach to supply chain design allowed savings of up to 6 % compared to the mean value problem solution for the considered supply chain design problem. The stochastic programming solution also exhibits substantially lower variability over testing scenarios, which is a desirable property during the results approbation phase of the supply chain configuration methodology.

4.3 Nonlinear Programming Models

Due to major computational difficulties, nonlinear configuration models have not been frequently encountered in the supply chain configuration literature (see Wu and O’Grady (2004) for a brief discussion of nonlinear programming models in supply chain configuration). The main nonlinear factors relevant to supply chain configuration, such as inventory and transportation costs, are usually represented using piece-wise linear functions (e.g., Tsiakis et al. 2001).

Explicitly, nonlinear constraints have been used in models solved using simulation-based optimization and other nonparametric optimization methods, which are discussed in Chap. 9.

5 Sample Application

To illustrate application of the generic mixed-integer programming model presented in Sect. 8.3.1, the SCC Bike supply chain configuration is optimized. The objective is to maximize profit by selecting suppliers, locating the assembly plants and allocating customers to distribution centers. Two product groups are considered during the configuration and they differ mainly by the type of frame used in their production. The quarterly demand exhibits seasonal variations (Table 8.2) and manufacturing capacity is not sufficient to handle demand peeks in a single period (Table 8.3). The revenues per product also account for costs not explicitly considered in the configuration model. The production cost is higher for carbon frame bikes and is randomized to induce differences among the productions sites. The fixed cost is determined according to industry data concerning investments made per plant of certain capacity¹ and adjusted to include operational costs.

Table 8.2 Quarterly demand for product groups

Table 8.3 Manufacturing capacity and production costs at plants

There are at least two suppliers for every set of materials. The suppliers vary by prices offered (generated by randomly around a specified mean value) and by location what affects the transportation cost. The transportation cost for materials is generated by assuming that a sea transport is used at the cost of $0.03 per thousand units per mile (twice as much for frames). The actual sea-link distances are used to calculate c ₁. The material purchasing prices are given in Table 8.4. Transportation costs for products are generated assuming that trucks are used for transportation at the cost of $2.8 and $5.6 per thousand units per mile. These transportation cost coefficients and actual distances between locations are used to calculate c ₂ and c ₃, respectively.

Table 8.4 Material prices offered by different suppliers ($/set)

The optimization is performed for the base scenario characterized by the data provided above. The optimized profit E = 74.4 mil.$ and the service level measured as a ratio between total sales and total demand is 90 %. The service level is below 100 % due to insufficient capacity to deal with seasonal variations in demand. Figure 8.1 shows the costs breakdown according to the measures used. The sourcing costs (purchasing plus material transportation) are the biggest expense and the transportation costs have relatively little impact on the total cost. Therefore, material prices and production costs have the most significant impact on configuration results. The inventory cost is negligible because only period-to-period inventory storage cost at the distribution centers is taken into account (work-in-process inventory and inter-period inventory are not explicitly accounted for). The resulting supply chain configuration is shown in Fig. 8.2. Only one supplier for each set of materials is selected and the purchasing cost is dominates the selection. The plants stock the distribution centers regardless of their location in order to deal with the capacity limitations.

Fig. 8.1

The supply chain configuration costs breakdown

Fig. 8.2

The optimized supply chain configuration

If the capacity restrictions are relaxed or sufficiently high plant to distribution center allocation fee is introduced, the plants would supply products only to their regional distribution centers. The current arrangement of flexible allocation requires distribution centers to cope with regional differences due to different assembly locations (i.e., multi-language user manuals). The seasonal character of the demand and incorporating of inventory management decisions substantially affects the configuration decisions.

6 Model Integration

The supply chain configuration methodology emphasizes the integration of decision-making models with information models. Therefore, the supply chain configuration model’s data model is used to develop the supply chain optimization model. Figure 8.3 elaborates the transition from information modeling to quantitative modeling. This figure represents implementation of the optimization-related functionality of the integrated decision support system presented in Chap. 5. The commercially available LINGO² mathematical programming system is used in this case, although the approach is similar to several other mathematical programming languages. The figure shows only one-way interactions for simplicity. Obviously, modeling outcomes can be sent back to the supply chain management information system in a similar manner.

Fig. 8.3

Development of optimization models on the basis of information models

The general data model discussed previously is developed using a general modeling method such as UML, while the mathematical model is implemented using a special-purpose modeling language, LINGO. The LINGO model includes data definitions in the form of data sets, data link definitions providing link to data sources, and a formalized representation of the mathematical program. Data definitions and data links are generated automatically using data provided in the general data model (Fig. 8.4). Transformations are informally listed as follows (numbers in the list correspond to the numbering of arrows in Fig. 8.4):

Fig. 8.4

Generation of the LINGO data definition from the general data model

• 1. A data set declaration instruction is generated for each class in the diagram. All attributes except dimension are also included in the instruction line to declare parameters and variables of the mathematical programming model.

• 2. A variable declaration instruction is generated for the dimension attribute of each class (in the example, the variable is named PD). The generated instruction also defines data reading from the data source (using the @POINTER function of the special purpose programming language).

• 3. An instruction for reading values of the declared parameters is generated for each attribute of the Parameter type in the class diagram.

• 4. Attributes of type DecisionVariable are only included in the data set declaration instruction line (see Transformation 1 above) and this arrow only signifies the representation of decision variables.

The mathematical program is composed in a semiautomated manner by a decision maker who indicates which constraints to include from the decision-modeling knowledge base. The modeling technique specific data model contains actual data to be passed from the decision-modeling system to the LINGO solver during the problem-solving process. LINGO supports two main data transfer mechanisms:

• Open Database Connectivity (ODBC) based data transfer. In this case, the separate modeling technique specific data model is not necessary because LINGO can directly request data from database tables using the standard database access protocol.

• Remote Procedure Call (RPC) based data transfer. This mode is necessary if LINGO is part of a more complex decision-making system and is invoked programmatically. In this case, LINGO receives two specially structured data arrays from the decision-modeling system. The first array contains meta-data about data being transferred. The second array contains actual values. The decision-modeling system is responsible for merging data from the general data model into these two arrays.

7 Summary

This chapter describes the generic supply chain configuration model, modifications of this model, and the integration of the mathematical programming model into the overall decision-modeling process.

Computational limitations still remain an important factor when considering practical application of mathematical programming for the supply chain configuration problem solving. Solving configuration models using computational approaches described in this chapter requires substantial expertise in mathematical programming, and algorithms are developed on a case-by-case basis. Therefore, commercial applications often rely on pure computational power or heuristic approaches. The former is not always sufficient for medium-size problems, while the latter cannot guarantee the quality of obtained solutions. Computational feasibility also restricts the development of nonlinear mathematical programming models.