### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The pre- and post-integration multi-product supply chain network models
- 3. Quantifying synergy associated with multi-product supply chain network integration
- 4. Numerical examples
- 5. Summary and conclusions
- Acknowledgments
- References

In this paper, we develop multi-product supply chain network models with explicit capacities, before and after their horizontal integration. In addition, we propose a measure, which allows one to quantify and assess, from a supply chain network perspective, the synergy benefits associated with the integration of multi-product firms through mergers/acquisitions. We utilize a system-optimization perspective for the model development and provide the variational inequality formulations, which are then utilized to propose a computational procedure which fully exploits the underlying network structure. We illustrate the theoretical and computational framework with numerical examples. This paper is a contribution to the literatures of supply chain integration and mergers and acquisitions.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The pre- and post-integration multi-product supply chain network models
- 3. Quantifying synergy associated with multi-product supply chain network integration
- 4. Numerical examples
- 5. Summary and conclusions
- Acknowledgments
- References

Today, supply chains are more extended and complex than ever before. At the same time, the current competitive economic environment requires that firms operate efficiently, which has spurred interest among researchers as well as practitioners to determine how to utilize supply chains more effectively and efficiently.

In this increasingly competitive economic environment, there is also a pronounced amount of merger activity. Indeed, according to Thomson Financial, in the first 9 months of 2007 alone, worldwide merger activity hit US$3.6 trillion, surpassing the total from all of 2006 combined (Wong, 2007). Interestingly, Langabeer and Seifert (2003) showed a compelling and direct correlation between the level of success of the merged companies and how effectively the supply chains of the merged companies are integrated. However, a survey of 600 executives involved in their companies' mergers and acquisitions (M&A), conducted by Accenture and the Economist Unit, found that less than half (45%) achieved expected cost-savings synergies (Byrne, 2007). It is, therefore, worthwhile to develop tools that can better predict the associated strategic gains associated with supply chain network integration, in the context of M&A, which may include, among others, possible cost savings (Eccles et al., 1999).

Furthermore, although there are numerous articles discussing multi-echelon supply chains, the majority deal with a homogeneous product (see, e.g., Dong et al., 2004; Nagurney, 2006a; Wang et al., 2007). Firms are seeing the need to spread their investment risk by building multi-product supply facilities, which also gives the advantage of flexibility to meet changing market demands. According to a study of the US supply output at the firm-product level between 1972 and 1997, on the average, two-thirds of the US supply firms altered their mix of products every 5 years (Bernard et al., 2009). By running a multi-use plant, costs of supply may be divided among different products, which may increase efficiencies.

Moreover, it is interesting to note the relationships between merger activity to multi-product output. For example, according to a study of the US supply output at the firm-product level between 1972 and 1997, <1% of a firm's product additions occurred due to M&A. Actually, 95% of the firms engaging in M&A were found to adjust their product mix, which can be associated with ownership changes (Bernard et al., 2009). The importance of the decision as to what to offer (e.g., products and services), as well as the ability of firms to realize synergistic opportunities of the proposed merger, if any, can add tremendous value. It should be noted that a successful merger depends on the ability to measure the anticipated synergy of the proposed merger (cf. Chang, 1988).

This paper is built on the recent work of Nagurney (2009) who developed a system-optimization perspective for supply chain network integration in the case of horizontal M&A. In this paper, we also focus on the case of horizontal mergers (or acquisitions) and we extend the contributions in Nagurney (2009) to the much more general and richer setting of multiple product supply chains. Our approach is most closely related to that of Dafermos (1973) who proposed transportation network models with multiple modes/classes of transportation. In particular, we develop a system-optimization approach to the modeling of multi-product supply chains and their integration and we explicitly introduce capacities on the various economic activity links associated with manufacturing/production, storage, and distribution. Moreover, in this paper, we analyze the synergy effects associated with horizontal multi-product supply chain network integration, in terms of the operational synergy, that is, the reduction, if any, in the cost of production, storage, and distribution. Finally, the proposed computational procedure fully exploits the underlying network structure of the supply chain optimization problems both pre- and post-integration.

We note that Min and Zhou (2002) provided a synopsis of supply chain modeling and the importance of planning, designing, and controlling the supply chain as a whole. Nagurney (2006b) subsequently proved that supply chain network equilibrium problems, in which there is cooperation between tiers, but competition among decision-makers within a tier, can be reformulated and solved as transportation network equilibrium problems. Cheng and Wu (2006) proposed a multi-product, and multi-criterion, supply-demand network equilibrium model. Davis and Wilson (2006), in turn, studied differentiated product competition in an equilibrium framework. Mixed integer linear programming models have been used to study synergy in supply chains, which has been considered by Soylu et al. (2006), who focused on energy systems, and by Xu (2007).

This paper is organized as follows. The pre-integration multi-product supply chain network model is developed in Section 2. Section 2 also introduces the horizontally merged (or integrated) multi-product supply chain model. The method of quantification of the synergistic gains, if any, is provided in Section 3, along with new theoretical results. In Section 4, we present numerical examples, which not only illustrate the richness of the framework proposed in this paper, but which also demonstrate quantitatively how the costs associated with horizontal integration affect the possible synergies. We conclude the paper with Section 5, in which we summarize the results and present suggestions for future research.

### 3. Quantifying synergy associated with multi-product supply chain network integration

- Top of page
- Abstract
- 1. Introduction
- 2. The pre- and post-integration multi-product supply chain network models
- 3. Quantifying synergy associated with multi-product supply chain network integration
- 4. Numerical examples
- 5. Summary and conclusions
- Acknowledgments
- References

We measure the synergy by analyzing the total costs before and after the supply chain network integration (cf. Eccles et al., 1999; Nagurney, 2009). For example, the synergy based on total costs and proposed by Nagurney (2009), but now in a multi-product context, which we denote here by *S*^{TC}, can be calculated as the percentage difference between the total cost pre vs the total cost post the integration:

- (14)

From (14), one can see that the lower the total cost *TC*^{1}, the higher the synergy associated with the supply chain network integration. Of course, in specific firm operations one may wish to evaluate the integration of supply chain networks with only a subset of the links joining the original two supply chain networks. In that case, Fig. 2 would be modified accordingly and the synergy as in (14) computed with *TC*^{1} corresponding to that new supply chain network topology.

We now provide a theorem which shows that if the total costs associated with the integration of the supply chain networks of the two firms are identically equal to zero, then the associated synergy can never be negative.

**Theorem 3.***If the total cost functions associated with the integration/merger links from node 0 to nodes A and B for each product are identically equal to zero, then the associated synergy, S*^{TC}, can never be negative.

*Proof*. We first note that the pre-integration supply chain optimization problem can be defined over the same expanded network as in Fig. 2 but with the cross-shipment links extracted and with the paths defined from node 0 to the retail nodes. In addition, the total costs from node 0 to nodes *A* and *B* must all be equal to zero. Clearly, the total cost minimization solution to this problem yields the same total cost value as obtained for *TC*^{0}. We must now show that *TC*^{0}−*TC*^{1}0.

Assume not, that is, that *TC*^{0}−*TC*^{1}<0, then, clearly, we have not obtained an optimal solution to the post-integration problem, since, the new links need not be used, which would imply that *TC*^{0}=*TC*^{1}, which is a contradiction.

Another interpretation of this theorem is that, in the system-optimization context (assuming that the total cost functions remain the same as do the demands), the addition of new links can never make the total cost increase; this is in contrast to what may occur in the context of user-optimized networks, where the addition of a new link may make everyone worse-off in terms of user cost. This is the well-known Braess paradox (1968); see, also, Braess et al. (2005).

### 4. Numerical examples

- Top of page
- Abstract
- 1. Introduction
- 2. The pre- and post-integration multi-product supply chain network models
- 3. Quantifying synergy associated with multi-product supply chain network integration
- 4. Numerical examples
- 5. Summary and conclusions
- Acknowledgments
- References

In this section, we present numerical examples for which we compute the solutions to the supply chains both pre and post the integration, along with the associated total costs and synergies as defined in Section 3. The examples were solved using the modified projection method (see, e.g., Korpelevich, 1977; Nagurney, 2009) embedded with the equilibration algorithm (cf. Dafermos and Sparrow, 1969; Nagurney, 1984). The modified projection method is guaranteed to converge if the function that enters the variational inequality is monotone and Lipschitz continuous (provided that a solution exists). Both these assumptions are satisfied under the conditions imposed on the multi-product total cost functions in Section 2 as well as by the total cost functions underlying the numerical examples below. Since we also assume that the feasible sets are non-empty, we are guaranteed that the modified projection method will converge to a solution of variational inequalities (7) and (13).

We implemented the computational procedure in FORTRAN and utilized a Unix system at the University of Massachusetts Amherst for the computations. The algorithm was considered to have converged when the absolute value of the difference between the computed values of the variables (the link flows; respectively, the Lagrange multipliers) at two successive iterations differed by no more than 10^{−5}. In order to fully exploit the underlying network structure, we first converted the multi-product supply chain networks, into single-product “extended” ones as discussed in Dafermos (1973) for multimodal/multi-class traffic networks. The link capacity constraints, which do not explicitly appear in the original traffic network models, were adapted accordingly. The modified projection method yielded subproblems, at each iteration, in flow variables and in price variables. The former were computed using the equilibration algorithm of Dafermos and Sparrow (1969) and the latter were computed explicitly and in closed form.

For all the numerical examples, we assumed that each firm *i; i*=*A*, *B*, was involved in the production, storage, and distribution of two products, and each firm had, before the integration/merger, two manufacturing plants, one distribution center, and supplied the products to two retail outlets.

After the integration of the two firms' supply chain networks, each retailer was indifferent as to which firm supplied the products and the integrated/merged firms could store the products at any of the two distribution centers and could supply any of the four retailers. Figure 3 depicts the pre-integration supply chain network(s), whereas Fig. 4 depicts the post-integration supply chain network for the numerical examples.

For all the examples, we assumed that the pre-integration total cost functions and the post-integration total cost functions were non-linear (quadratic), of the form:

- (15)

with convexity of the total cost functions being satisfied (except, where noted, for the top-most merger links from node 0).

**Example 1.** Example 1 served as the baseline for our computations. Example 1 data are now described. The pre- and post-integration total cost functions for products 1 and 2 are listed in Table 1. The links post-integration that join the node 0 with nodes *A* and *B* had associated total costs equal to zero for each product *j*=1, 2, for Examples 1–3. The demands at the retail outlets for firm *A* and firm *B* were set to 5 for each product. Hence, for *i*=*A*, *B; j*=1, 2, and *k*=1, 2. The capacity on each link was set to 25 both *pre-* and *post*-integration, so that: *u*_{a}=25 for all links *a*∈*L*^{0}; *a*∈*L*^{1}. The weights: *α*_{j}=1 were set to 1 for both products *j*=1, 2, both pre- and post-integration; thus, we assumed that the products are equal in volume.

Table 1. Definition of links and associated total cost functions for Example 1 Link *a* | From Node | To Node | | |
---|

1 | *A* | *M*_{1}^{A} | 1(*f*_{1}^{1})^{2}+2*f*_{1}^{2}*f*_{1}^{1}+11*f*_{1}^{1} | 2(*f*_{1}^{2})^{2}+2*f*_{1}^{1}*f*_{1}^{2}+8*f*_{1}^{2} |

2 | *A* | *M*_{2}^{A} | 2(*f*_{2}^{1})^{2}+2*f*_{2}^{2}*f*_{2}^{1}+8*f*_{2}^{1} | 1(*f*_{2}^{2})^{2}+2*f*_{2}^{1}*f*_{2}^{2}+6*f*_{2}^{2} |

3 | *M*_{1}^{A} | *D*_{1,1}^{A} | 3(*f*_{3}^{1})^{2}+2.5*f*_{3}^{2}*f*_{3}^{1}+7*f*_{3}^{1} | 4(*f*_{3}^{2})^{2}+2.5*f*_{3}^{1}*f*_{3}^{2}+7*f*_{3}^{2} |

4 | *M*_{2}^{A} | *D*_{1,1}^{A} | 4(*f*_{4}^{1})^{2}+1.5*f*_{4}^{2}*f*_{4}^{1}+3*f*_{4}^{1} | 3(*f*_{4}^{2})^{2}+1.5*f*_{4}^{1}*f*_{4}^{2}+11*f*_{4}^{2} |

5 | *D*_{1,1}^{A} | *D*_{1,2}^{A} | 1(*f*_{5}^{1})^{2}+*f*_{5}^{2}*f*_{5}^{1}+6*f*_{5}^{1} | 4(*f*_{5}^{2})^{2}+*f*_{5}^{1}*f*_{5}^{2}+11*f*_{5}^{2} |

6 | *D*_{1,2}^{A} | *R*_{1}^{A} | 3(*f*_{6}^{1})^{2}+1.5*f*_{6}^{2}*f*_{6}^{1}+4*f*_{6}^{1} | 4(*f*_{6}^{2})^{2}+1.5*f*_{6}^{1}*f*_{6}^{2}+10*f*_{6}^{2} |

7 | *D*_{1,2}^{A} | *R*_{2}^{A} | 4(*f*_{7}^{1})^{2}+2*f*_{7}^{2}*f*_{7}^{1}+7*f*_{7}^{1} | 2(*f*_{7}^{2})^{2}+2*f*_{7}^{1}*f*_{7}^{2}+8*f*_{7}^{2} |

8 | B | *M*_{1}^{B} | 4(*f*_{8}^{1})^{2}+3*f*_{8}^{2}*f*_{8}^{1}+5*f*_{8}^{1} | 4(*f*_{8}^{2})^{2}+3*f*_{8}^{1}*f*_{8}^{2}+6*f*_{8}^{2} |

9 | B | *M*_{2}^{B} | 1(*f*_{9}^{1})^{2}+1.5*f*_{9}^{2}*f*_{9}^{1}+4*f*_{9}^{1} | 4(*f*_{9}^{2})^{2}+1.5*f*_{9}^{1}*f*_{9}^{2}+6*f*_{9}^{2} |

10 | *M*_{1}^{B} | *D*_{1,1}^{B} | 2(*f*_{10}^{1})^{2}+3*f*_{10}^{2}*f*_{10}^{1}+3.5*f*_{10}^{1} | 3(*f*_{10}^{2})^{2}+3*f*_{10}^{1}*f*_{10}^{2}+4*f*_{10}^{2} |

11 | *M*_{2}^{B} | *D*_{1,1}^{B} | 1(*f*_{11}^{1})^{2}+2.5*f*_{11}^{2}*f*_{11}^{1}+4*f*_{11}^{1} | 4(*f*_{11}^{2})^{2}+2.5*f*_{11}^{1}*f*_{11}^{2}+5*f*_{11}^{2} |

12 | *D*_{1,1}^{B} | *D*_{1,2}^{B} | 4(*f*_{12}^{1})^{2}+3*f*_{12}^{2}*f*_{12}^{1}+6*f*_{12}^{1} | 2(*f*_{12}^{2})^{2}+3*f*_{12}^{1}*f*_{12}^{2}+5*f*_{12}^{2} |

13 | *D*_{1,2}^{B} | *R*_{1}^{B} | 3(*f*_{13}^{1})^{2}+3*f*_{13}^{2}*f*_{13}^{1}+7*f*_{13}^{1} | 4(*f*_{13}^{2})^{2}+3*f*_{13}^{1}*f*_{13}^{2}+10*f*_{13}^{2} |

14 | *D*_{1,2}^{B} | *R*_{2}^{B} | 4(*f*_{14}^{1})^{2}+0.5*f*_{14}^{2}*f*_{14}^{1}+4*f*_{14}^{1} | 4(*f*_{14}^{2})^{2}+0.5*f*_{14}^{1}*f*_{14}^{2}+12*f*_{14}^{2} |

15 | *M*_{1}^{A} | *D*_{1,1}^{B} | 4(*f*_{15}^{1})^{2}+2*f*_{15}^{2}*f*_{15}^{1}+6*f*_{15}^{1} | 4(*f*_{15}^{2})^{2}+2*f*_{15}^{1}*f*_{15}^{2}+7*f*_{15}^{2} |

16 | *M*_{2}^{A} | *D*_{1,1}^{B} | 4(*f*_{16}^{1})^{2}+2*f*_{16}^{2}*f*_{16}^{1}+6*f*_{16}^{1} | 3(*f*_{16}^{2})^{2}+2*f*_{16}^{1}*f*_{16}^{2}+7*f*_{16}^{2} |

17 | *M*_{1}^{B} | *D*_{1,1}^{A} | 1(*f*_{17}^{1})^{2}+3.5*f*_{17}^{2}*f*_{17}^{1}+4*f*_{17}^{1} | 4(*f*_{17}^{2})^{2}+3.5*f*_{17}^{1}*f*_{17}^{2}+5*f*_{17}^{2} |

18 | *M*_{2}^{B} | *D*_{1,1}^{A} | 4(*f*_{18}^{1})^{2}+3*f*_{18}^{2}*f*_{18}^{1}+9*f*_{18}^{1} | 4(*f*_{18}^{2})^{2}+3*f*_{18}^{1}*f*_{18}^{2}+ 7*f*_{18}^{2} |

19 | *D*_{1,2}^{A} | *R*_{1}^{B} | 4(*f*_{19}^{1})^{2}+3.5*f*_{19}^{2}*f*_{19}^{1}+7*f*_{19}^{1} | 1(*f*_{19}^{2})^{2}+3.5*f*_{19}^{1}*f*_{19}^{2}+9*f*_{19}^{2} |

20 | *D*_{1,2}^{A} | *R*_{2}^{B} | 2(*f*_{20}^{1})^{2}+3*f*_{20}^{2}*f*_{20}^{1}+5*f*_{20}^{1} | 4(*f*_{20}^{2})^{2}+3*f*_{20}^{1}*f*_{20}^{2}+6*f*_{20}^{2} |

21 | *D*_{1,2}^{B} | *R*_{1}^{A} | 4(*f*_{21}^{1})^{2}+2.5*f*_{21}^{2}*f*_{21}^{1}+3*f*_{21}^{1} | 3(*f*_{21}^{2})^{2}+2.5*f*_{21}^{1}*f*_{21}^{2}+9*f*_{21}^{2} |

22 | *D*_{1,2}^{B} | *R*_{2}^{A} | 3(*f*_{22}^{1})^{2}+2*f*_{22}^{2}*f*_{22}^{1}+4*f*_{22}^{1} | 4(*f*_{22}^{2})^{2}+2*f*_{22}^{1}*f*_{22}^{2}+3*f*_{22}^{2} |

The pre-integration optimal solutions for the product flows for each product for Examples 1–3 are given in Table 2. We note that Example 1, pre-integration, was used as the basis from which variants post-integration were constructed, yielding Examples 2 and 3, as described below.

Table 2. Pre-integration optimal product flow solutions to Examples 1–3 Link *a* | From Node | To Node | | |
---|

1 | *A* | *M*_{1}^{A} | 8.50 | 0.80 |

2 | *A* | *M*_{2}^{A} | 1.50 | 9.20 |

3 | *M*_{1}^{A} | *D*_{1,1}^{A} | 8.50 | 0.80 |

4 | *M*_{2}^{A} | *D*_{1,1}^{A} | 1.50 | 9.20 |

5 | *D*_{1,1}^{A} | *D*_{1,2}^{A} | 10.00 | 10.00 |

6 | *D*_{1,2}^{A} | *R*_{1}^{A} | 5.00 | 5.00 |

7 | *D*_{1,2}^{A} | *R*_{2}^{A} | 5.00 | 5.00 |

8 | *B* | *M*_{1}^{B} | 0.00 | 8.03 |

9 | *B* | *M*_{2}^{B} | 10.00 | 1.97 |

10 | *M*_{1}^{B} | *D*_{1,1}^{B} | 0.00 | 8.03 |

11 | *M*_{2}^{B} | *D*_{1,1}^{B} | 10.00 | 1.97 |

12 | *D*_{1,1}^{B} | *D*_{1,2}^{B} | 10.00 | 10.00 |

13 | *D*_{1,2}^{B} | *R*_{1}^{B} | 5.00 | 5.00 |

14 | *D*_{1,2}^{B} | *R*_{2}^{B} | 5.00 | 5.00 |

The post-integration optimal solutions are reported in Table 3 for product 1 and in Table 4 for product 2.

Table 3. Post-integration optimal flow solutions to the examples for product 1 Link *a* | From Node | To Node | Ex. 1 | Ex. 2 | Ex. 3 |
---|

1 | *A* | *M*_{1}^{A} | 5.94 | 0.76 | 5.36 |

2 | *A* | *M*_{2}^{A} | 0.53 | 0.00 | 1.98 |

3 | *M*_{1}^{A} | *D*_{1,1}^{A} | 5.94 | 0.00 | 5.36 |

4 | *M*_{2}^{A} | *D*_{1,1}^{A} | 0.53 | 0.00 | 1.98 |

5 | *D*_{1,1}^{A} | *D*_{1,2}^{A} | 18.27 | 19.24 | 17.34 |

6 | *D*_{1,2}^{A} | *R*_{1}^{A} | 5.00 | 5.00 | 5.00 |

7 | *D*_{1,2}^{B} | *R*_{2}^{A} | 3.27 | 4.24 | 4.27 |

8 | *B* | *M*_{1}^{B} | 6.25 | 1.67 | 5.00 |

9 | *B* | *M*_{2}^{B} | 7.29 | 17.57 | 7.66 |

10 | *M*_{1}^{B} | *D*_{1,1}^{B} | 0.00 | 0.00 | 0.00 |

11 | *M*_{2}^{B} | *D*_{1,1}^{B} | 1.73 | 0.00 | 2.66 |

12 | *D*_{1,1}^{B} | *D*_{1,2}^{B} | 1.73 | 0.76 | 2.66 |

13 | *D*_{1,2}^{B} | *R*_{1}^{B} | 0.00 | 0.00 | 0.00 |

14 | *D*_{1,2}^{B} | *R*_{2}^{B} | 0.00 | 0.00 | 1.93 |

15 | *M*_{1}^{A} | *D*_{1,1}^{B} | 0.00 | 0.76 | 0.00 |

16 | *M*_{2}^{A} | *D*_{1,1}^{B} | 0.00 | 0.00 | 0.00 |

17 | *M*_{1}^{B} | *D*_{1,1}^{A} | 6.25 | 1.67 | 5.00 |

18 | *M*_{2}^{B} | *D*_{1,1}^{A} | 5.55 | 17.57 | 5.00 |

19 | *D*_{1,2}^{A} | *R*_{1}^{B} | 5.00 | 5.00 | 5.00 |

20 | *D*_{1,2}^{A} | *R*_{2}^{B} | 5.00 | 5.00 | 3.07 |

21 | *D*_{1,2}^{B} | *R*_{1}^{A} | 0.00 | 0.00 | 0.00 |

22 | *D*_{1,2}^{B} | *R*_{2}^{A} | 1.73 | 0.76 | 0.73 |

Table 4. Post-integration optimal flow solutions to the examples for product 2 Link *a* | From node | To node | Ex. 1 | Ex. 2 | Ex. 3 |
---|

1 | *A* | *M*_{1}^{A} | 3.44 | 4.66 | 5.00 |

2 | *A* | *M*_{2}^{A} | 11.81 | 11.88 | 8.74 |

3 | *M*_{1}^{A} | *D*_{1,1}^{A} | 0.00 | 0.88 | 0.00 |

4 | *M*_{2}^{A} | *D*_{1,1}^{A} | 4.91 | 0.48 | 3.74 |

5 | *D*_{1,1}^{A} | *D*_{1,2}^{A} | 4.91 | 4.82 | 3.74 |

6 | *D*_{1,2}^{A} | *R*_{1}^{A} | 1.52 | 0.00 | 0.61 |

7 | *D*_{1,2}^{A} | *R*_{2}^{A} | 2.58 | 0.00 | 1.20 |

8 | *B* | *M*_{1}^{B} | 2.34 | 3.46 | 3.58 |

9 | *B* | *M*_{2}^{B} | 2.42 | 0.00 | 2.68 |

10 | *M*_{1}^{B} | *D*_{1,1}^{B} | 2.34 | 0.00 | 3.58 |

11 | *M*_{2}^{B} | *D*_{1,1}^{B} | 2.42 | 0.00 | 2.68 |

12 | *D*_{1,1}^{B} | *D*_{1,2}^{B} | 15.09 | 15.18 | 16.26 |

13 | *D*_{1,2}^{B} | *R*_{1}^{B} | 4.88 | 2.72 | 5.00 |

14 | *D*_{1,2}^{B} | *R*_{2}^{B} | 4.30 | 2.46 | 3.07 |

15 | *M*_{1}^{A} | *D*_{1,1}^{B} | 3.44 | 3.78 | 5.00 |

16 | *M*_{2}^{A} | *D*_{1,1}^{B} | 6.89 | 11.40 | 5.00 |

17 | *M*_{1}^{B} | *D*_{1,1}^{A} | 0.00 | 3.46 | 0.00 |

18 | *M*_{2}^{B} | *D*_{1,1}^{A} | 0.00 | 0.00 | 0.00 |

19 | *D*_{1,2}^{A} | *R*_{1}^{B} | 0.12 | 2.28 | 0.00 |

20 | *D*_{1,2}^{A} | *R*_{2}^{B} | 0.70 | 2.54 | 1.93 |

21 | *D*_{1,2}^{B} | *R*_{1}^{A} | 3.48 | 5.00 | 4.39 |

22 | *D*_{1,2}^{B} | *R*_{2}^{A} | 2.42 | 5.00 | 3.80 |

Since none of the link flow capacities were reached, either pre- or post-integration, the vectors and had all their components equal to zero. The total cost, pre-merger, *TC*^{0}=5702.58. The total cost, post-merger, *TC*^{1}=4240.86. Please also refer to Table 5 for the total cost and synergy values for this example as well as for the next two examples. The synergy *S*^{TC} for the supply chain network integration for Example 1 was =25.63%.

Table 5. Total costs and synergy values for the examples Measure | Example 1 | Example 2 | Example 3 |
---|

Pre-integration *TC*^{0} | 5702.58 | 5702.58 | 5702.58 |

Post-integration *TC*^{1} | 4240.86 | 2570.27 | 3452.34 |

Synergy calculations *S*^{TC} | 25.63% | 54.93% | 39.46% |

It is interesting to note that, since the distribution center associated with the original firm *A* has total storage costs that are lower for product 1, whereas firm *B*'s distribution center has lower costs associated with the storage of product 2, that firm *A*'s original distribution center, after the integration/merger, stores the majority of the volume of product 1, while the majority of the volume of product 2 is stored, post-integration, at firm *B*'s original distribution center. It is also interesting to note that, post-integration, the majority of the production of product 1 takes place in firm *B*'s original manufacturing plants, whereas the converse holds true for product 2. This example, hence, vividly illustrates the types of supply chain cost gains that can be achieved in the integration of multi-product supply chains.

**Example 2.** Example 2 was constructed from Example 1 but with the following modifications. We now considered an idealized situation in which we assumed that the total costs associated with the new integration links (see Table 1, links 15–22) for each product were identically equal to zero.

Post-integration, the optimal flow for each product, for each firm, has now changed; see Tables 3 and 4. It is interesting to note that now the second manufacturing plant associated with the original firm *B* produces the majority of product 1 but the majority of product 1 is still stored at the original distribution center of firm *A*. Indeed, the zero costs associated with distribution between the original supply chain networks lead to further synergies as compared with those obtained for Example 1.

Since, again, none of the link flow capacities were reached, either pre- or post-integration, the vectors and had all their components equal to zero. The total cost, post-merger, *TC*^{1}=2570.27. The synergy *S*^{TC} for the supply chain network integration for Example 2 was 54.93%. Observe that this obtained synergy is, in a sense, the maximum possible for this example since the total costs for both products on all the new links are all equal to zero.

**Example 3.** Example 3 was constructed from Example 2 but with the following modifications. We now assumed that the capacities associated with the links that had zero costs between the two original firms had their capacities reduced from 25 to 5. The computed optimal flow solutions are given in Table 3 for product 1 and in Table 4 for product 2.

The total cost, post-merger, was now *TC*^{1}=3452.34. The synergy *S*^{TC} for the supply chain network integration for Example 3 was 39.46%. Hence, even with substantially lower capacities on the new links, given the zero costs, the synergy associated with the supply chain network integration in Example 3 was quite high, although not as high as obtained in Example 2.

Firm *B*'s original distribution center now stores more of product 1 and 2 than it did in Example 2 (post-integration). Also, because of capacity reductions associated with the cross-shipment links there is a notable reduction in the volume of shipment of product 1 from the second manufacturing plant of firm *B* to firm *A*'s original distribution center and in the shipment of product 2 from firm *A*'s original second manufacturing plant to firm *B*'s original distribution center.

#### 4.1. Additional computations/examples

We then proceeded to ask the following question: assuming that the links, post-merger, joining node 0 to nodes *A* and *B* no longer had zero associated total cost for each product but, rather, reflected a cost associated with merging the two firms. We further assumed that the cost (cf. (15)) was linear and of the specific form given by

for the upper-most links (cf. Fig. 4). Hence, we assumed that all the terms were identical and equal to an *h*. At what value would the synergy then for Examples 1, 2, and 3 become negative? Through computational experiments we were able to determine these values. In the case of Example 1, if *h*=36.52, then the synergy value would be approximately equal to zero since the new total cost would be approximately equal to *TC*^{0}=5702.58. For any value larger than the above *h*, one would obtain negative synergy. This has clear implications for mergers in terms of supply chain network integration and demonstrates that the total costs associated with the integration/merger itself have to be carefully weighed against the cost benefits associated with the integrated supply chain activities. In the case of Example 2, the *h* value was approximately 78.3. A higher value than this *h* for each such merger link would result in the total cost exceeding *TC*^{0} and, hence, negative synergy would result.

Finally, for completeness, we also determined the corresponding *h* in the case of Example 3 and found the value to be *h*=78.3, as in Example 2.

### 5. Summary and conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. The pre- and post-integration multi-product supply chain network models
- 3. Quantifying synergy associated with multi-product supply chain network integration
- 4. Numerical examples
- 5. Summary and conclusions
- Acknowledgments
- References

In this paper, we developed multi-product supply chain network models, which allow one to evaluate the total costs associated with manufacturing/production, storage, and distribution of firms' supply chains both pre- and post-integration. Such horizontal integrations can take place, for example, in the context of M&A, an activity which has garnered much interest and momentum recently. The model(s) utilize a system-optimization perspective and allow for explicit upper bounds on the various links associated with manufacturing, storage, and distribution. The models are formulated and solved as variational inequality problems.

In addition, we utilized a proposed multi-product synergy measure to identify the potential cost gains associated with such horizontal supply chain network integrations. We proved that, in the case of zero “merging” costs, that the associated synergy can never be negative. We computed solutions to several numerical examples for which we determine the optimal product flows and Lagrange multipliers/shadow prices associated with the capacity constraints both before and after the integration. The computational approach allows one to explore many issues regarding supply chain network integration and to effectively ascertain the synergies before any implementation of a potential merger. In addition, we determined, computationally, for several examples, what identical linear costs would yield zero synergy, with higher values resulting in negative synergy.

There are numerous questions that remain and that will be considered for future research. It would be interesting to develop competitive variants of the models in a game theoretic context and to also explore elastic demands. Also, this paper does consider the time dimension in that it models the supply chain networks before and after the proposed merger and, hence, it considers two distinct points in time. For certain applications it may be useful to have a more detailed time discretization with accompanying network structure. Finally, it would be very interesting to explicitly incorporate the risks associated with supply chain network integration within our framework.