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Keywords:

  • multi-product supply chains;
  • horizontal integration;
  • mergers and acquisitions;
  • total cost minimization;
  • synergy

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The pre- and post-integration multi-product supply chain network models
  5. 3. Quantifying synergy associated with multi-product supply chain network integration
  6. 4. Numerical examples
  7. 5. Summary and conclusions
  8. Acknowledgments
  9. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The pre- and post-integration multi-product supply chain network models
  5. 3. Quantifying synergy associated with multi-product supply chain network integration
  6. 4. Numerical examples
  7. 5. Summary and conclusions
  8. Acknowledgments
  9. 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.

2. The pre- and post-integration multi-product supply chain network models

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

This section develops the pre- and post-integration supply chain network multi-product models using a system-optimization approach (based on the Dafermos (1973) multi-class model) but with the inclusion of explicit capacities on the various links. Moreover, here, we provide a variational inequality formulation of multi-product supply chains and their integration, which enables a computational approach which fully exploits the underlying network structure. We also identify the supply chain network structures both before and after the merger and construct a synergy measure.

Section 2.1 describes the underlying pre-integration supply chain network associated with an individual firm and its respective economic activities of manufacturing, storage, distribution, and retailing. Section 2.2 develops the post-integration model. The models are extensions of the Nagurney (2009) models to the more complex, and richer, multi-product domain.

2.1. The pre-integration multi-product supply chain network model

We first formulate the pre-integration multi-product decision-making optimization problem faced by firms A and B and we refer to this model as Case 0. We assume that each firm is represented as a network of its supply chain activities, as depicted in Fig. 1. Each firm i; i=AB, has inline image manufacturing facilities; inline image distribution centers, and serves inline image retail outlets. Let inline image denote the graph consisting of nodes [Ni] and directed links [Li] representing the supply chain activities associated with each firm i; i=AB. Let L0 denote the links: LALB as in Fig. 1. We assume that each firm is involved in the production, storage, and distribution of J products, with a typical product denoted by j.

image

Figure 1.  Supply chains of firms A and B before the integration.

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The links from the top-tiered nodes i; i=AB, in each network in Fig. 1 are connected to the manufacturing nodes of the respective firm i, which are denoted, respectively, by: inline image. These links represent the manufacturing links. The links from the manufacturing nodes, in turn, are connected to the distribution center nodes of each firm i; i=AB, which are denoted by inline image. These links correspond to the shipment links between the manufacturing facilities and the distribution centers where the products are stored. The links joining nodes inline image with nodes inline image for i=AB, correspond to the storage links for the products. Finally, there are shipment links joining the nodes inline image for i=AB with the retail nodes: inline image for each firm i=AB. Each firm i, for simplicity, and, without loss of generality, is assumed to have its own individual retail outlets for delivery of the products, as depicted in Fig. 1, before the integration.

The demands for the products are assumed as given and are associated with each product, and each firm and retail pair. Let inline image denote the demand for product j; j=1, …, J, at retail outlet inline image associated with firm i; i=AB; inline image. A path consists of a sequence of links originating at a node i; i=AB, and denotes supply chain activities comprising manufacturing, storage, and distribution of the products to the retail nodes. Let inline image denote the non-negative flow of product j, on path p. Let inline image denote the set of all paths joining an origin node i with (destination) retail node inline image. Clearly, since we are first considering the two firms before any integration, the paths associated with a given firm have no links in common with paths of the other firm. This changes (see also Nagurney, 2009) when the integration occurs, in which case the number of paths and the sets of paths also change, as do the number of links and the sets of links, as described in Section 2.2. The following conservation of flow equations must hold for each firm i, each product j, and each retail outlet inline image:

  • image(1)

that is, the demand for each product must be satisfied at each retail outlet.

Links are denoted by a, b, etc. Let inline image denote the flow of product j on link a. We must have the following conservation of flow equations satisfied:

  • image(2)

where δap=1 if link a is contained in path p and δap=0, otherwise. Here P0 denotes the set of all paths in Fig. 1, that is, inline image. The path flows must be non-negative, that is,

  • image(3)

We group the path flows into the vector x.

Note that the different products flow on the supply chain networks depicted in Fig. 1 and share resources with one another. To capture the costs, we proceed as follows. There is a total cost associated with each product j; j=1, …, J, and each link (cf. Fig. 1) of the network corresponding to each firm i; i=AB. We denote the total cost on a link a associated with product j by inline image. The total cost of a link associated with a product, be it a manufacturing link, a shipment/distribution link, or a storage link is assumed to be a function of the flow of all the products on the link; see, for example, Dafermos (1973). Hence, we have that

  • image(4)

The top tier links in Fig. 1 have total cost functions associated with them that capture the manufacturing costs of the products; the second tier links have multi-product total cost functions associated with them that correspond to the total costs associated with the subsequent distribution/shipment to the storage facilities, and the third tier links, since they are the storage links, have associated with them multi-product total cost functions that correspond to storage. Finally, the bottom-tiered links, since they correspond to the shipment links to the retailers, have total cost functions associated with them that capture the costs of shipment of the products.

We assume that the total cost function for each product on each link is convex, continuously differentiable, and has a bounded third order partial derivative. Since the firms' supply chain networks, pre-integration, have no links in common (cf. Fig. 1), their individual cost minimization problems can be formulated jointly as follows:

  • image(5)

subject to: constraints (1)–(3) and the following capacity constraints:

  • image(6)

The term αj denotes the volume taken up by product j, whereas ua denotes the non-negative capacity of link a.

Observe that this problem is, as is well known in the transportation literature (cf. Beckmann et al., 1956; Dafermos and Sparrow, 1969; Dafermos, 1973), a system-optimization problem but in capacitated form. Under the above-imposed assumptions, the optimization problem is a convex optimization problem. If we further assume that the feasible set underlying the problem represented by the constraints (1)–(3) and (6) is non-empty, then it follows from the standard theory of non-linear programming (cf. Bazaraa et al., 1993) that an optimal solution exists.

Let K0 denote the set where inline image, where f is the vector of link flows. We assume that the feasible set K0 is non-empty. We associate the Lagrange multiplier βa with constraint (6) for each aL0. We denote the associated optimal Lagrange multiplier by inline image. This term may be interpreted as the price or value of an additional unit of capacity on link a; it is also sometimes referred to as the shadow price. We now provide the variational inequality formulation of the problem. For convenience, and since we are considering Case 0, we denote the solution of variational inequality (7) below as inline image and we refer to the corresponding vectors of variables with superscripts of 0.

Theorem 1.The vector of link flowsinline imageis an optimal solution to the pre-integration problem if and only if it satisfies the following variational inequality problem with the vector of non-negative Lagrange multipliersinline image:

  • image(7)

2.2. The post-integration multi-product supply chain network model

We now formulate the post-integration case, referred to as Case 1. Figure 2 depicts the post-integration supply chain network topology. Note that there is now a supersource node 0 which represents the integration of the firms in terms of their supply chain networks with additional links joining node 0 to nodes A and B, respectively.

image

Figure 2.  Supply chain network after firms A and B merge.

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As in the pre-integration case, the post-integration optimization problem is also concerned with total cost minimization. Specifically, we retain the nodes and links associated with the network depicted in Fig. 1 but now we add the additional links connecting the manufacturing facilities of each firm and the distribution centers of the other firm as well as the links connecting the distribution centers of each firm and the retail outlets of the other firm. We refer to the network in Fig. 2, underlying this integration, as G1=[N1L1], where N1N0∪ node 0 and L1L0∪ the additional links as in Fig. 2. We associate total cost functions as in (4) with the new links, for each product j. Note that if the total cost functions associated with the integration/merger links connecting node 0 to node A and node 0 to node B are set equal to zero, this means that the supply chain integration is costless in terms of the supply chain integration/merger of the two firms. Of course, non-zero total cost functions associated with these links may be utilized to also capture the risk associated with the integration. We will explore such issues numerically in Section 4.

A path p now (cf. Fig. 2) originates at the node 0 and is destined for one of the bottom retail nodes. Let inline image, in the post-integrated network configuration given in Fig. 2, denote the flow of product j on path p joining (origin) node 0 with a (destination) retail node. Then, the following conservation of flow equations must hold:

  • image(8)

where inline image denotes the set of paths connecting node 0 with retail node inline image in Fig. 2. Owing to the integration, the retail outlets can obtain each product j from any manufacturing facility, and any distributor. The set of paths inline image.

In addition, as before, let inline image denote the flow of product j on link a. Hence, we must also have the following conservation of flow equations satisfied:

  • image(9)

Of course, we also have that the path flows must be non-negative for each product j, that is,

  • image(10)

We assume, again, that the supply chain network activities have non-negative capacities, denoted as ua, ∀aL1, with αj representing the volume factor for product j. Hence, the following constraints must be satisfied:

  • image(11)

Consequently, the optimization problem for the integrated supply chain network is:

  • image(12)

subject to constraints: (8)–(11).

The solution to the optimization problem (12) subject to constraints (8) through (11) can also be obtained as a solution to a variational inequality problem akin to (7) where now aL1. The vectors f and β have identical definitions as before, but are re-dimensioned/expanded accordingly and superscripted with a 1. Finally, instead of the feasible set K0 we now have inline image. We assume that K1 is non-empty. We denote the solution to the variational inequality problem (13) below governing Case 1 by inline image and denote the vectors of corresponding variables as (1β1). We now, for completeness, provide the variational inequality formulation of the Case 1 problem. The proof is immediate.

Theorem 2.The vector of link flowsinline imageis an optimal solution to the post-integration problem if and only if it satisfies the following variational inequality problem with the vector of non-negative Lagrange multipliersinline image:

  • image(13)

We let TC0 denote the total cost, inline image, evaluated under the solution inline image to (7) and we let TC1, inline image denote the total cost evaluated under the solution inline image to (13). Owing to the similarity of variational inequalities (7) and (13) the same computational procedure can be utilized to compute the solutions. Indeed, we utilize the variational inequality formulations of the respective pre- and post-integration supply chain network problems since we can then exploit the simplicity of the underlying feasible sets K0 and K1 which include constraints with a network structure identical to that underlying multimodal system-optimized transportation network problems.

It is worthwhile to distinguish the multi-product supply chain network models developed above from the single product models in Nagurney (2009). First, we note that the total cost functions in the objective functions (5) and (12) are not separable as they were, respectively, in the single product models in Nagurney (2009). In addition, since we are dealing now with multiple products, which can be of different physical dimensions, the corresponding capacity constraints [cf. (6) and (11)] are also more complex than was the case for their single product counterparts. We also emphasize that the above multi-product framework contains, as a special case, the merger of firms that produce (pre-merger) distinct products, which is captured by assigning a demand of zero to those products at the respective demand markets. Of course, in such a case, the total cost functions would also be adapted accordingly.

Finally, the multi-product models developed in this paper allow for non-zero total costs associated with the top-most merger links (cf. Fig. 2), which join node 0 to nodes A and B. In Nagurney (2009), it was assumed that the corresponding total costs, in the single product case, were zero. Of course, it would also be interesting to explore the issue of “retooling” a manufacturing facility, post-merger, for it to be able to produce the other firm's product(s) in its original manufacturing facilities.

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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The pre- and post-integration multi-product supply chain network models
  5. 3. Quantifying synergy associated with multi-product supply chain network integration
  6. 4. Numerical examples
  7. 5. Summary and conclusions
  8. Acknowledgments
  9. 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 STC, can be calculated as the percentage difference between the total cost pre vs the total cost post the integration:

  • image(14)

From (14), one can see that the lower the total cost TC1, 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 TC1 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, STC, 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 TC0. We must now show that TC0TC1geqslant R: gt-or-equal, slanted0.

Assume not, that is, that TC0TC1<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 TC0=TC1, 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

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

image

Figure 3.  Pre-integration supply chain network topology for the numerical examples.

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image

Figure 4.  Post-integration supply chain network topology for the examples.

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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:

  • image(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, inline image for i=AB; j=1, 2, and k=1, 2. The capacity on each link was set to 25 both pre- and post-integration, so that: ua=25 for all links aL0; aL1. The weights: αj=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 aFrom NodeTo Nodeinline imageinline image
1AM1A1(f11)2+2f12f11+11f112(f12)2+2f11f12+8f12
2AM2A2(f21)2+2f22f21+8f211(f22)2+2f21f22+6f22
3M1AD1,1A3(f31)2+2.5f32f31+7f314(f32)2+2.5f31f32+7f32
4M2AD1,1A4(f41)2+1.5f42f41+3f413(f42)2+1.5f41f42+11f42
5D1,1AD1,2A1(f51)2+f52f51+6f514(f52)2+f51f52+11f52
6D1,2AR1A3(f61)2+1.5f62f61+4f614(f62)2+1.5f61f62+10f62
7D1,2AR2A4(f71)2+2f72f71+7f712(f72)2+2f71f72+8f72
8BM1B4(f81)2+3f82f81+5f814(f82)2+3f81f82+6f82
9BM2B1(f91)2+1.5f92f91+4f914(f92)2+1.5f91f92+6f92
10M1BD1,1B2(f101)2+3f102f101+3.5f1013(f102)2+3f101f102+4f102
11M2BD1,1B1(f111)2+2.5f112f111+4f1114(f112)2+2.5f111f112+5f112
12D1,1BD1,2B4(f121)2+3f122f121+6f1212(f122)2+3f121f122+5f122
13D1,2BR1B3(f131)2+3f132f131+7f1314(f132)2+3f131f132+10f132
14D1,2BR2B4(f141)2+0.5f142f141+4f1414(f142)2+0.5f141f142+12f142
15M1AD1,1B4(f151)2+2f152f151+6f1514(f152)2+2f151f152+7f152
16M2AD1,1B4(f161)2+2f162f161+6f1613(f162)2+2f161f162+7f162
17M1BD1,1A1(f171)2+3.5f172f171+4f1714(f172)2+3.5f171f172+5f172
18M2BD1,1A4(f181)2+3f182f181+9f1814(f182)2+3f181f182+ 7f182
19D1,2AR1B4(f191)2+3.5f192f191+7f1911(f192)2+3.5f191f192+9f192
20D1,2AR2B2(f201)2+3f202f201+5f2014(f202)2+3f201f202+6f202
21D1,2BR1A4(f211)2+2.5f212f211+3f2113(f212)2+2.5f211f212+9f212
22D1,2BR2A3(f221)2+2f222f221+4f2214(f222)2+2f221f222+3f222

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 aFrom NodeTo Nodeinline imageinline image
1AM1A8.500.80
2AM2A1.509.20
3M1AD1,1A8.500.80
4M2AD1,1A1.509.20
5D1,1AD1,2A10.0010.00
6D1,2AR1A5.005.00
7D1,2AR2A5.005.00
8BM1B0.008.03
9BM2B10.001.97
10M1BD1,1B0.008.03
11M2BD1,1B10.001.97
12D1,1BD1,2B10.0010.00
13D1,2BR1B5.005.00
14D1,2BR2B5.005.00

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 aFrom NodeTo NodeEx. 1 inline imageEx. 2 inline imageEx. 3 inline image
1AM1A5.940.765.36
2AM2A0.530.001.98
3M1AD1,1A5.940.005.36
4M2AD1,1A0.530.001.98
5D1,1AD1,2A18.2719.2417.34
6D1,2AR1A5.005.005.00
7D1,2BR2A3.274.244.27
8BM1B6.251.675.00
9BM2B7.2917.577.66
10M1BD1,1B0.000.000.00
11M2BD1,1B1.730.002.66
12D1,1BD1,2B1.730.762.66
13D1,2BR1B0.000.000.00
14D1,2BR2B0.000.001.93
15M1AD1,1B0.000.760.00
16M2AD1,1B0.000.000.00
17M1BD1,1A6.251.675.00
18M2BD1,1A5.5517.575.00
19D1,2AR1B5.005.005.00
20D1,2AR2B5.005.003.07
21D1,2BR1A0.000.000.00
22D1,2BR2A1.730.760.73
Table 4.  Post-integration optimal flow solutions to the examples for product 2
Link aFrom nodeTo nodeEx. 1 inline imageEx. 2 inline imageEx. 3 inline image
1AM1A3.444.665.00
2AM2A11.8111.888.74
3M1AD1,1A0.000.880.00
4M2AD1,1A4.910.483.74
5D1,1AD1,2A4.914.823.74
6D1,2AR1A1.520.000.61
7D1,2AR2A2.580.001.20
8BM1B2.343.463.58
9BM2B2.420.002.68
10M1BD1,1B2.340.003.58
11M2BD1,1B2.420.002.68
12D1,1BD1,2B15.0915.1816.26
13D1,2BR1B4.882.725.00
14D1,2BR2B4.302.463.07
15M1AD1,1B3.443.785.00
16M2AD1,1B6.8911.405.00
17M1BD1,1A0.003.460.00
18M2BD1,1A0.000.000.00
19D1,2AR1B0.122.280.00
20D1,2AR2B0.702.541.93
21D1,2BR1A3.485.004.39
22D1,2BR2A2.425.003.80

Since none of the link flow capacities were reached, either pre- or post-integration, the vectors inline image and inline image had all their components equal to zero. The total cost, pre-merger, TC0=5702.58. The total cost, post-merger, TC1=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 STC for the supply chain network integration for Example 1 was =25.63%.

Table 5.  Total costs and synergy values for the examples
MeasureExample 1Example 2Example 3
Pre-integration TC05702.585702.585702.58
Post-integration TC14240.862570.273452.34
Synergy calculations STC25.63%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 inline image and inline image had all their components equal to zero. The total cost, post-merger, TC1=2570.27. The synergy STC 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.

We now also provide the computed vector of Lagrange multipliers inline image. All terms were equal to zero except those for links 15–20 since the sum of the corresponding product flows on each of these links was equal to the imposed capacity of 5. In particular, we now had: inline image, inline image, inline image, inline image, inline image, and inline image.

The total cost, post-merger, was now TC1=3452.34. The synergy STC 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

  • image

for the upper-most links (cf. Fig. 4). Hence, we assumed that all the inline image 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 TC0=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 TC0 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The pre- and post-integration multi-product supply chain network models
  5. 3. Quantifying synergy associated with multi-product supply chain network integration
  6. 4. Numerical examples
  7. 5. Summary and conclusions
  8. Acknowledgments
  9. 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.

Acknowledgments

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

This research was supported by the John F. Smith Memorial Fund at the Isenberg School of Management. This support is gratefully acknowledged. The authors would also like to thank Professor June Dong for helpful discussions. The authors also acknowledge the helpful comments and suggestions received during the review process.

References

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