a method for designing a strategy map using ahp and linear programming

A method for designing a strategy map using AHPand linear programming

Luis E. Quezada a,n, Héctor A. López-Ospina b

a Department of Industrial Engineering, Faculty of Engineering, University of Santiago of Chile (USACH), Avenida Ecuador 3769,Estación Central, Santiago, Chileb Faculty of Engineering and Applied Science, University of Los Andes, Monseñor Álvaro del Portillo 12445, Las Condes, Santiago, Chile

a r t i c l e i n f o

Article history:Received 20 January 2014Accepted 11 August 2014Available online 23 August 2014

Keywords:Balanced scorecardStrategy mapAnalytic Hierarchy ProcessLinear programming

a b s t r a c t

This paper presents a method to support the identification of the cause-effect relationships of strategicobjectives of a strategy map of a balanced scorecard. A strategy map contains the strategic objectives ofan organization, grouped into four perspectives (a) finances, (b) clients, (c) internal processes and(d) growth and learning, all of them linked through cause-effect relationships. The issue addressed inthis paper is the identification of those relationships, topic in which the existing literature is scarce. Aprevious work was revisited, which uses the Analytic Hierarchy Process (AHP) to establish the“importance” of the arcs (relationships) of a strategy map. That work then deletes those arcs with animportance lower than a given threshold level defined by the authors. This paper goes beyond byselecting the arcs using a multi-objective linear programming model (LP). The model has two objectives(a) to minimize the number of selected relationships and (b) to maximize the total importance of theselected relationships. It is interested to see that a trade-off between both objectives is produced, so acontrol variable is used to incorporate the importance given to both objectives by managers. The paperalso shows some applications of the method and their analysis.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The Balanced Scorecard (BSC) developed by Kaplan and Norton(1992, 1996) in the 1990s is a performance measurement systemthat has evolved to become a complete strategic managementsystem (Kaplan and Norton, 2001a, 2001b). The BSC has been verypopular among practitioners as well as in the literature (Gomeset al., 2004; Neely, 2005; Nudurupati et al., 2010).

The main contribution of the BSC is that it includes strategicobjectives and performance measures that are not solely financial.The BSC considers four perspectives, where strategic objectivesand performance measures are defined: (a) Financial, (b) Clients,(c) Internal processes, and (d) Learning and growth. There is acausal relationship among these perspectives: If the learning &growth perspective is improved, then the internal processesperspective will be improved. There is also a positive effect onclient's perspective which will ultimately have an impact on thefinancial perspective. Measures that are a result of past actions orevents and measures representing actions that will have an impactin the future are also included. The return on investment is an

example of the first situation while the amount invested inincreasing workers competences is an example of the second. Amore detailed description of those relationships is included inwhat is called a strategy map, in which strategic objectives areconnected to represent the causal relationship between them(Kaplan and Norton, 2004). Some authors such as Nørreklit(2000, 2003) and Bessire and Baker (2005) have criticized thevalidity of strategy maps, but Banker et al. (2011), through anexperiment with students enrolled in an MBA program, found thata strategy map had a positive impact in the effective use of a BSC.

As in this study, the Analytic Hierarchy Process (AHP) and itsextension, the Analytic Network Process (ANP) (Saaty, 2001, 2002),to model a Balanced Scorecard have been applied to model a BSCin many studies. One of the advantages of AHP and ANP is thatthey allow the combination of tangible and intangible factors,which is a characteristic of a BSC.

Examples of these applications are the studies carried out byLeung et al. (2006), who developed AHP and ANP models of a BSC,taking into account the time-dependency of the measure and theone developed by Sarkis (2003), who utilized ANP to model a BSCto establish all the factors influencing performance measures. Inthe same way, Yuan and Chiu (2011) used a case-based reasoning(CBR) system to obtain weights for the elements of a BSC. Theycompared their results with the results obtained using AHP. Theyargued that their method reached to more effective performance

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijpe

Int. J. Production Economics

http://dx.doi.org/10.1016/j.ijpe.2014.08.0080925-5273/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ56 2 27180090.E-mail addresses: [email protected] (L.E. Quezada),

[email protected] (H.A. López-Ospina).

Int. J. Production Economics 158 (2014) 244–255

www.sciencedirect.com/science/journal/09255273

www.elsevier.com/locate/ijpe

http://dx.doi.org/10.1016/j.ijpe.2014.08.008



http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijpe.2014.08.008&domain=pdf



mailto:[email protected]

mailto:[email protected]


measurements. Yüksel and Dagdeviren (2010) designed a fuzzyANP to model the BSC of a manufacturing company to evaluate theperformance of a business. Hsu et al. (2011) used an ANP toincorporate the sustainability issue in a BSC. Huang et al. (2011)used AHP to prioritize the strategic objectives of a BSC in apharmaceutical firm. Bentes et al. (2012) built an AHP model of aBSC of a company to evaluate the performance of three businessunits. Tjader et al. (2014) designed a BSC using an ANP for a casecompany, in which the alternatives of the model were IT out-sourcing strategies.

The issue addressed in this paper is how to identify the causalrelationships of a strategy map. Only a few papers were foundrelative to this subject. One of the papers is the work reported byKunc (2008), who used a method based on strategic thinking(Senge, 1999) to create a strategy map. Quezada et al. (2009)developed a method for designing this map based on the wayfirms design this map in practice. Other methods are quantitativeand use multi-criteria methods to identify the relationshipsbetween the strategic objectives. Yang et al. (2008) used theDecision Making Trial and Evaluation Laboratory (DEMATEL)technique to establish causal-effect relationships between policiesaffecting innovative companies. Jassbi et al. (2011) used fuzzyDEMATEL to model the relationships within a strategy map, eventhough they did not describe how to derive the strategy map intheir analysis. Wu (2012) used DEMATEL to create a strategy map.It is interested to note that in the case presented, some keyperformance indicators (KPI) have reciprocal feedbacks, i.e., theyaffect themselves, which is because DEMATEL is able to capturecomplex relationships that may exist in a BSC. This is because, thistool allows the identification of both direct and indirect impacts.Other authors such as Tsen (2010) and Chen et al. (2011) combinedANP and DEMATEL to model and analyze a BSC. These two authorsdo not explain how to obtain a strategy map from their analysis,even though both argue that it is possible to build one using theirproposed methods.

In all the papers found in the literature regarding BSC in whichAHP or ANP is used, the causal relationships are already pre-defined. However, the objective of the method presented in thispaper is to identify those relationships. When DEMATEL is used,authors have to specify a “threshold level” to establish when arelationship is important enough to be considered within astrategy map, something that is not required in the methodpresented here.

This work combines AHP and linear programming (LP) todesign a strategy map of a BSC. AHP is used to assign prioritiesto all the possible relationships within a strategy map and LP isused to select those that are “important” that will ultimately beincluded in the strategy map.

The combination of AHP/ANP and LP has been applied in manystudies, particularly to the subject of evaluation and selection ofsuppliers. AHP and ANP have been used mainly to prioritize thefactors affecting the decision, while LP has been used to optimizethe evaluation or selection, incorporating constraints or condi-tions that are necessary to meet at the same time (Ghodsypourand O’Brien, 1998; Amid et al., 2011; Ghodsypour and O’Brien,2001; Demirtas and Üstün, 2008; Demirtas and Ustun, 2009;Lin, 2012; Lin et al., 2011; Kokangul and Susuz, 2009; Shaw et al.,2012).

An important conclusion reported by Leung et al. (2006) isexpressed as “We show that the AHP and the ANP can be tailor-made for specific situations and can be used to overcome some ofthe traditional problems of BSC implementation”. This conclusionis important because, in practice, a strategy map does notnecessarily fit the structure of the four dimensions defined byKaplan and Norton (1992), so the method presented here can alsobe tailored to specific characteristics of a firm.

The work by Quezada and Quintero (2011) who used the AHPto model a strategy map serves as the basis for the methodproposed in this study. In their model, nodes represent strategicobjectives and directed arcs represent causal relationships. Theyuse the AHP to estimate the priority of each one of the arcs, andthen they select those arcs (for inclusion in the strategy map) thathave a priority higher than a given threshold value. This method issummarized in Appendix A. The contribution of this work is theutilization of a linear programming model to select the links for astrategy map.

2. The method

In general, the method uses the AHP to estimate the impor-tance of the relationships between the strategic objectives, andthen it selects those relationships that are important enough to beincluded in a strategy map. The purpose of the AHP depends onthe problem. It can be used to set up relationships and calculateprobabilities, etc., but it is most commonly used to identifypriorities. In this study, we use it to model relationships.

The proposed method has three main stages:

Stage 1: Modeling the strategy map as a hierarchical model.Stage 2: Estimating of the importance of each one of therelationships.Stage 3: Selecting the relationships which are “important”

Stage 1 and stage 2 are the same as the stages used by Quezadaand Quintero (2011) in their method. This paper proposes adifferent method for carrying out the third stage.

2.1. Stage 1

The first stage corresponds to the representation of a strategymap in the form of a hierarchical model. In that model, all thenodes between consecutive hierarchical levels are connectedamong them. Fig. 1 depicts an example of a hierarchical modelrepresenting a strategy map. Level 1, level 2, level 3 and level4 represent the financial, clients, internal processes and learning &growth perspectives, respectively, even though the levels may betailored to the situation of a specific company. A level 0 with onlyone node has been added for the purpose of the estimation of theimportance of the nodes within the level 1. The nodes represent

Level 0

Level l

Level 2

Level 3

Level 4

Fig. 1. Initial representation of a strategy map using a hierarchical model.

L.E. Quezada, H.A. López-Ospina / Int. J. Production Economics 158 (2014) 244–255 245

the strategic objectives. However, any perspective may includemore than one hierarchical level.

2.2. Stage 2

The second stage is the estimation of the importance of each ofthe relationships in the hierarchical model, which is accomplishedusing the AHP. This estimation corresponds to the priority (orimportance) of all the nodes at a given level in relation to eachnode of the preceding level.

Consider Fig. 2 that shows two consecutive levels of thehierarchical model: n and nþ1, n¼0, 1, 2, 3 and Jn; Jnþ1;¼a setof nodes in levels n and nþ1, respectively. Node i belongs to leveln and nodes j and k belong to level nþ1.

Using the AHP, all the nodes of level nþ1 are pairwisecompared to each node i. Nodes j and k are pairwise-comparedby answering the following question: How much more important isthe strategic objective j than the strategic objective k for achieving thestrategic objective i? The comparison is carried out using Saaty'sfundamental scale (Saaty, 1994) that ranges from 1 to 9, in which1 means that both objectives are equally important and 9 meansthat the first objective is extremely more important that thesecond one. The resulting values are input into a comparisonmatrix. The importance of every node of level nþ1 with regard toeach node i is obtained by computing the principal righteigenvector of the comparison matrix (Saaty, 1994). After this,the consistency ratio (CR) is calculated to check the consistency ofthe responses. This procedure is repeated for every node of level n.

The result of the described process is that the importance ofevery node of one level with regard to every node of the precedinglevel is quantified. Using the AHP, the global importance of everynode within each hierarchical level can also be computed.

Letwn

ij¼ importance of node j of level nþ1 in relation to node i oflevel n (obtained by pairwise comparison). 8 i є Jn; 8 j є Jnþ1

Clearly,

∑jA Jnþ 1

wnij ¼ 1 8 iє Jn ð1Þ

Letf ni ¼global importance of node i in level n 8 i є JnThe proposed method uses the local importance of the nodes,

which is computed as anij¼ local importance of node j of level nþ1in relation to node i of level n, 8 i є Jn; 8 j є Jnþ1

anij ¼wnij f

ni 8 i є Jn; 8 j є Jnþ1 ð2Þ

The local importance of node j (with regard to node i) is theimportance of node j, derived from the comparison matrix, multi-plied by the global importance of node i.

Note that

∑jA Jnþ 1

anij ¼ f ni 8 iє Jn ð3Þ

∑iA Jn

∑jA Jnþ 1

anij ¼ ∑iA Jn

f ni ¼ 1 ð4Þ

The level 0 has only one node, with 100% of importance.

Thus f 01 ¼ 1 ð5ÞThen, the sum of all local importance of the nodes of level

1 must be 1.0.

∑iA J1

a0i1 ¼ f 01 ¼ 1:0 ð6Þ

The linear programming model utilizes the parameter anij as themain input.

2.3. Stage 3

The third stage is the selection of the relationships that are“important”. Quezada and Quintero (2011) define a thresholdvalue to make this selection. In this way, if the importance of arelationship is higher than that value, then that relationship isselected to be included in the strategy map. A brief description ofthe method developed by Quezada and Quintero is presented inAppendix A.

The method presented here uses a linear programming modelto execute this stage.

3. The linear programming model

3.1. Overview

The input of the linear programming model is the importanceof every arc of the hierarchy of the AHP model. The objective is toselect those arcs that are “important”. Managers would like tohave few relationships; otherwise, the strategy map would not beuseful in representing the strategy of the organization. However,the selected arcs should account for a high importance. Obviously,these objectives are contradictory. In effect, the minimum numberof selected arcs would be achieved if none of the arcs is selected,and the maximum importance would be achieved if all of the arcsare selected. For this reason, the solution should be somethingbetween both extremes.

3.2. Decision variables

Ynij ¼

1; if the relationshipði; jÞbetween levelsðn;nþ1Þis selected

O; otherwise

(

dþn is the number of relationships selected from level n to level

nþ1. d�n is the difference between 1 and the accumulated local

importance of relationships selected from level n to level nþ1.

3.3. Parameters

anij is the local importance of node j of level nþ1 in relation tonode i in level n. cn;nþ1 is the total number of arcs between level nand level nþ1.

3.4. Constraints

Definition of dþn

∑ði;jÞ

Ynij�dþ

n ¼ 0 8 n ð7Þ

dþn is the number of selected arcs within a level. We would like

this variable to be minimum, because we want to minimize thenumber of selected arcs (i.e., the relationships to be included in thestrategy map).

i

kj

Level n(with p nodes)

Level n+1(with q nodes)

1 p

1 q

Fig. 2. Example of two hierarchical levels.

L.E. Quezada, H.A. López-Ospina / Int. J. Production Economics 158 (2014) 244–255246

Definition of d�n

∑ði;jÞ

anijYnijþd�

n ¼ 1 8 n ð8Þ

d�n is the difference between 1 (the maximum accumulated

importance of level n) and the accumulated importance of theselected arcs of level n. This variable should also be as low aspossible, which is equivalent to maximizing the accumulatedimportance of the arcs selected. This is because; one of theobjective is to include only the “most important” relationships inthe strategy map.

Feasibility conditions

∑kA Jnþ 1

YnikZ1; 8 i;n; Jnþ1 ¼ set of nodes of level nþ1 ð9Þ

∑iA Jn

YnikZ1; 8k;n; Jn ¼ set of nodes of level n ð10Þ

The first set of feasibility conditions specifies that each nodemust be linked to at least one node of the subsequent level.Obviously, these conditions do not apply for the lowest level. Inthe same way, the second set specifies that each node must beconnected to at least one node of the preceding level. Again, theseconditions do not apply for the first level (level 0).

These conditions come from the fact that each node (represent-ing a strategy objective) should affect some node of the upperlevel. On the other hand, each node has to be affected by at leastone node from the lower level.

Conditions on the decision variables

YnikAf0;1g; dþ

n ; d�n Z0 8 i; k;n ð11Þ

3.5. Objective function

min Z ¼ θ∑n

dþn

Cn;nþ1þð1�θÞ∑

nd�n ð12Þ

There are two objectives. The first objective is to reduce thenumber of relationships selected, and the second objective is toincrease the accumulated importance of those relationshipsselected. Both objectives go in different directions. If we want toincrease the number of relationships, then the accumulatedimportance of the relationships also increases, so it is not possibleto improve both at the same time. For this reason, a controlvariable θ is used to represent the importance given to theobjective of minimizing the number of relationships selected.

The variable dþn is divided by the total number of relationships

between levels ðcn;nþ1Þ. In this way, both objectives are on thesame scale (0 to 1), so they can be compared.

Clearly, if θ¼0, then the model will select only a few arcs tomeet what was called the feasibility condition. If θ¼1, then themodel will select all the arcs of the graph, which is somethingundesirable because the strategy of the company would not berepresented. What is interesting about this formulation is thatthrough the parametric variation of θ, the minimum number ofimportant arcs will be selected.

4. Illustrative example

4.1. The case

To illustrate this method, the same case described in Quezadaand Quintero (2011) is used. The company has the strategicobjectives shown in Table 1, grouped according to the perspectiveof a Balanced Scorecard. It is important to remember that Quezadaand Quintero (2011) modeled the strategy map for the company asa hierarchical model, and then they used the AHP to estimate thepriority (importance) of all the relationships. In this work, thesepriorities were used to build the linear programming model.Because the managers of the company were previously inter-viewed to obtain the priorities of the relationships, this step wasout of the scope of this investigation; so its description wasomitted.

Fig. 3 depicts the hierarchical model. The application of AHP inthe company had the results shown in Tables 2–5. The columns“Local Importance” are the input to the linear programming model(parameters aij

n).Table 2 indicates that Profits, Return on Investment and

Financial Stability have an importance of 0.55; 0.21 and 0.24,respectively within the Financial Perspective. Table 3 indicates thatthe local importance of Price, Delivery and Freshness in relation toProfits is 0.07, 0.39 and 0.09, respectively.

Table 1Strategic objectives.

Perspective Strategic objective

Financial 1. Increase profits2. Increase return on investment (ROI)3. Improve financial stability

Clients 1. Reduce prices2. Improve delivery3. Improve product's freshness

Internal processes 1. Improve production process2. Improve supply chain3. Increase capacity

Learning and growth 1. Increase employer's competences2. Improve motivation

1 2 3

1

3

2 3

1 2

21

Level 2

Level 1

Level 3

1Level 0

Level 4

Fig. 3. Hierarchical model of company.


4.2. The linear programming model

The linear programming model for the case is summarized asThe objective function is

min z¼ θdþ1

9þdþ

2

9þdþ

3

6

� �þð1�θÞðd�

1 þd�2 þd�

3 Þ

The constraints of the number of relationships are

Level 1 : Y111þY1

21þY131þY1

12þY122þY1

32þY113þY1

23þY133�dþ

1 ¼ 0

Level 2 : Y211þY2

21þY231þY2

12þY222þY2

32þY213þY2

23þY233�dþ

2 ¼ 0

Level 3 : Y311þY3

21þY331þY3

12þY322þY3

32�dþ13 ¼ 0

The constraints of the importance of relationships are

Level 1 : 0:07Y111þ0:14Y1

21þ…þ0:09Y113þ0:02Y1

23þ0:07Y133þd�

1 ¼ 1

Level 2 : 0:23Y211þ0:33Y2

21þ…þ0:04Y213þ0:05Y2

23þ0:11Y233þd�

2 ¼ 1

Level 3 : 0:47Y311þ0:14Y3

21þ…þ0:07Y322þ0:05Y3

32þd�3 ¼ 1

The feasibility conditions are

Y111þY1

12þY113Z1

Y121þY1

22þY123Z1

Y131þY1

32þY133Z1

Y111þY1

21þY131Z1

Y112þY1

22þY132Z1

Y113þY1

23þY133Z1

⋮

Y312þY3

22Z1

Y321þY3

22Z1

Y331þY3

32Z1

Y311þY3

21þY331Z1

Y321þY3

22þY332Z1

Table 6 shows the optimal solution for the case θ¼0.5, whenmanagers give equal importance to both objectives (minimizenumber of relationships and maximize importance). The optimalsolution is the same as the one obtained by Quezada and Quintero(2011). However, it is not possible to conclude which method isbetter because there is not a “best” solution for the problem. Thelinear programming model offers managers the possibility ofanalyzing various values of θ to choose the one that is moreappropriate for the company. Fig. 4 depicts the optimal solutiongraphically.

Fig. 5 shows the percent of the number the relationshipsselected in the optimal solution as a function of θ. Fig. 6 showsthe percent of the accumulated importance of the relationshipsselected as a function of θ.

As expected, as the value of the control variable θ increases,both the number of relationships and the percent of accumulatedimportance of those relationships decrease because a low θ meansthat managers give low importance to reducing the number ofrelationships and a high importance to maximizing the accumu-lated importance. The LP model increases the number of relation-ships, which lead to a higher accumulated importance. A high θmeans that managers give high importance to reducing thenumber of relationships and low importance to maximizing theaccumulated importance. The model reduces the number ofrelationships, which leads to a lower accumulated importance.

Table 2Importance of nodes of level 1 in relation to level 0.

Level 0 Node 0 (100%)

Priority Local importance

Profits 0.55 0.55ROI 0.21 0.21Financial stability 0.24 0.24Total 1.0 1.0


Level 1 Profits (0.55) ROI (0.21) Financial stability(0.24)

Priority Localimportance



Price 0.14 0.07 0.68 0.14 0.62 0.15Delivery 0.71 0.39 0.24 0.05 0.10 0.02Freshness 0.15 0.09 0.08 0.02 0.28 0.07Total 1.0 0.55 1.0 0.21 1.0 0.24


Level 2 Price (0.37) Delivery (0.46) Freshness (0.17)




Production 0.61 0.23 0.71 0.32 0.21 0.04Supply

chain0.27 0.10 0.19 0.09 0.13 0.02

Capacity 0.12 0.04 0.10 0.05 0.66 0.11Total 1.0 0.37 1.0 0.46 1.0 0.17


Level 3 Production (0.59) Supply chain (0.21) Capacity (0.20)

Priority Local importance Priority Local importance Priority Local importance

Competences 0.80 0.47 0.67 0.14 0.75 0.15Motivation 0.20 0.12 0.33 0.07 0.25 0.05Total 1.0 0.59 1.0 0.21 1.0 0.24

Table 6Optimal solution for θ¼0.5.

Variable Value Variable Value Variable Valuen¼1 n¼2 n¼3

Y111 0 Y2

11 1 Y311 1

Y112 1 Y2

12 1 Y312 1

Y113 1 Y2

13 0 Y321 1

Y121 1 Y2

21 1 Y322 0

Y122 0 Y2

22 0 Y331 1

Y123 0 Y2

23 0 Y332 0

Y131 1 Y2

31 0Y132 0 Y2

32 0Y133 0 Y2

33 1


4.3. Applications of the programming model and analysis

In this section, other real-world cases are analyzed. Quezadaet al. (2013) applied the method developed by Quezada andQuintero (2011) to seven companies to evaluate the method. Fiveof these firms already had a strategy map. In these cases, theactual strategy maps were compared with those obtained usingtheir method. Their conclusion was that the method selects thoserelationships that have a higher importance than those included inthe actual strategy maps. The complete information about thefirms and the detailed calculations can be found in González(2011).

To analyze the method proposed in this paper, the data fromfour of the cases studied by Quezada et al. (2013) were used.Strategy maps were first developed using the proposed method,and they were then compared with those obtained with theapplication of the method developed by Quezada and Quintero(2011) and the actual strategy maps of the companies. Theproposed method was not actually applied to the companies butwas instead used as a theoretical analysis.

Table 7 shows the number of strategic objectives of each of thecompanies.

This section presents a comparison between the actual strategymap (map), our method using a linear programming model (mod)and the algorithm by Quezada and Quintero (2011) (alg). Toaccomplish this, a measure of the degree of similarity was defined.This measure calculates the difference between the relationshipsselected by the methods in relation to total number of relation-ships. The comparison is made for each level and for theoverall model.

� Similarity between optimization model and strategy map, bylevel of the hierarchy.

Similarity between levelsðn;nþ1Þ

¼ 1�∑ijA ðn;nþ1Þ y

n;nþ1ijðmodÞ �yn;nþ1

ijðmapÞ

�� Cn;nþ1

ð13Þ

� Similarity between optimization model and strategy map

Similarity¼ 1�∑n∑ijA ðn;nþ1Þ y


ijðmapÞ

�� ∑n Cn;nþ1

ð14Þ

� Similarity between optimization model and algorithm, by levelof the hierarchy

Similarity between levelsðn;nþ1Þ

¼ 1�∑ijA ðn;nþ1Þ y


ijðalgÞ

�� Cn;nþ1

ð15Þ

� Similarity between optimization model and algorithm

Similarity¼ 1�∑n∑ijA ðn;nþ1Þ y


ijðalgÞ

�� ∑n Cn;nþ1

ð16Þ

where the binary variable yn;nþ1ijðmodÞ corresponds to the solution of

the optimization model described in this paper, the binary

variable yn;nþ1ijðalgÞ corresponds to the solution of the algorithm by

Quezada and Quintero (2011) and the binary variable yn;nþ1ijðmapÞ

corresponds to the real strategy map of the company. Asexplained in the model, those variables take the value “1” ifthe relationship is selected and “0” otherwise. As expected, thesevalues vary depending on the value of θ. As an illustration of thecalculation, the case of company B is shown in Appendix B.

1 2 3

1

3

2 3

1 2

21

Level 2

Level 1

Level 3

1Level 0

Level 4

Fig. 4. Optimal solution for θ¼0.5.

0%

20%

40%

60%

80%

100%

120%

0 0.2 0.4 0.6 0.8 1

% re

latio

nshi

ps s

elec

ted

% Number of Relationships selected

Level 1

Level 2

Level 3

mean

Fig. 5. Percent of number of relationships selected as a function of θ.

0%

20%

40%

60%

80%

100%

120%

0 0.2 0.4 0.6 0.8 1

% Im

port

ance

% of importance

Level 1

Level 2

Level 3

mean

Fig. 6. Percent of accumulated importance of relationships selected.

Table 7Number of strategy objectives.

CompanyLevel A B C D

1 3 4 3 12 2 4 4 33 4 4 3 44 3 3 2 2


% of Similarity between levels. Optimization model and algorithm. Company A

% of global Similarity. Optimization model and algorithm. Company A

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

level 1-level 2 level 2-level 3level 3-level 4

-5

10

25

40

55

70

85

100

0.0 0.2 0.4 0.6 0.8 1.0

θ

Fig. 7. Percent of similarity between optimization model and algorithm for Company A for various values of θ.

% of Similarity between levels. Optimization model and algorithm. Company B

% of global Similarity. Optimization model and algorithm. Company B

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1θ

Fig. 8. Percent of similarity between optimization model and algorithm for Company B for various values of θ.

% of Similarity between levels. Optimization model and algorithm.

Company C% of global Similarity. Optimization model

and algorithm. Company C

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

Fig. 9. Percent of similarity between optimization model and algorithm for Company C for various values of θ.


% of Similarity between levels. Optimization model and algorithm. Company D

% of global Similarity. Optimization model and algorithm. Company D

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Fig. 10. Percent of similarity between optimization model and algorithm for Company D for different values of θ.

% of Similarity between levels. Optimization model and strategy map. Company A

% of global Similarity. Optimization model and strategy map. Company A

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Fig. 11. Percent of similarity between optimization model and strategic map for Company A for different values of θ.

% of Similarity between levels. Optimization model and strategy map. Company B

% of global Similarity. Optimization model and strategy map. Company B

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1


0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Fig. 12. Percent of similarity between optimization model and strategic map for Company B for various values of θ.


Figs. 7-14 show the levels or percentage of similarity betweenthe optimization model and the algorithm and the levels ofsimilarity between the optimization model and the strategy mapfor the four companies studied.

In all the cases studied, for a value of θ¼0.5, the algorithm andthe linear programming model obtain the same result. The resultfor both methods is the same when the value of θ is higher than orequal to 0.7 and lower than 0.9 because of the inclusion of thefeasibility condition and the high valuation of having a reducednumber of relationships. With those values of θ; the optimizationmodel finds the minimum number of relationships that meets thefeasibility condition.

If the algorithm is compared to the optimization model, theresults are similar when θ is approximately 0.5. However, this isnot always true, so it is possible to conclude that the resultsdepend on the structure of the graph and the importance of therelationships.

There is not a structure 100% similar when the result of theoptimization model is compared with the actual strategy map.However, it is interesting to analyze each case separately. In thecase of company A, between levels 1 and 2, similarities of 100% areobtained when θ is lower than or equal to 0.3, which indicates that

between those levels (Finances and Clients), managers of companyA prefer to have more relationships to obtain an importance of therelationships close to 1. The same result is obtained in the case ofcompany D, between the levels 2 and 3 (Clients and InternalProcesses) and a value of θ of approximately 0.4. In the case ofcompany C, a similarity of 100% is obtained for a value of θ equal orlower than 0.2.

In some cases, the same level of similarity between theoptimization model and the actual strategy map is obtained, butwith different solutions of the optimization model. For example,for company A, the same similarity is achieved for values of θ¼0.4, 0.5 and 0.6, i.e., with different solutions, the same similaritymay be obtained because the measure is an aggregated value ofthe differences between both graphs.

5. Conclusions

This paper addressed the issue of determining the cause-effectrelationships within a strategy map of a balanced scorecard (BSC).A model that combines the Analytic Hierarchy Process (AHP) andLinear Programming (LP) is presented. The method proposed by

% of Similarity between levels. Optimization model and strategy map. Company C

% of global Similarity. Optimization model and strategy map. Company C

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

level 1- level 2 level 2- level 3level 3- level 4

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1θ

Fig. 13. Percent of similarity between optimization model and strategic map for Company C for various values of θ.

% of Similarity between levels. Optimization model and strategy map. Company D

% of global Similarity. Optimization model and strategy map. Company D

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

level 1-level 2 level 2-level 3

level 3-level 4

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Fig. 14. Percent of similarity between optimization model and strategic map for Company D for various values of θ.


Quezada and Quintero (2011) in their paper uses AHP to obtain theimportance of all possible relationships within a strategy map, andthen it selects some of them according to a criterion. The proposedmodel aims to replace that selection criterion by a more robustmathematical model. The combination of AHP and LP is not new,but no application of the combined approach on building strategymap was found. There are many applications in the area ofevaluation and selection of suppliers.

The linear programming model considers two conflictingobjectives. A low number of relationships should be included inthe strategy map, but at the same time the relationships should beimportant enough to capture the essence of the strategy of thecompany. To find a balance between both, a control variable θ ϵ[0, 1] is used, which represents the importance given to theobjective of reducing the number of relationships to be includedin the strategy map. In this way, 1- θ represents the importancegiven to the objective of maximizing the importance of thoserelationships included in the strategy map. The use of a linearprogramming model allows managers to find, varying the para-meter θ, a strategy map that produces the right balance betweenthe two objectives defined above.

Using the same example shown in Quezada and Quintero(2011), the same solution was obtained when θ¼0.5, i.e., whenmanagers give equal importance to both objectives. Other caseswere analyzed, and a similar result was found.

The results of the linear programming model were alsocompared with actual strategy maps. To perform this comparison,a measure of similarity was defined. The similarity measure wasfound to vary along the levels of the network (perspectives of thestrategy map).

The advantage of modeling the problem as an analyticalhierarchy process is that the hierarchical model can be tailor-made to the characteristics of the particular company. For thisreason, the method is not limited to the case of the fourperspectives defined by Kaplan and Norton (1992): financial,clients, internal processes and growth & learning.

This work has two main limitations. First, the proposed methodis only valid when the strategy map of the company can be

modeled hierarchically. To investigate this in the future, the useof the Analytic Network Process (ANP) could be explored. Thiswould allow the inclusion of dependence and feedback, whichwould be more realistic for modeling a strategy map. The secondlimitation is that the proposed method has not yet been validatedas a management tool. To accomplish this, the proposed methodshould also be applied to companies without a strategy map. Inthose cases, managers could be interviewed about the utility of theproposed method.

Acknowledgments

This work was supported by the University of Santiago of Chile(Project DICYT- USACH, No. 060517QLL).

Appendix A. Algorithm proposed by Quezada and Quintero(2011)

The authors use the same procedure in the stages 1 and 2 toobtain the local importance of the relationships of the hierarchicalmodel. They use a different method for stage 3.

Using the same notationanij¼ local importance of node j of level nþ1 in relation to node i

in level nCn;nþ1¼total number of arcs between level n and level nþ1.A threshold value is defined for each hierarchical level:

tn ¼ threshold value of level n¼ 1Cn; nþ1

ðA:1Þ

The solution is

Ynij ¼

1; if anijZtn

0; if anijotn

(

8 i; j;n ðA:2ÞHowever, the resulting values may not necessarily satisfy the

feasibility conditions (6) and (7) of the linear programming model.

Welfare of company (0)

Product-ServicePrice-Quality

Increase client’s portfolio (7)

Recognition from strategic partners (6)

Improve client’s profitability (5)

Improve cost structure (12)

Comply regulations (11)

Efficient client’s management (10)

Knowledge about Customers (9)

Improving information

Strengthen organizational culture

Generate integral development of

Increase revenues current lines (1)

Increase incomes from international

New projects (3) Improve ROI (4)

market (2)

relationships (8)

technology (15)culture (14)people (13)

Fig. B.1. Hierarchical Model for Company B. (Adapted from Gonzalez (2011) with permission).


In case a node is disconnected, the following procedure isapplied:

If node i of level n is not connected to any of the nodes j of levelnþ1, then a connection is added to that node with the highestimportance of the relationship, i.e.:

Assign ynij ¼ 1; if anij ¼maxk

fankjg ðA:3Þ

If node j of level n is not connected to any of the nodes i of leveln�1, then a connection is added to that node with the highestimportance of the relationship, i.e.:

Assign yn�1ij ¼ 1; if anij ¼max

pfanjpg ðA:4Þ

Appendix B. Calculation of similarity

As an illustration, the calculation of the level of similaritybetween the solution of the linear programming model(with θ¼ 0;4) and the actual strategy map is presented (in thecase of company B, presented by Gonzalez (2012)). In the case ofthe actual strategy map of the company, the variables “y” take the

value 1 if a relationship exists and 0 otherwise. The samecalculation is carried out in the case of the similarity betweenthe solution of the linear programming model and the algorithmby Quezada and Quintero (2011). Fig B.1 depicts the hierarchicalmodel with all the links. For simplicity, the values of the localrelationships been have omitted, and each objective has beenidentified by only one index (0 to 15).

Table B.1 shows the calculation of similarity between thestrategy map and the solution of the LP model for θ¼ 0:4. Thenumber of arcs with the same value is counted, which is equiva-lent to counting the number of “0s” in the absolute differencecolumn of the table. As an example, the similarity between level1 and level 2 is 63% (10 identical relationships out of 16) and theoverall similarity is 66% (29 identical relationships out of 44).

References

Amid, A., Ghodsypour, S.H., O’Brien, C., 2011. A weighted max–min model for fuzzymulti-objective supplier selection in a supply chain. Int. J. Prod. Econ. 13,139–145.

Banker, R.D., Chang, H., Pizzini, M., 2011. The judgmental effects of strategy maps inbalanced scorecard performance evaluations. Int. J. Account. Inf. Syst. 12,259–279.

Table B.1Calculation of similarity between strategy map and solution of LP model, for θ¼ 0:4.

Variable Solution LP modelðθ¼ 0;4Þ

Actual strategymap

Absolutedifference

Similarity by level Overall similarity

Relationships between level1 and level 2

Y15 1 1 0 10 relationships identical out of16¼63%

29 relationships identical out of44¼66%Y16 1 0 1

Y17 1 1 0Y18 1 0 1Y25 0 1 1Y26 1 1 0Y27 1 1 0Y28 0 0 0Y35 1 0 1Y36 0 0 0Y37 1 1 0Y38 1 0 1Y45 1 0 1Y46 0 0 0Y47 0 0 0Y48 1 1 0


Y59 0 1 1 9 relationships identical out of16¼56%Y510 1 1 0

Y511 1 1 0Y512 1 0 1Y69 1 1 0Y610 0 0 0Y611 1 0 1Y612 0 0 0Y79 1 1 0Y710 1 0 1Y711 1 0 1Y712 1 0 1Y89 1 0 1Y810 0 0 0Y811 1 1 0Y812 1 1 0


Y913 1 1 0 9 relationships identical out of12¼75%Y914 1 1 0

Y915 0 1 1Y10,13 1 1 0Y10,14 1 1 0Y10,15 1 1 0Y11,13 1 1 0Y11,14 0 1 1Y11,15 1 1 0Y12,13 0 1 1Y12,14 1 1 0Y12,15 1 1 0


http://refhub.elsevier.com/S0925-5273(14)00266-7/sbref1






Bentes, A.V., Carneiro, J., Ferreira da Silva, J., Kimura, H., 2012. Multidimensionalassessment of organizational performance: integrating BSC and AHP. J. Bus. Res.65, 1790–1799.

Bessire, D., Baker, C.R., 2005. The French Tableau de Bord and the AmericanBalanced Scorecard: a critical analysis. Crit. Perspect. Account. 16, 645–664.

Chen, F.H., Hsu, T.S., Tzeng, G.H., 2011. A balanced scorecard approach to establish aperformance evaluation and relationship model for hot spring hotels based ona hybrid MCDM model combining DEMATEL and ANP. Int. J. Hosp. Manag. 30,908–932.

Demirtas, E., Üstün, O., 2008. An integrated multiobjective decision making processfor supplier selection and order allocation. Omega 36, 76–90.

Demirtas, E.A., Ustun, O., 2009. Analytic network process and multi-period goalprogramming integration in purchasing decisions. Comput. Ind. Eng. 56,677–690.

González, M.A., 2011. Validation of an algorithm for the selection of importantcause-effect relationships in a balanced scorecard model based on AHP (M.Sc.thesis). University of Santiago of Chile.

Ghodsypour, S.H., O’Brien, C., 1998. A decision support system for supplier selectionusing an integrated analytic hierarchy process and linear programming. Int. J.Prod. Econ. 56–57, 192–212.

Ghodsypour, S.H., O’Brien, C., 2001. The total cost of logistics in supplier selection,under conditions of multiple sourcing, multiple criteria and capacity constraint.Int. J. Prod. Econ. 73 (1), 15–27.

Gomes, C.F., Yasin, M.M., Lisboa, J.V., 2004. Literature review of manufacturingperformance measures and measurement in an organizational context: aframework and direction for future research. J. Technol. Manag. 15 (6), 511–530.

Hsu, C.W., Hu, A.H., Chiou, C., Chen, T.C., 2011. Using the FDM and ANP to constructa sustainability balanced scorecard for the semiconductor industry. Expert Syst.Appl. 38, 12891–12899.

Huang, H.C., Lai, M.C., Lin, L.H., 2011. Developing strategic measurement andimprovement for the biopharmaceutical firm: using the BSC hierarchy. ExpertSyst. Appl. 38, 4875–4881.

Jassbi, J., Mohamadnejad, F., Nasrollahzadeh, H., 2011. A Fuzzy DEMATEL frameworkfor modeling cause and effect relationships of strategy map. Expert Syst. Appl.38, 5967–5973.

Kaplan, R.S., Norton, D.P., 2001a. Transforming the balanced scorecard fromperformance measurement to strategic management: Part I. Account. Horiz.15 (1), 87–106.

Kaplan, R.S., Norton, D.P., 2001b. Transforming the balanced scorecard fromperformance measurement to strategic management: Part II. Account. Horiz.15; b, pp. 147–162.

Kaplan, R.S., Norton, D.P., 1992. The balanced scorecard – measures that driveperformance. Harv. Bus. Rev. 71–79.

Kaplan, R.S., Norton, D.P., 1996. The Balanced Scorecard. Harvard Business SchoolPress, Boston MA, USA.

Kaplan, R.S., Norton, D.P., 2004. Strategy Maps: Converting Intangible Assets intoTangible Outcomes. Harvard Business School Press, Boston, MA.

Kunc, M., 2008. Using systems thinking to enhance strategy maps. Manag. Decis. 46(5), 761–778.

Leung, L.C., Lam, K.C., Cao, D., 2006. Implementing the balanced scorecard using theanalytic hierarchy process & the analytic network process. J. Oper. Res. Soc. 57,682–691.

Lin, C.T., Chen, C.B., Ting, Y.C., 2011. An ERP model for supplier selection inelectronics industry. Expert Syst. Appl. 38, 1760–1765.

Lin, R.H., 2012. An integrated model for supplier selection under a fuzzy situation.Int. J. Prod. Econ. 138, 55–61.

Kokangul, A., Susuz, Z., 2009. Integrated analytical hierarch process and mathema-tical programming to supplier selection problem with quantity discount. Appl.Math. Model. 33, 1417–1429.

Neely, A., 2005. The evolution of performance measurement research: develop-ments in the last decade and a research agenda for the next. Int. J. Oper. Prod.Manag. 25 (12), 1264–1277.

Nørreklit, H., 2000. The balanced on the balanced scorecard – a critical analysis ofsome of its assumptions. Manag. Account. Res. 11, 65–88.

Nørreklit, H., 2003. The Balanced Scorecard: what is the score? A rhetorical analysisof the Balanced Scorecard. Account. Organ. Soc. 28, 591–619.

Nudurupati, S.S., Bititci, U.S., Kumar, V., Chan, F.T.S., 2010. State of the art literaturereview on performance measurement. Comput. Ind. Eng. 60 (2), 1–12.

Quezada, L., Cordova, F., Palominos, P., Godoy, K., Ross, J., 2009. Method foridentifying strategic objectives in strategy maps. Int. J. Prod. Econ. 122 (1),122–500.

Quezada, L., Quintero, D., 2011. Quantitative model for the design of a strategy map.In: Proceedings of the 21th International Conference on Production Research,July 31–August 4, Stuttgart, Germany.

Quezada, L., Palominos, P., Gonzalez, M.A., 2013. Application of AHP in the design ofa strategy sap. J. iBus. 5 (3B), 133–137.

Saaty, T.L., 1994. Fundamentals of Decision Making and Priority Theory with theAnalytical Hierarchy Process. RWS Publications, Pittsburgh, PA, USA.

Saaty, T.L., 2001. Decision making with dependence and feedback: the AnalyticNetwork Process, 2nd edition. RWS Publications, Pittsburgh, PA.

Saaty, R., 2002. Decision Making in Complex Environment. RWS Publications,Pittsburgh, PA, USA.

Sarkis, J., 2003. Quantitative models for performance measurement systems-alternate considerations. Int. J. Prod. Econ. 86, 81–90.

Senge, P.M., 1999. The Fifth Discipline: The Art & Practice of the LearningOrganization. Random House, London.

Shaw, H., Shankar, R., Yadav, S.S., Thakur, L.S., 2012. Supplier selection using fuzzyAHP and fuzzy multi-objective linear programming for developing low carbonsupply chain. Expert Syst. Appl. 39, 8182–8192.

Tjader, Y., May, H.J., Shang, J.L., Vargas, G., Gao, N., 2014. Firm-level outsourcingdecision making: a balanced scorecard-based analytic network process model.Int. J. Prod. Econ. 147 (C), 614–623.

Tsen, M.L., 2010. Implementation and performance evaluation using the fuzzynetwork balanced scorecard. Comput. Educ. 55, 188–201.

Wu, H.Y., 2012. Constructing a strategy map for banking institutions with keyperformance indicators of the balanced scorecard. Eval. Prog. Plan. 35, 303–320.

Yang, C.H., Chen, J.C., Shyu, J.Z., Tzeng, G.H., 2008. Causal relationship analysis basedon DEMATEL technique for innovative policies in SMEs. In: PICMET 2008Proceedings, 27–31 July, Cape Town, South Africa (c).

Yuan, F.C., Chiu, C., 2011. A hierarchical design of case-based reasoning in thebalanced scorecard application. Expert Syst. Appl. 6, 333–342.

Yüksel, I., Dagdeviren, M., 2010. Using the fuzzy analytic network process (ANP) forBalanced Scorecard (BSC): a case study for a manufacturing firm. Expert Syst.Appl. 37, 1270–1278.






































































































a method for designing a strategy map using ahp and linear programming

Documents