1. Introduction

The Lexicographic Goal Programming Model;

The arrangement of the undesirable deviations was caused by the initial goal programming formulations. Minimizing deviations at higher priority levels is much more crucial than at lower priority levels. Hence, lexicographic (preemptive) or non-Archimedean goal programming are examples of preemptive paradigms.

(1)

Subject to:

$$\begin{array}{l}Constra\mathrm{int}\\ Goa{l}_1={\displaystyle \sum _{j=1}^n{\alpha }_{ij}{\lambda }_j}-{\partial }_i^{+}+{\partial }_i^{-}={\gamma }_{{}_i}\forall i=1,2,\mathrm{..............}n\end{array}$$

(2)

$$\begin{array}{l}Constra\mathrm{int}\\ System={\displaystyle \sum _{j=1}^n{\alpha }_{ij}{\lambda }_j}={\vartheta }_i,\forall i=m+1,\mathrm{..............}p\end{array}$$

(3)

$$\begin{array}{l}{\partial }_i^{+},{\partial }_i^{-},{\lambda }_j\ge 0,\forall i=1,\mathrm{..............}m\mathrm{and}\forall j=1,\mathrm{..............}n\\ {\partial }_i^{+}({\partial }_i^{-})=0\end{array}$$

(4)

2. Methodology

2.2. Case Study

Goal programming was used in this study to track the performance of the Lagos State Government from 2017 to 2021. Her financial records for the specified years were used to compile information on a number of project portfolios that were carried out in the state in order to monitor or improve performance. This study applies a goal-programming methodology to ten projects carried out over a five-year period by the Lagos State Government (2017-2021).

Table 1. Project Portfolio of Lagos State, Nigeria.

Table 1. Project Portfolio of Lagos State, Nigeria.

Portfolio	2017	2018	2019	2020	2021
Agric Project	2,950,699	2,114,882	1,341,008	7,814,527	10,175,949
Construction and Rehabilitation	19,787,898	22,976,320	8,993,492	5,672,168	15,384,376
LAMATA BRT Project	25,354,578	36,353,883	4,039,138	14,145,225	8,097,857
Health Projects	0	0	0	484,298	777,923
Multilateral Funding Projects	7,716,605	0	1,469,547	1,300,311	5,535,337
Conservation Projects	0	0	2,278	6,796	33,926
Oil and Gas Project	117,504	73,582	95,249	651,505	42,062
Schools Furniture	0	0	0	665,496	927,309
Entrepreneurial Skill	409,444	476,876	214,336	1,691,054	1,594,278
Emergency Rescue Equipment	4,163,105	1,582,244	2,968,086	1,859,292	2,245,517

Source: Lagos State, Annual Report.

The Model Targets

λ₁: Is the total quantity of project portfolio in 2017 financial statements of Lagos State

λ₂: Is the total quantity of project portfolio in 2018 financial statements of Lagos State

λ₃: Is the total quantity of project portfolio in 2019 financial statements of Lagos State

λ₄: Is the total quantity of project portfolio in 2020 financial statements of Lagos State

λ₅: Is the total quantity of project portfolio in 2021 financial statements of Lagos State

$$\begin{array}{l}\mathrm{Agric}\mathrm{Project}\mathrm{Goal}\mathrm{Constraint}\\ \mathrm{The}\mathrm{following}\mathrm{equation}\mathrm{is}\mathrm{developed}\mathrm{for}\mathrm{agric}\mathrm{project}\mathrm{goal}\mathrm{constraint}\\ ={\displaystyle \sum _{j=1}^n{\alpha }_{ij}{\lambda }_j}-{\partial }_i^{+}+{\partial }_i^{-}={\gamma }_{{}_i}\forall i=1,2,\mathrm{..............}n\\ ={\alpha }_{11}{\lambda }_1+{\alpha }_{12}{\lambda }_2+{\alpha }_{13}{\lambda }_3+{\alpha }_{14}{\lambda }_4+{\alpha }_{15}{\lambda }_5+{\partial }_i^{-}-{\partial }_i^{+}={\tau }_1\end{array}$$

(8)

$$\begin{array}{l}ConstructionandRehabilitationGoalConstraint\\ \mathrm{The}\mathrm{equation}\mathrm{determines}\mathrm{the}\mathrm{construction}\mathrm{and}\mathrm{rehabilittion}\mathrm{goal}\mathrm{constraint}\\ ={\alpha }_{21}{\lambda }_1+{\alpha }_{22}{\lambda }_2+{\alpha }_{23}{\lambda }_3+{\alpha }_{24}{\lambda }_4+{\alpha }_{25}{\lambda }_5++{\partial }_2^{-}-{\partial }_2^{+}={\tau }_2\end{array}$$

(9)

$$\begin{array}{l}LAMATABRTProjectGoalConstraint\\ \mathrm{The}\mathrm{following}\mathrm{equation}\mathrm{is}\mathrm{developed}\mathrm{for}\mathrm{LAMATA}\mathrm{BRT}\mathrm{goal}\mathrm{constraint}\\ ={\alpha }_{31}{\lambda }_1+{\alpha }_{32}{\lambda }_2+{\alpha }_{33}{\lambda }_3+{\alpha }_{34}{\lambda }_4+{\alpha }_{35}{\lambda }_5++{\partial }_3^{-}-{\partial }_3^{+}={\tau }_3.\end{array}$$

(10)

$$\begin{array}{l}HealthProjectsGoalConstraint\\ \mathrm{The}\mathrm{following}\mathrm{equation}\mathrm{determines}\mathrm{the}\mathrm{health}\mathrm{project}\mathrm{goal}\mathrm{constraint}\\ ={\alpha }_{41}{\lambda }_1+{\alpha }_{42}{\lambda }_2+{\alpha }_{43}{\lambda }_3+{\alpha }_{44}{\lambda }_4+{\alpha }_{45}{\lambda }_5+{\partial }_4^{-}-{\partial }_4^{+}={\tau }_4\end{array}$$

(11)

$$\begin{array}{l}MultilateralFundingProjectsGoalConstraint\\ \mathrm{The}\mathrm{equation}\mathrm{developes}\mathrm{for}\mathrm{multilateral}\mathrm{funding}\mathrm{goal}\mathrm{constraint}\\ ={\alpha }_{51}{\lambda }_1+{\alpha }_{52}{\lambda }_2+{\alpha }_{53}{\lambda }_3+{\alpha }_{54}{\lambda }_4+{\alpha }_{55}{\lambda }_5+{\partial }_5^{-}-{\partial }_5^{+}={\tau }_5\end{array}$$

(12)

$$\begin{array}{l}ConservationProjectsGoalConstraint\\ \mathrm{The}\mathrm{equation}\mathrm{developes}\mathrm{for}\mathrm{conservation}\mathrm{project}\mathrm{goal}\mathrm{constraint}\\ ={\alpha }_{61}{\lambda }_1+{\alpha }_{62}{\lambda }_2+{\alpha }_{63}{\lambda }_3+{\alpha }_{64}{\lambda }_4+{\alpha }_{65}{\lambda }_5+{\partial }_6^{-}-{\partial }_6^{+}={\tau }_6.\end{array}$$

(13)

$$\begin{array}{l}OilandGasProjectGoalConstraint\\ ={\alpha }_{71}{\lambda }_1+{\alpha }_{72}{\lambda }_2+{\alpha }_{73}{\lambda }_3+{\alpha }_{74}{\lambda }_4+{\alpha }_{75}{\lambda }_5+{\partial }_7^{-}-{\partial }_7^{+}={\tau }_7\end{array}$$

(14)

$$\begin{array}{l}SchoolsFurnitureGoalConstraint\\ ={\alpha }_{81}{\lambda }_1+{\alpha }_{82}{\lambda }_2+{\alpha }_{83}{\lambda }_3+{\alpha }_{84}{\lambda }_4+{\alpha }_{85}{\lambda }_5+{\partial }_8^{-}-{\partial }_8^{+}={\tau }_8.\end{array}$$

(15)

$$\begin{array}{l}EntrepreneurialSkillGoalConstraint\\ ={\alpha }_{91}{\lambda }_1+{\alpha }_{92}{\lambda }_2+{\alpha }_{93}{\lambda }_3+{\alpha }_{94}{\lambda }_4+{\alpha }_{95}{\lambda }_5+{\partial }_9^{-}-{\partial }_9^{+}={\tau }_9\end{array}$$

(16)

$$\begin{array}{l}EmergencyRescueEquipmentGoalConstraint\\ ={\alpha }_{10,1}{\lambda }_1+{\alpha }_{10,2}{\lambda }_2+{\alpha }_{10,3}{\lambda }_3+{\alpha }_{10,4}{\lambda }_4+{\alpha }_{10,5}{\lambda }_5+{\partial }_{10}^{-}-{\partial }_{10}^{+}={\tau }_{10}\end{array}$$

(17)

As was previously stated, the objective of this study is to maximize the project portfolio of the Lagos State Government for the specified period while minimizing costs, so it is necessary to add positive and negative deviations to the constraints in order to assess whether the objectives are expanding or contracting.

Objective Function

The objective function is defined in the following manner:

(18)

Based on the established goal constraints, the GP model (18) is created and formulated as follows

$$\begin{array}{l}Subjectto:\\ ={\alpha }_{11}{\lambda }_1+{\alpha }_{12}{\lambda }_2+{\alpha }_{13}{\lambda }_3+{\alpha }_{14}{\lambda }_4+{\alpha }_{15}{\lambda }_5+{\partial }_i^{-}-{\partial }_i^{+}={\tau }_1\\ ={\alpha }_{21}{\lambda }_1+{\alpha }_{22}{\lambda }_2+{\alpha }_{23}{\lambda }_3+{\alpha }_{24}{\lambda }_4+{\alpha }_{25}{\lambda }_5++{\partial }_2^{-}-{\partial }_2^{+}={\tau }_2\\ ={\alpha }_{31}{\lambda }_1+{\alpha }_{32}{\lambda }_2+{\alpha }_{33}{\lambda }_3+{\alpha }_{34}{\lambda }_4+{\alpha }_{35}{\lambda }_5++{\partial }_3^{-}-{\partial }_3^{+}={\tau }_3\\ ={\alpha }_{41}{\lambda }_1+{\alpha }_{42}{\lambda }_2+{\alpha }_{43}{\lambda }_3+{\alpha }_{44}{\lambda }_4+{\alpha }_{45}{\lambda }_5+{\partial }_4^{-}-{\partial }_4^{+}={\tau }_4\\ ={\alpha }_{51}{\lambda }_1+{\alpha }_{52}{\lambda }_2+{\alpha }_{53}{\lambda }_3+{\alpha }_{54}{\lambda }_4+{\alpha }_{55}{\lambda }_5+{\partial }_5^{-}-{\partial }_5^{+}={\tau }_5\\ ={\alpha }_{61}{\lambda }_1+{\alpha }_{62}{\lambda }_2+{\alpha }_{63}{\lambda }_3+{\alpha }_{64}{\lambda }_4+{\alpha }_{65}{\lambda }_5+{\partial }_6^{-}-{\partial }_6^{+}={\tau }_6\\ ={\alpha }_{71}{\lambda }_1+{\alpha }_{72}{\lambda }_2+{\alpha }_{73}{\lambda }_3+{\alpha }_{74}{\lambda }_4+{\alpha }_{75}{\lambda }_5+{\partial }_7^{-}-{\partial }_7^{+}={\tau }_7\\ ={\alpha }_{81}{\lambda }_1+{\alpha }_{82}{\lambda }_2+{\alpha }_{83}{\lambda }_3+{\alpha }_{84}{\lambda }_4+{\alpha }_{85}{\lambda }_5+{\partial }_8^{-}-{\partial }_8^{+}={\tau }_8\\ ={\alpha }_{91}{\lambda }_1+{\alpha }_{92}{\lambda }_2+{\alpha }_{93}{\lambda }_3+{\alpha }_{94}{\lambda }_4+{\alpha }_{95}{\lambda }_5+{\partial }_9^{-}-{\partial }_9^{+}={\tau }_9\\ ={\alpha }_{10,1}{\lambda }_1+{\alpha }_{10,2}{\lambda }_2+{\alpha }_{10,3}{\lambda }_3+{\alpha }_{10,4}{\lambda }_4+{\alpha }_{10,5}{\lambda }_5+{\partial }_{10}^{-}-{\partial }_{10}^{+}={\tau }_{10}\\ \\ \forall {\lambda }_n,{\partial }_n^{-},{\partial }_n^{+}\ge 0\end{array}$$

(19)

Table 2. Target achievement.

Table 2. Target achievement.

	Goal	Outcomes	Target
Min/Max Agric Project	ɤ1	${\partial }_1^{-},{\partial }_1^{+}$	Achievement
Min/Max Construction and Rehabilitation	ɤ2	${\partial }_2^{-},{\partial }_2^{+}$	Achievement
Min/Max LAMATA BRT Project	ɤ3	${\partial }_3^{-},{\partial }_3^{+}$	Achievement
Min/Max Health Projects	ɤ4	${\partial }_4^{-},{\partial }_4^{+}$	Achievement
Min/Max Multilateral Funding Projects	ɤ5	${\partial }_5^{-},{\partial }_5^{+}$	Achievement
Min/Max Conservation Projects	ɤ6	${\partial }_6^{-},{\partial }_6^{+}$	Achievement
Min/Max Oil and Gas Project	ɤ7	${\partial }_7^{-},{\partial }_7^{+}$	Achievement
Min/Max Schools Furniture	ɤ8	${\partial }_8^{-},{\partial }_8^{+}$	Achievement
Min/Max Entrepreneurial Skill	ɤ9	${\partial }_9^{-},{\partial }_9^{+}$	Achievement
Min/Max Emergency Rescue Equipment	ɤ10	${\partial }_{10}^{-},{\partial }_{10}^{+}$	Achievement

Source: Self-developed.

Project Portfolio of Lagos State, Nigeria

In the table below, the cost of each project appears in Billion Naira. But for convenience, the values are further divided by 1,000,000 and rounded to 3-significant to have the current state.

Table 3. Lagos State Project Cost (Billion of Naira).

Table 3. Lagos State Project Cost (Billion of Naira).

Targets	2017	2018	2019	2020	2021	Total
Agric Project	2.951	2.115	1.341	7.815	10.176	24.398
Construction and Rehabilitation	19.788	22.976	8.994	5.672	15.384	72.814
LAMATA BRT Project	25.355	36.354	4.039	14.145	8.098	87.991
Health Projects	0	0	0	0.484	0.778	1.262
Multilateral Funding Projects	7.717	0	1.470	1.300	5.535	16.022
Conservation Projects	0	0	0.002	0.007	0.034	0.043
Oil and Gas Project	0.118	0.074	0.095	0.652	0.042	0.981
Schools Furniture	0	0	0	0.666	0.927	1.593
Entrepreneurial Skill	0.410	0.477	0.214	1.691	1.594	4.386
Emergency Rescue Equipment	4.163	1.582	2.968	1.859	2.246	12.818
Total	60,499,833	63,577,787	19,123,134	34,290,672	44,814,534

Source: Lagos State, Annual Report.

Then, the project portfolio of Lagos State government for the period of 5 years can now be applied to the proposed model (19) above:

$\begin{array}{l}Subjectto:\\ 2.951{\lambda }_1+2.115{\lambda }_2+1.341{\lambda }_3+7.815{\lambda }_4+10.176{\lambda }_5+{\partial }_i^{-}-{\partial }_i^{+}=24.398\\ 19.788{\lambda }_1+22.976{\lambda }_2+8.994{\lambda }_3+5.672{\lambda }_4+15.384{\lambda }_5+{\partial }_2^{-}-{\partial }_2^{+}=72.814\\ 25.355{\lambda }_1+36.354{\lambda }_2+4.039{\lambda }_3+14.145{\lambda }_4+8.098{\lambda }_5+{\partial }_3^{-}-{\partial }_3^{+}=87.991\\ 0{\lambda }_1+0{\lambda }_2+0{\lambda }_3+0.484{\lambda }_4+0.778{\lambda }_5+{\partial }_4^{-}-{\partial }_4^{+}=1.262\\ 7.717{\lambda }_1+0{\lambda }_2+1.470{\lambda }_3+1.300{\lambda }_4+5.535{\lambda }_5+{\partial }_5^{-}-{\partial }_5^{+}=16.022\\ 0{\lambda }_1+0{\lambda }_2+0.002{\lambda }_3+0.007{\lambda }_4+0.034{\lambda }_5+{\partial }_6^{-}-{\partial }_6^{+}=0.043\\ 0.118{\lambda }_1+0.074{\lambda }_2+0.095{\lambda }_3+0.652{\lambda }_4+0.042{\lambda }_5+{\partial }_7^{-}-{\partial }_7^{+}=0.981\\ 0{\lambda }_1+0{\lambda }_2+0{\lambda }_3+0.666{\lambda }_4+0.927{\lambda }_5+{\partial }_8^{-}-{\partial }_8^{+}=1.593\\ =0.410{\lambda }_1+0.477{\lambda }_2+0.214{\lambda }_3+1.691{\lambda }_4+1.594{\lambda }_5+{\partial }_9^{-}-{\partial }_9^{+}=4.386\\ =4.163{\lambda }_1+1.582{\lambda }_2+2.968{\lambda }_3+1.859+2.246{\lambda }_5+{\partial }_{10}^{-}-{\partial }_{10}^{+}=12.818\end{array}$

$\forall {\lambda }_n,{\partial }_n^{-},{\partial }_n^{+}\ge 0$ (Non-negative constraint)

The goal programming model defined above was solved using LINGO 17.0 x32 versions (20). Goal-setting is also covered in the sections that follow:

3. Results

Table 4. Goal Accomplishment.

Table 5. Outcomes of Deviational Variables.

4. Discussion of Result

5. Conclusions

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