# Optimizing Design of Smart Workplace through Multi-Objective Programming
## 4 Proposed Smart Workplace Design Optimization Procedure
### Fuzzy Goal Programming  
Traditional goal programming (GP) is a branch of multi-objective optimization. It is an extension of linear programming to manipulate multiple, usually conflicting objective measures. Each of these measures is given a goal or target value required to be achieved. Deviations both above and below from these goals are measured. undesired deviations from this set of target values are then minimized in an objective function called the achievement function.  
In GP [36], the simple additive model for m goals, \(G_{i}\ (x)\ (i=1,2..,\ m)\) with deviational variables \(p_{i}\), \(n_{i}\) is defined as:  
\[\sum_{i=1}^{m}(p_{i}+n_{i}\ ) \tag{17}\]  
Subject to  
\(G_{i}(x)+n_{i}-p_{i}=g_{i}\)  
\(n_{i}\bullet p_{i}=0\),  
\(n_{i}\), \(p_{i}\), \(g_{j}\geq 0\), \(I=1,2\)..., \(m\)  
where: \(g_{i}\), represents the aspiration level of the \(i\)-th goal.  
However, in ambiguous decision-making situations, goal levels can be difficult to be specified precisely using exact values. In these situations of ill-defined multi-objective optimization, the fuzzy goal programming (FGP) [37] fits well. In FGP, we use a membership function instead of deviational variables to model deviations from the average levels desired to be achieved. The FGP problem is formulated as:  
Find X to satisfy:  
\[G_{i}x\ \widetilde{>}\ g_{i},\ i\ =\ 1,2\ldots,\ \text{m} \tag{18}\]  
Subject to  
\(AX\leq b\)\(X\geq 0\)  
where: \(X\) is an \(n\)-vector with components \(x_{1},x_{2},...\ x_{n}\). and \(AX\leq b\), _are_ system constraints in vector notation. The symbol '\(\stackrel{{\sim}}{{>}}\)' refers to the fuzzification of the aspiration level (i.e., approximately greater than or equal to). The \(i\)-\(th\) fuzzy goal \(G_{i}(x)\stackrel{{>}}{{>}}g_{i}\) indicates that the decision-maker is satisfied even if less than \(g_{i}\) is achieved up to a specific tolerance limit. To solve the FGP problem, we use a linear membership function for the \(i\)-th fuzzy goal, \(\mu_{i}\) (see [PERSON] (1976), (1978) [35; 38]), which can be expressed as follows:  
\[\mu_{i}=\left\{\begin{array}{rl}1&\mbox{if }G_{i}(x)\geq g_{i}\\ \frac{G_{i}(x)-L_{i}}{g_{i}-L_{i}}&\mbox{if }L_{i}\leq G_{i}(x)\leq g_{i}\\ 0&\mbox{if }G_{i}(x)\leq L_{i}\end{array}\right. \tag{19}\]  
where: \(L_{i}\) is the lower tolerance limit for the fuzzy goal.  
In the case of the goal \(G_{i}(x)\stackrel{{<}}{{<}}g_{i}\), the membership function is defined as:  
\[\mu_{i}=\left\{\begin{array}{rl}1&\mbox{if }G_{i}(x)\leq g_{i}\\ \frac{U_{i}-G_{i}(x)}{U_{i}-g_{i}}&\mbox{if }g_{i}\leq G_{i}(x)\leq U_{i}\\ 0&\mbox{if }G_{i}(x)\geq U_{i}\end{array}\right. \tag{20}\]  
where: \(L_{i}\) is the Upper tolerance limit for the fuzzy goal.  
Now, the additive model of the FGP problem is formulated by adding the membership functions together as:  
\[\mbox{Maximize }V(\mu)=\sum_{i=1}^{m}\mu_{i} \tag{21}\]  
Subject to \(\mu_{i}=\frac{G_{i}(x)-L_{i}}{g_{i}-L_{i}}\)  
\(AX\leq b\)  
\(\mu_{i}\leq 1\)  
\(X\), \(\mu_{i}\geq 0\), \(I=1\),\(2\)..., \(m\)  
where: \(V(\mu)\) is called the fuzzy achievement function.  
In this research, we utilize FGP to optimize the impact relations values of the SETs on operational objectives of the smart workplace (i.e., OOs). It means that every OO of the workplace will be considered as an individual goal, and the coefficient of the FGP are the positive impact values of each of the SET on that goal, in such a way that we seek to optimize the total positive impact of these SETs on these goals, OOs.  
Next section, we present a case example for illustrating the application of the proposed smart workplace optimization procedure.