Project.Rmd

---
title: "Project"
output: pdf_document
---

### [**Employees Classifications (basic case)**]{.ul}

1.  PM
2.  Eng,
3.  Surveyor,
4.  Designer

$i$ is set of people {employee 1, employee 2, }

$j$ is set of projects

$k$ is set of classifications {PM, Engineer, Designer, Surveyor}

[**Decision variables:**]{.ul}

1.   $X_i,_j,_k$ [Continuous]{.ul}: Hours that an employee $i$ work in functional class $k$ in project $j$

2.   $Y_i,_j,_k$ = ${0, 1}$ Binary variable, If employee $i$ is assigned to classification $k$ for project $j$ then $y_i,_j,_k$is 1 otherwise it is 0.

    ## $A_i,_k$ = ${0, 1}$ binary, can employee $i$ do task $k$

3.   $S_{i,k}$ =1 if employee is qualified to do task $k$

4.   $P_i,_k$: Profit, Hourly rate for employee $i$ classification $k$ in project $j$ \* 10% \*

5.  $E_j$ : project budget = Not to Exceed (NTE) cost per project

6.  $R_{i,k}$: billing rate for $i$ when performing task classification $k$

7.  $T_{j,k}$ is Total hours per project for task classification $k$

[**Objective Function:**]{.ul}

$$
 \begin{split}
 \begin{aligned}
    \text{Maximize  }   &  \sum_{i} \sum_{j} \sum_{k} X_{i,j,k} P_{i,k} \\
 \end{aligned}
 \end{split}
$$

[**Constraints**]{.ul}

[Utilization]{.ul}:total hours assigned to a staff who classified as engineer/ designer/ surveyor should not exceed 1664 hours per year.

1.  $\sum_{j=1}^{j=NProj} \sum_{k=1}^{k=3} X_{i,j,k} \leq1664 \,\text { hours } \forall i$

2.  $\sum_{j=1}^{j=NProj} \sum_{k=1}^{k=3} R_{i,k}\cdot X_{i,j,k} \leq E_j \,\ \forall j$

3.  $X_{i,j,k}\leq BigM * Y_{i,j,k} \,\forall i,j,k$

4.  $Y_{i,j,k}\leq S_{i,k} \forall i j k$

5.  $\sum_{i=1}^{i=NP} X_{i,j,k} \geq T_{j,k} \forall i$

Including the libraries:

```{r}
library(ompr, quietly = TRUE)
library(magrittr, quietly = TRUE)
library(pander, quietly = TRUE)
library(ROI, quietly = TRUE)
library(ROI.plugin.glpk, quietly = TRUE)
library(ompr.roi, quietly = TRUE)
library(pander, quietly = TRUE)
library(TRA)
library(Benchmarking, quietly=TRUE)
```


```{r Data}

Classification<-c("PM", "Engineer", "Designer", "Surveyor")
##Classification of staff
BillingRate <-matrix(c(200, 186, 120, 135), ncol = 4, nrow = 4,
                     dimnames = c(list(c("P1", "P2", "P3", "P4")),
                                  list(c("PM", "Engineer", "Designer", "Surveyor")))
                     )
## Billing Rates per above classiciation in order
print (BillingRate)
MaxHours<-1664
## 2080 hours per year multiplied by 80% utilization per staff = 1664
## we will need to change this to be a variable so each class has its ulitzation factor
ProfitPerStaff<-0.1*BillingRate
## profit is assumed 10% per the hourly rate
Staff<-matrix(c("St1","St2", "St3", "St4", "St5", "St6", "St7", "St8", "St9", "St10"), ncol = 10)
## These are all roster of technical staff workinng at the firm 
n<-10
## Number of Staff
p<-4
##Number of projects
c<-4
## Number of classifications



```

```{r model Variables, constraints, objhective}
model <- MIPModel()%>%
add_variable(Vx[i,j,k], i=1:n, j=1:p, k=1:c, type= "continuous") %>%
  ## hours assigned to staff i on project j, in a classification k
    add_variable(Vy[i,j,k], i=1:n, j=1:p, k=1:c, type = "binary") %>%
  ## whether staff i is assigned to project j in a  classification k
  add_variable(S[i,k], i=1:n, k=1:c, type = "binary") %>%
  ##can staff i, do task k
  add_variable(R[i,k], i=1:n, k=1:c, type = "continuous") %>%
  ##  hours needed for prjtec i to perform task in classification k
    add_variable (T[j,k], j=1:p, k=1:c, type = "continuous")%>%
    ## profit per staff
    add_variable (P[i,k], i=1:n, k=1:c, type = "continuous")%>%
    ## Not to exceed amout of each project j
    add_variable (E[j], j=1:p, type="continuous") %>%
    
    
    set_objective(sum_expr((Vx[i,j,k]*P[i,j]), i=1:n, j=1:p, k=1:c), "max") %>%
    
    add_constraint (sum_expr (sum_expr(X[i,j,k])))
    
  
  
```

```{r output}

```