# Developing Talent from a Supply–Demand Perspective: An Optimization Model for Managers

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

## 2. Fundamentals

#### 2.1. TOPSIS

#### 2.2. Chance-Constrained Programming; The Basic Theory

**ξ**is a stochastic vector and ${g}_{j}(\mathbf{x},\mathit{\xi})$;$j=1,\dots ,p$ are stochastic constraint functions. Since stochastic parameters are involved in characterizing all given constraints, we cannot reasonably define the maximization term in the objective function and the direction ≤ in the given constraints. In order to treat this type of problem correctly, Liu [37] suggested using the following CCP model:

## 3. Problem Description

**Remark**

**1.**

**Remark**

**2.**

- (I)
- # attritions for job position j = # current employees for job position j × attrition rate for job position j
- (II)
- # desired employees for job position j = # current employees for job position j × (1 + growth rate for job position j)
- (III)
- # promotions to job position j = sum of (# current employees of any job position l × the advancement rate from job l to job j)
- (IV)
- # promotions from job position j = # current employees of job position j × sum of the advancement rates from job position j to any job position k
- (V)
- # gained/lost positions for job position j = # promotions to job position j − # attritions for job position j − # promotions from job position j = # promotions to job position j − # current employees for job position j × (attrition rate for job position j + sum of the advancement rates from job j to any job position k).It is clear that if the value of the latter relation is positive, we have some employees gained for job position j; otherwise, some employees have been lost for that job position. Moreover, we have the following key relationships:
- (VI)
- # hires needed for job position j = # desired employees for job position j − # gained/lost positions for job position j − # current employees for job position j = # current employees of job position j × (growth rate for job position j + attrition rate for job position j + sum of the advancement rates from job j to any job position k) − # promotions to job position j
- (VII)
- # of available employees in job position j at the end of a period = # current employees for job position j at the beginning of that period + # gained/lost positions for job position j + # of hired employees for job position j.

## 4. Model Development

i | index of recruitment channels; $i=1,\dots ,m$. |

j | index of job positions; $j=1,\dots ,n$. |

t | index of time periods; $t=1,\dots ,T$. |

${\tilde{b}}_{j}$ | the acceptance rate by candidates for job position j. |

${\iota}_{j}$ | the number of current employees for job position j at the beginning of planning horizon. |

${e}_{j}^{t}$ | the unit cost per hour of having extra hires for job position j in period t than it is required. |

${\overline{e}}_{j}^{t}$ | the unit cost per hour of having less hires for job position j in period t than it is required. |

${\overline{\delta}}_{j}^{t}$ | the maximum application capacity for job position j in period t. |

${\overline{\beta}}_{j}^{t}$ | the total interview rate for job position j in period t. |

${\lambda}_{j}^{t}$ | the maximum offering rate for job position j in period t. |

${g}_{j}^{t}$ | the maximum expected development for job position j in period t. |

${\tilde{k}}_{j}$ | the time (hour) required for considering initial documents and selecting candidates for being interviewed for job position j. |

${\tilde{\overline{k}}}_{j}$ | the time (hour) required for analyzing the results of interviews for job position j. |

${r}_{j}^{t}$ | the unit revenue per hour yielded by each employee in job position j in period t. |

${\psi}_{j}^{t}$ | the unit salary per hour paid to each employee in job position j in period t. |

${o}_{j}^{t}$ | the unit cost of interview per hour for job position j in period t. |

${\vartheta}_{j}^{t}$ | the maximum number of employee changes for job position j in period t. |

${w}_{i}$ | the relative closeness of recruitment channel i derived from TOPSIS. |

${\delta}_{i}^{t}$ | the maximum application capacity of recruitment channel i in period t. |

${\beta}_{i}^{t}$ | the total amount of interview rate in recruitment channel i in period t. |

${u}_{jk}$ | the indicator parameter which is equal to 1 if someone can be promoted from job position j to job position k and vice versa, otherwise; is 0. |

$R{T}^{t}$ | total recruitment process time (person-hour) in period t. |

$\epsilon $ | a very small positive number. |

M | a big positive number. |

${A}_{ij}^{t}$ | the number of candidates for job position j applied via recruitment channel i in period t. |

${X}_{ij}^{t}$ | the interview rate per candidate for job position j in recruitment channel i in period t. |

${Y}_{j}^{t}$ | the offering rate per candidate for job position j in period t. |

${\mathsf{\Phi}}_{j}^{t}$ | the attrition rate per employee for job position j in period t. |

${V}_{lj}^{t}$ | the advancement rate per employee from job position l to job position j in period t. |

${G}_{j}^{t}$ | the growth need by role for job position j in period t. |

${S}_{j}^{t}$ | the number of employees for job position j at the end of period t. |

${C}_{j}^{t}$ | the number of current employees for job position j at the beginning of period t. |

${Z}_{j}^{t}$ | the number of candidates hired for job position j in period t. |

${H}_{j}^{t}$ | the number of hires needed for job position j in period t. |

${P}_{j}^{t}$ | the binary variable which is one (zero) if we have less (more) hires for job position j in period t. |

#### 4.1. Objective Function

#### 4.2. Constraints

## 5. Solution Procedure

#### 5.1. Linearization

#### 5.2. Stochastic Constraints

**Lemma**

**1.**

## 6. Empirical Study

#### 6.1. Data Acquisition

#### 6.2. Results Analysis

#### 6.3. Stochastic Analysis

#### 6.4. Sensitivity Analysis

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Part (a):**In this part, we introduce the related functions of the Lingo 16 optimization software to model stochasticity in a programming problem, as follows:

- @SPSTGRNDV(1,RANDOM_VAR Name) for identifying the random variables, which are ${\tilde{k}}_{j}$ and ${\tilde{\overline{k}}}_{j}$ in our stochastic program.
- @SPDIST<TYPE>(PARAM_1[,...,PARAM_N],RANDOM_VAR Name) for declaring parametric distributions such as lognormal distribution; @SPDISTLOGN(MU,SIGMA, RNDVAR), exponential distribution; @SPDISTEXPO(LAMDA,RNDVAR) and so forth.
- @SPCHANCE('Set_Name','>='|'<=',Probability) for identifying Chance-Constraint set; here Constraint (33).
- @SPSAMPSIZE (1,SIZE) for setting sample sizes in generating random numbers of random variables.

**Part (b):**In order to define the stochastic parameters ${\tilde{k}}_{j}$ and ${\tilde{\overline{k}}}_{j}$ in the program body, the following commands are utilized:

- @for(job_position(j): @SPSTGRNDV(1,k(j)));
- @for(job_position(j): @SPSTGRNDV(1,kp(j)));

`k(j)`and

`kp(j)`denote ${\tilde{k}}_{j}$ and ${\tilde{\overline{k}}}_{j}$, respectively. Furthermore, we write the following commands to determine the associated PDFs with ${\tilde{k}}_{j}$ and ${\tilde{\overline{k}}}_{j}$; $j=1,\dots ,5$:

- @SPDISTEXPO(2.6570,k(1)); @SPDISTEXPO(1.3422,k(2));
- @SPDISTEXPO(1.1328,k(3)); @SPDISTEXPO(0.9961,k(4));
- @SPDISTLOGN(0.777,0.521,k(5));
- @SPDISTEXPO(1.2091,kp(1)); @SPDISTEXPO(0.8482,kp(2));
- @SPDISTEXPO(0.7617,kp(3)); @SPDISTEXPO(0.6957,kp(4));
- @SPDISTLOGN(1.019,0.467,kp(5));

- @SPCHANCE('CCP_TIME','>=',0.95);
- @SPCHANCE('CCP_TIME',C_TIME);
- @SPSAMPSIZE(1,60);

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**Figure 6.**Sensitivity analysis on the mean of ${\tilde{k}}_{j}$ (blue) and ${\tilde{\overline{k}}}_{j}$ (red) and ${\tilde{b}}_{j}$ (green).

Job Positions (j) | Time Period (t) | ${\mathit{\iota}}_{\mathit{j}}$ | ${\mathit{e}}_{\mathit{j}}^{\mathit{t}}$ | ${\overline{\mathit{e}}}_{\mathit{j}}^{\mathit{t}}$ | ${\overline{\mathit{\beta}}}_{\mathit{j}}^{\mathit{t}}$ | ${\overline{\mathit{\delta}}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{\lambda}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{r}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{\psi}}_{\mathit{j}}^{\mathit{t}}$ |
---|---|---|---|---|---|---|---|---|---|

Coordinator | 1 | 125 | $5.90 | $47.73 | 0.2 | 1000 | 0.9 | $30.53 | $29.00 |

2 | - | $8.40 | $50.73 | 0.2 | 1000 | 0.9 | $32.20 | $30.68 | |

3 | - | $9.40 | $53.78 | 0.2 | 1100 | 0.9 | $34.35 | $31.28 | |

Analyst | 1 | 96 | $7.40 | $49.50 | 0.2 | 700 | 0.9 | $37.80 | $36.25 |

2 | - | $10.40 | $52.50 | 0.2 | 700 | 0.9 | $39.48 | $37.93 | |

3 | - | $11.15 | $54.22 | 0.2 | 770 | 0.9 | $41.63 | $38.53 | |

Senior analyst | 1 | 43 | $10.68 | $60.75 | 0.2 | 450 | 0.66 | $48.05 | $46.98 |

2 | - | $14.68 | $64.25 | 0.2 | 450 | 0.66 | $49.73 | $48.66 | |

3 | - | $16.53 | $67.30 | 0.2 | 495 | 0.66 | $51.88 | $50.40 | |

Manager | 1 | 16 | $24.81 | $142.6 | 0.25 | 100 | 0.5 | $74.10 | $72.40 |

2 | - | $29.93 | $146.1 | 0.25 | 100 | 0.5 | $75.78 | $74.08 | |

3 | - | $31.08 | $149.2 | 0.25 | 110 | 0.5 | $77.43 | $75.33 | |

Senior manager | 1 | 6 | $47.90 | $245.9 | 0.33 | 30 | 0.4 | $96.18 | $90.62 |

2 | - | $50.60 | $252.9 | 0.33 | 30 | 0.4 | $97.85 | $92.30 | |

3 | - | $52.18 | $256.0 | 0.33 | 33 | 0.4 | $98.55 | $93.79 |

Recruitment Channels (i) | Time Period (t) | ${\mathit{\beta}}_{\mathit{i}}^{\mathit{t}}$ | ${\mathit{\delta}}_{\mathit{i}}^{\mathit{t}}$ |
---|---|---|---|

Career fair | 1 | 0.65 | 1000 |

2 | 0.65 | 1080 | |

3 | 0.65 | 1166 | |

Company website | 1 | 0.7 | 1000 |

2 | 0.6 | 1080 | |

3 | 0.5 | 1166 | |

Social media | 1 | 0.8 | 500 |

2 | 0.75 | 540 | |

3 | 0.75 | 583 |

Recruitment Channels | ${\mathit{C}}_{1}$ (Year) | ${\mathit{C}}_{2}$ ($) | ${\mathit{C}}_{3}$ |
---|---|---|---|

Career fair | 1.85 | $56,000 | 7.61 |

Company website | 3.10 | $64,400 | 5.42 |

Social media | 2.36 | $69,300 | 5.80 |

Job Positions | ${\tilde{\mathit{k}}}_{\mathit{j}}$ | ${\tilde{\overline{\mathit{k}}}}_{\mathit{j}}$ | ${\tilde{\mathit{b}}}_{\mathit{j}}$ |
---|---|---|---|

Coordinator | exp (2.6570) | exp (1.2091) | uniform (0.06,1.00) |

Analyst | exp (1.3422) | exp (0.8482) | uniform (0.16,0.87) |

Senior analyst | exp (1.1328) | exp (0.7617) | uniform (0.42,0.82) |

Manager | exp (0.9961) | exp (0.6957) | uniform (0.72,1.00) |

Senior manager | Lognormal (0.777,0.521) | Lognormal (1.019,0.467) | uniform (0.83,1.00) |

Coordinator | Analyst | Senior Analyst | Manager | Senior Manager | |
---|---|---|---|---|---|

Coordinator | - | 1 | 0 | 0 | 0 |

Analyst | 1 | - | 1 | 0 | 0 |

Senior analyst | 0 | 1 | - | 1 | 0 |

Manager | 0 | 0 | 1 | - | 1 |

Senior manager | 0 | 0 | 0 | 1 | - |

Original MINLP | Linearized MILP | |
---|---|---|

# of decision variables | 103 | 839 |

# of constraints | 113 | 1153 |

Solution time (sec) | - | 89.3 |

Job Positions | Recruitment Channels | ||||||||
---|---|---|---|---|---|---|---|---|---|

Career Fair | Company Website | Social Media | |||||||

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

Coordinator | 0 | 92 | 78 | 82 | 0 | 0 | 0 | 0 | 0 |

Analyst | 0 | 88 | 66 | 92 | 0 | 0 | 0 | 0 | 0 |

Senior analyst | 15 | 0 | 0 | 0 | 27 | 21 | 0 | 0 | 0 |

Manager | 0 | 0 | 0 | 0 | 49 | 56 | 27 | 0 | 0 |

Senior manager | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 14 | 33 |

Job Positions | Recruitment Channels | ||||||||
---|---|---|---|---|---|---|---|---|---|

Career Fair | Company Website | Social Media | |||||||

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

Coordinator | 0 | 0.092 | 0.079 | 0.198 | 0 | 0 | 0 | 0 | 0 |

Analyst | 0 | 0.198 | 0.067 | 0.194 | 0 | 0 | 0 | 0 | 0 |

Senior analyst | 0.187 | 0 | 0 | 0 | 0.104 | 0.053 | 0 | 0 | 0 |

Manager | 0 | 0 | 0 | 0 | 0.204 | 0.223 | 0.092 | 0 | 0 |

Senior manager | 0 | 0 | 0 | 0 | 0 | 0 | 0.158 | 0.217 | 0.237 |

**Table 9.**The optimal solutions of operational rates, ${Z}_{j}^{t}$, ${H}_{j}^{t}$ and ${S}_{j}^{t}$ in three time periods.

Job Positions | ${\mathbf{\Phi}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{G}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{Y}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{Z}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{H}}_{\mathit{j}}^{\mathit{t}}$ | ${\mathit{S}}_{\mathit{j}}^{\mathit{t}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

Coordinator | 0 | 0 | 0 | 0 | 0 | 0 | 0.90 | 0.69 | 0.90 | 5 | 2 | 1 | 5 | 2 | 1 | 125 | 125 | 125 |

Analyst | 0 | 0 | 0 | 0.073 | 0.049 | 0.046 | 0.90 | 0.90 | 0.90 | 6 | 5 | 1 | 6 | 5 | 1 | 103 | 108 | 113 |

Senior analyst | 0 | 0 | 0 | 0 | 0 | 0 | 0.66 | 0.66 | 0.66 | 1 | 1 | 1 | 1 | 1 | 1 | 43 | 43 | 43 |

Manager | 0 | 0 | 0 | 0.313 | 0.286 | 0 | 0.50 | 0.50 | 0.50 | 1 | 4 | 5 | 1 | 4 | 5 | 21 | 27 | 27 |

Senior manager | 0 | 0 | 0 | 0.333 | 0.250 | 0.500 | 0.40 | 0.37 | 0.40 | 1 | 1 | 2 | 1 | 1 | 2 | 8 | 10 | 15 |

Job Positions | Coordinator | Analyst | Senior Analyst | Manager | Senior Manager | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

Coordinator | 0 | 0 | 0 | 0.040 | 0.016 | 0.008 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Analyst | 0 | 0 | 0 | 0 | 0 | 0 | 0.042 | 0.019 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Senior analyst | 0 | 0 | 0 | 0 | 0 | 0.069 | 0 | 0 | 0 | 0.116 | 0.07 | 0 | 0 | 0 | 0 |

Manager | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.074 | 0 | 0 | 0 | 0.063 | 0.047 | 0.111 |

Senior manager | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Job Positions | ${\tilde{\mathit{k}}}_{\mathit{j}}$ | ${\tilde{\overline{\mathit{k}}}}_{\mathit{j}}$ | ||||
---|---|---|---|---|---|---|

Sample Mean | Sample StdDev | p-Value | Sample Mean | Sample StdDev | p-Value | |

Coordinator | 0.3725 | 0.3597 | 1.00 | 0.8718 | 1.0136 | 0.997 |

Analyst | 0.7382 | 0.7177 | 1.00 | 1.1606 | 1.0972 | 1.00 |

Senior analyst | 0.9083 | 1.0021 | 1.00 | 1.3012 | 1.2729 | 1.00 |

Manager | 1.0033 | 1.0017 | 1.00 | 1.4028 | 1.3258 | 1.00 |

Senior manager | 2.4700 | 1.2948 | 1.00 | 3.1183 | 1.6386 | 1.00 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moheb-Alizadeh, H.; Handfield, R.B. Developing Talent from a Supply–Demand Perspective: An Optimization Model for Managers. *Logistics* **2017**, *1*, 5.
https://doi.org/10.3390/logistics1010005

**AMA Style**

Moheb-Alizadeh H, Handfield RB. Developing Talent from a Supply–Demand Perspective: An Optimization Model for Managers. *Logistics*. 2017; 1(1):5.
https://doi.org/10.3390/logistics1010005

**Chicago/Turabian Style**

Moheb-Alizadeh, Hadi, and Robert B. Handfield. 2017. "Developing Talent from a Supply–Demand Perspective: An Optimization Model for Managers" *Logistics* 1, no. 1: 5.
https://doi.org/10.3390/logistics1010005