# The Assignment Problem in Human Resource Project Management under Uncertainty

## Abstract

**:**

## 1. Introduction

#### 1.1. The Assignment Problem as an Element of Human Project Resource Management

#### 1.2. From Deterministic AP to AP under Uncertainty—Motivation of the Research

#### 1.3. The Gap in the Literature

- -
- would be easy to use for managers,
- -
- would not require such data that are too difficult to estimate in the case of projects with a high degree of novelty and a fast-evolving environment (like fuzzy numbers, probabilities),
- -
- would be useful for any kind of decision-maker (pessimists, moderate decision-makers, and optimists)

#### 1.4. The Structure of the Paper

## 2. Materials and Methods

#### 2.1. Basic Variants of the Deterministic Assignment Problem

_{i,j}, t

_{i,j}, or c

_{i,j}denoting the performance (efficiency, time, or cost) connected with agent A

_{i}and task T

_{j}. The agents may be assigned to perform any task. Table 1 presents the assignment problem on the assumption that the efficiency parameters are given.

_{i,j}is the binary variable. It is equal to 1 if task T

_{j}is performed by agent A

_{i}.

_{i}and b

_{i}signify the minimal desired and maximal possible number of tasks that can be executed by particular agents.

#### 2.2. Selected Extensions of the Deteministic Assignment Problem

#### 2.3. Uncertainty in HRPM—From Probabilities and Fuzzy Numbers to Scenario Planning

_{i,j}, t

_{i,j}, or c

_{i,j}are one of those that may be nondeterministic in the assignment problem (other uncertain information can be related, for instance, to the number of available workers or the number of significant tasks, but such aspects are not taken into consideration in our research). Let us recall that these coefficients describe the performance (efficiency, time, or cost) connected with agent A

_{i}and task T

_{j}. Usually, they can be treated as deterministic data, but, in the paper, we mainly investigate the case of new projects (innovative or innovation projects) and projects performed in very stormy periods (e.g., economic crisis, pandemic, tense political situation, lack of raw materials, unforeseen fluctuations in demand, unstable law). In such circumstances, the performance of a given task depends not only on the employees’ competence, skills, physical fitness, and mental capabilities but also on numerous additional internal (ability to cope with stressful situations, ability to cope with failure, openness to changes, creativity) and external factors (technology, regulations, demand).

- the number of agents and tasks is known;
- the characteristics describing the agent-task performance are uncertain;
- the experts are able to define possible scenarios concerning performance quality;
- the number of scenarios for particular cases (agent–task) may be different (the sets of scenarios are independent),

#### 2.4. Should Scenario Planning be Applied with Probabilities?

- there are enough resources, budget, and time to prepare analytical scenarios with assigned probabilities;
- “the corporate culture prefers quantitative or quasi-quantitative procedures to purely qualitative reasoning”;
- “the scenario team is familiar with the concept of Bayesian probabilities”.

## 3. Results

#### 3.1. AP Algorithm under Uncertainty—Applications and Steps

- The number of tasks and agents is known and deterministic. Depending on the problem, the worker is supposed to execute one task or several tasks, but, in each case, the work is performed by one agent only.
- The tasks are independent, which means that the performance of a given task does not affect the performance of other tasks.
- At least one agent–task performance is described by means of nondeterministic parameters.
- The set of scenarios for each uncertain agent–task performance can be different since it may depend on different factors.
- The algorithm is designed for one-shot decisions (projects) and pure strategies (i.e., strategies where only one decision variant is chosen, not a combination of them).
- The decision-maker (manager) can declare his/her state of mind and soul (predictions) on the basis of optimism (β) and pessimism (α) coefficients that reflect the attitude towards risk within a given problem. β is close to one for extreme optimists and close to zero for extreme pessimists. These parameters may be different for each agent–task performance. Note that the sum of the pessimism coefficient and the optimism coefficient is equal to one (α + β = 1), and both parameters belong to the interval [0,1].
- The aforementioned parameters are used to calculate the so-called H + B index for each nondeterministic agent–task performance (that measure is briefly explained below).
- The final optimization model is solved after transforming the initial scenario parameters into weighted values.
- The algorithm allows us to find a solution to all the optimization problems presented in Section 2.1.
- The method gives the possibility of changing the final solution in the case of too-high standard deviations connected with the chosen assignments. A high standard deviation indicates a significant dispersion of estimated scenario results, which is treated as an unwanted feature.

- For optimists (β > 0.5), the optimism coefficient is multiplied by the best value and the pessimism coefficient is multiplied by all the remaining outcomes. All the products are added. Then, the weighted value is divided by the sum of all the applied coefficients.
- For pessimists (α > 0.5), the pessimism coefficient is multiplied by the worst value and the optimism coefficient is multiplied by all the remaining outcomes. All the products are added. Then, the weighted value is divided by the sum of all the applied coefficients.
- For moderate DMs (α = β = 0.5), it is possible to use any formula described above since the H + B index is equal to the Bayes index in this case.

- Define the target, the set of agents, the set of tasks, the constraints, and the possible scenarios for the particular uncertain agent–task performance.
- Estimate the optimism and pessimism coefficients for each nondeterministic agent–task performance.
- Apply the H + B rule separately for each uncertain agent–task performance.
- Formulate the optimization model using H + B indices.
- Solve the model.
- If the recommended assignments are connected with too-high standard deviations, correct the solution by eliminating the recommended assignment with the highest standard deviation and solving a new amended optimization model.

#### 3.2. Illustrative Examples

_{1,1}= [0.7 × 10 + 0.3 × (11 + 15 + 18)]/(0.7 + 3 × 0.3) = 12.63

_{2,3}equals 1 if Agent A2 performs task T3, and 0 otherwise), one objective function (with 80 products), 8 constraints concerning the agents, 10 constraints related to the tasks and 80 variable constraints. We only show a reduced version of the objective function, the first agent condition, and the first task condition (decision variables are binary).

_{2,5}= 0 (such a constraint should also be added for all the potential agent–job assignments characterized by a standard deviation level equal to at least 8.54). Then again, the PM would have the opportunity to accept the solution or to change it.

- the PM is interested in maximizing the efficiency of the least efficient worker,
- it is required that all tasks be accomplished by assigning exactly one agent to each task and exclusively one job to each agent,

_{2,5}= 0 and similar conditions for each potential assignment with a dispersion not lower than 8.54. It is up to the project manager to evaluate the first solution and possibly change it.

## 4. Discussion

- (1)
- According to Point 5 presented in Section 3.1, the goal of the new algorithm is to indicate the best pure strategy (not mixed strategy). In the context of the assignment problem, a pure strategy occurs when particular jobs are entirely assigned to one worker. However, as mentioned in Section 3.2, sometimes a given agent might not finish the task due to numerous possible chance events (e.g., illness). In such circumstances, the initial optimal assignment plan has to be updated by assigning the rest of the job to another agent. Note that the new strategy should still fulfill the constraints of the primary model (i.e., the maximal number of tasks for particular agents must not be exceeded). Imagine that Agent A8 has just accomplished T1 and T10. He has also executed a part of Task T3. Unfortunately, he is not able to continue this activity due to some unexpected events. Hence, the PM may assign the rest of Task T3 to A6 because A6 has executed only one activity and, among the remaining agents, this worker is the best for Task T3. A6-T3 performance is worse than the A8-T3, so total expected efficiency will certainly decrease; however, the dispersion connected with A6-T3 is equal to 0, which means that its efficiency is guaranteed. Note that when the plan has to be modified during project realization at moment t, total expected efficiency may be calculated as the sum of the real efficiency observed from the beginning of the project to t and the expected efficiency related to the period starting at moment t and ending with the project completion.
- (2)
- According to Points 1 and 3, the only uncertainty factor considered in the algorithm is related to agent–work performance. This element is very often difficult to predict. Furthermore, depending on the agent–work case, particular performances can be determined by different aspects. That is why the procedure gives the opportunity to diversify the set of possible scenarios (Point 4) and the level of pessimism and optimism coefficients (point 6) for each agent–task performance. The possibility of applying diverse parameters is crucial since, in real problems, the PM may have a different perception of particular human abilities.
- (3)
- In a sense, the AP algorithm under uncertainty is similar, for instance, to PERT (a project evaluation and review technique), which is a tool used in project time management (another area distinguished by PMI). PERT gives the opportunity to estimate three durations (pessimistic, most likely, optimistic) for each activity belonging to the project. The nondeterministic AP procedure also enables us to consider several scenarios, but this time (and this is a significant advantage), the number of possible scenarios for each agent–task performance may vary from 1 to any positive integer value (see Point 4).
- (4)
- Note that the AP algorithm under uncertainty is not the first AP procedure referring to scenarios. Numerous papers have been devoted to the min–max assignment problem and the min–max regret assignment problem (e.g., Aissi et al. 2005; Deineko and Woeginger 2006; Wu et al. 2018), where the scenarios are also applied. However, the assumptions and methodology adopted in those papers differ significantly from the idea presented in this article. In the cited papers, the construction of the problem allows us only to analyze the case of decision-makers being extreme pessimists since the aim of the applied methods is to find “a solution with the best worst-case value across all scenarios” (which is characteristic of the Wald rule) or to find “a feasible solution minimizing, over all possible scenarios, the maximum deviation of the value of the solution from the optimal value of the corresponding scenario” (which is characteristic of the Savage rule). Here, the DM’s attitude towards risk is taken into consideration in a quite simple way, and the use of the optimism coefficient enables us to solve the assignment problem for any decision-maker (pessimist, moderate, optimist). Additionally, the number of scenarios for each agent–task performance may be different, which is not the case for the procedures described in the aforementioned papers.
- (5)
- The novel approach (similarly to the original AP algorithm) still enables the project manager to find the best assignment for various optimization models (see Point 9). He or she can focus on the following objectives: total efficiency maximization, entire cost/time minimization, maximization of the efficiency of the least efficient worker, minimization of the time of the lowest worker, or minimization of the cost of the most expensive worker. Note that in the case of a cost objective function, there is no need to take into consideration all the cost factors because some of them may be common for each worker. It is better to focus on the elements that differ for particular agents (e.g., remuneration).
- (6)
- It is worth underlining that sometimes the optimization model may contain more than one optimal solution (i.e., solutions that have the best value of the objective function); however, when solving the model, only one optimal strategy is generated by the optimization tool. The remaining optimal plans may be obtained after subsequent use of this tool, but there is no guarantee that the whole set of optimal solutions will be found. Nevertheless, even if the manager does not know the number of optimal strategies and cannot generate all of them, such knowledge is not crucial in the decision-making process since each optimal plan satisfies all the constraints of the model and has the same level of objective function. The significant difference between particular optimal solutions may concern varied maximal dispersions, but Step 6 of the algorithm allows the DM to control this factor. If the dispersion connected with a given optimal solution is too high for the manager, there is an opportunity to generate a new strategy that does not contain the assignment with the unwanted dispersion level. Hence, if the optimization model consists of several optimal plans, and the DM is not willing to perform the first generated solution, Step 6 of the procedure can lead him or her to another optimal solution.
- (7)
- Point 7 of the AP algorithm under uncertainty concerns the use of the H + B index. Note that this measure gives the possibility of assigning the highest weight to the value that is the most expected; however, importantly, the remaining values connected with a given agent–task assignment are also considered because the less-essential weight is assigned to all of them. Such an approach enables the PM to control not only the extreme scenarios but each intermediate scenario as well, which is extremely vital when dealing with uncertainty. Let us add that thanks to the denominator of the H + B index (being the sum of all the applied weights), the final H + B measurement is never lower than the outcome of the worst scenario and never higher than the outcome of the best scenario.

## 5. Conclusions

- it allows the project managers to apply scenario planning (a relatively simple tool) to parameter estimation,
- it gives the opportunity to include information on the decision-maker’s (manager’s) nature (state of mind/soul) and attitude towards risk within a given problem—this is possible thanks to the pessimism and optimism coefficients, which are also used in other well-known decision rules),
- it does not require the use of objective probabilities (due to a high degree of uncertainty), and
- it enables the project managers to accept or correct the obtained solution (thanks to the standard deviation analysis).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Agent | Task | ||||
---|---|---|---|---|---|

T_{1} | … | T_{j} | … | T_{n} | |

A_{1} | e_{1,1} | … | e_{1,j} | … | e_{1,n} |

⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮ |

A_{i} | e_{i,}_{1} | ⋮ | e_{i,j} | ⋮ | e_{i,n} |

⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮ |

A_{m} | e_{m,}_{1} | … | e_{m}_{,j} | … | e_{m,n} |

_{i}—agent I; T

_{j}—task j; m—number of agents; n—number of tasks; e

_{i,j}—efficiency connected with agent A

_{i}and task T

_{j}. Source: Prepared by the author.

Agent | Task | ||||
---|---|---|---|---|---|

T_{1} | … | T_{j} | … | T_{n} | |

A_{1} | e_{1,1(a);}e_{1,1(b);}e _{1,1(c);}e_{1,1(d);} | … | e_{1,j} | … | e_{1,n(a); 1,n(b);}e _{1,n(c)} |

⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮ |

A_{i} | e_{i}_{,1} | ⋮ | e_{i}_{,j(a);}e_{i,j}_{(b)} | ⋮ | e_{i}_{,n(a);}e_{i}_{,n(b);}e _{i}_{,n(c);}e_{i}_{,n(d);}e_{i}_{,n(e);}e_{i}_{,n(f);} |

⋮ | ⋮ | ⋱ | ⋮ | ⋱ | ⋮ |

A_{m} | e_{m}_{,1(a);}e_{m}_{,1(b)} | … | e_{m}_{,j} | … | e_{m}_{,n} |

_{i}—Agent I; T

_{j}—task j; m—number of agents; n—number of tasks; e

_{i,j}—efficiency connected with agent A

_{i}and task T

_{j}. Source: Prepared by the author.

A/T | T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | T10 |
---|---|---|---|---|---|---|---|---|---|---|

A1 | 10, 11, 15, 18 | 4 | 7 | 1, 6, 9, 12, 17, 24 | 11 | 14 | 8, 9, 25 | 15, 18, 25 | 22 | 5 |

A2 | 6 | 15 | 8 | 7 | 15, 25, 30, 35 | 2 | 1 | 20 | 13, 14, 15 | 3, 9, 10, 11 |

A3 | 3, 4, 8, 9, 10 | 2, 5, 10 | 9, 13, 26 | 19 | 16 | 10 | 10 | 8 | 3, 7, 15, 25 | 1, 5, 10, 16 |

A4 | 6 | 8 | 1 | 10 | 16 | 19 | 25 | 6, 8, 16, 18, 25 | 3, 4, 7, 9, 15 | 14, 15, 17, 19 |

A5 | 3 | 3, 8, 17, 18, 30 | 1, 4, 7, 10, 18 | 24, 25, 27, 30 | 20 | 18 | 15 | 22, 25, 26 | 20 | 15 |

A6 | 24 | 15 | 21 | 7 | 15, 25, 30, 40 | 20, 21, 22, 23 | 5 | 25 | 13, 14, 20 | 3, 12, 13, 16 |

A7 | 15, 25 | 30 | 4 | 5 | 2 | 7 | 12, 15, 18, 20 | 2 | 6 | 2 |

A8 | 34, 36 | 22, 23, 24, 25 | 40 | 8 | 7 | 2 | 9 | 1, 2, 3, 4 | 19, 20, 25 | 23 |

A/T | T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | T10 |
---|---|---|---|---|---|---|---|---|---|---|

A1 | 12.63 | 4.00 | 7.00 | 9.59 | 11.00 | 14.00 | 12.15 | 18.00 | 22.00 | 5.00 |

A2 | 6.00 | 15.00 | 8.00 | 7.00 | 23.44 | 2.00 | 1.00 | 20.00 | 13.69 | 6.94 |

A3 | 6.45 | 5.14 | 15.00 | 19.00 | 16.00 | 10.00 | 10.00 | 8.00 | 11.44 | 7.22 |

A4 | 6.00 | 8.00 | 1.00 | 10.00 | 16.00 | 19.00 | 25.00 | 13.82 | 7.18 | 16.00 |

A5 | 3.00 | 15.20 | 8.00 | 26.50 | 20.00 | 18.00 | 15.00 | 24.33 | 20.00 | 15.00 |

A6 | 24.00 | 15.00 | 21.00 | 7.00 | 27.50 | 21.50 | 5.00 | 25.00 | 15.67 | 11.00 |

A7 | 24.00 | 30.00 | 4.00 | 5.00 | 2.00 | 7.00 | 18.75 | 2.00 | 6.00 | 2.00 |

A8 | 35.80 | 24.50 | 40.00 | 8.00 | 7.00 | 2.00 | 9.00 | 3.50 | 24.00 | 23.00 |

A/T | T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | T10 |
---|---|---|---|---|---|---|---|---|---|---|

A1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

A2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

A3 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

A4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

A5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

A6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

A7 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

A8 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

A/T | T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 |
---|---|---|---|---|---|---|---|---|

A1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

A2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

A3 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

A4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

A5 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

A6 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

A7 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

A8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Gaspars-Wieloch, H.
The Assignment Problem in Human Resource Project Management under Uncertainty. *Risks* **2021**, *9*, 25.
https://doi.org/10.3390/risks9010025

**AMA Style**

Gaspars-Wieloch H.
The Assignment Problem in Human Resource Project Management under Uncertainty. *Risks*. 2021; 9(1):25.
https://doi.org/10.3390/risks9010025

**Chicago/Turabian Style**

Gaspars-Wieloch, Helena.
2021. "The Assignment Problem in Human Resource Project Management under Uncertainty" *Risks* 9, no. 1: 25.
https://doi.org/10.3390/risks9010025