Metric estimation approach for managing uncertainty in resource leveling problem

Abstract

Real-life applications in project planning often involve grappling with inaccurate data or unexpected events, which can impact the project duration and cost. The delay in the project execution can be overcome by investing in additional resources to avoid compromising the project duration. The goal of the resource leveling problems (RLP) is to determine the optimal amount of resources to invest in, aiming to minimize the associated complementary costs and adhere to the fixed deadline. To tackle data uncertainty in the RLP, the literature has predominantly focused on developing robust and stochastic approaches. In contrast, sensitivity analysis and reactive approaches have received comparatively less attention, especially concerning the generalized RLP with flexible job durations. In this problem, the duration of each job depends on the amount of resources available for its execution. Therefore, utilizing more resources may help reduce the project duration but at an additional cost. This paper introduces a novel approach that addresses the generalized RLP with uncertain job and resource parameters, incorporating reactive and sensitivity-based methodologies. The proposed approach extends the concept of evaluation metrics from machine scheduling to the domain of the RLP with flexible job durations. It is based on a metric-based function that estimates the impact of changes in input data on the solution quality, considering both optimality and feasibility for the new problem instance. The approach is tested through numerical experiments conducted on benchmark instance sets to investigate the impact of variations in different problem parameters. The obtained results demonstrated a meaningful accuracy in estimating the impact on the value of the objective function. Additionally, they underscored the importance of utilizing optimality/feasibility preservation conditions, as for a significant portion of the tested instances, the use of these conditions gave a satisfactory outcome.

Lemmas with proofs

Lemma 1

Let instances A and B differ only in parameters $e_{r}$. If we apply the same solution $\sigma$ to the both instances, the upper bound for objective function values difference is

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| \le \rho _e^\sigma (A,B), \end{aligned}$$

(A.1)

where $\rho _e^\sigma (A,B)$ is represented in the following form

$$\begin{aligned} \rho _e^\sigma (A,B) = \max \left\{ \sum \limits _{r\in R} [e_r^A - e_r^B]^-\sum \limits _{j\in J} W_{jr},\sum \limits _{r\in R} [e_r^A - e_r^B]^+ \sum \limits _{j\in J} W_{jr}\right\} . \end{aligned}$$

(A.2)

We use $[.. ]^-$ and $[\ldots ]^+$ to define $[x]^- = \min \{x,0\},\text { and } [x]^+ = \max \{x,0\}.$

Proof

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| = |\sum \limits _{r\in R}\sum \limits _{t\in T}e_r^A o_{rt}^A - \sum \limits _{r\in R}\sum \limits _{t\in T}e_r^B o_{rt}^B|, \end{aligned}$$

here $o_{rt} = \max \{0,\sum \limits _{j\in J} c_{jrt} - L_{rt}\}$,

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| =\sum \limits _{r\in R} |e_r^A - e_r^B| \sum \limits _{t\in T}\max \left\{ 0,\sum \limits _{j\in J} c_{jrt}- L_{rt}\right\} . \end{aligned}$$

The right side is still solution-dependent, for each resource type the extra cost difference is multiplied by actual overload volume in the solution $\sigma$. We can form the solution-independent estimation with an upper bound for each $r\in R$

$$\begin{aligned} \sum \limits _{t\in T}\max \left\{ 0,\sum \limits _{j\in J} c_{jrt}- L_{rt}\right\} \le \sum \limits _{j\in J} W_{jr}. \end{aligned}$$

Then we have a solution-independent aggregated upper bound:

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| \le \sum \limits _{r\in R} |e_r^A - e_r^B| \sum \limits _{j\in J} W_{jr}. \end{aligned}$$

(A.3)

Moreover, we note that with this objective function form the aggregated positive and negative values of the difference $e_r^A - e_r^B$ compensate each other. If there exist two resource types $r_1,r_2\in R$, and $e_{r_1}^A < e_{r_1}^B$, $e_{r_2}^A > e_{r_2}^B$, then the total objective function values difference will be in a range

$$\begin{aligned} \left[ (e_{r_2}^B-e_{r_2}^A)\sum \limits _{j\in J} W_{jr_2},(e_{r_1}^B-e_{r_1}^A)\sum \limits _{j\in J} W_{jr_1}\right] . \end{aligned}$$

In a general case with an arbitrary set R, we can estimate the difference as

$$\begin{aligned} \rho _e^\sigma (A,B) = max\left\{ \sum \limits _{r\in R} \min \left\{ e_r^A - e_r^B,0\right\} \sum \limits _{j\in J} W_{jr},\sum \limits _{r\in R} \max \left\{ e_r^A - e_r^B,0\right\} \sum \limits _{j\in J} W_{jr}\right\} . \end{aligned}$$

$\square$

Lemma 2

Let instances A and B differ only in parameters $e_{r}$. If we apply the optimal solution of instance A $\sigma ^A$ as a solution to instance B, then the upper bound for objective function values difference is

$$\begin{aligned} V^B(\sigma ^A) - V^B(\sigma ^B) \le \Delta _e^\sigma (A,B) = \sum \limits _{r\in R} |e_r^A - e_r^B| \sum \limits _{j\in J} W_{jr}. \end{aligned}$$

(A.4)

Proof

We note that $V^B(\sigma ^A) \ge V^B(\sigma ^B)$ and $V^A(\sigma ^B) \ge V^A(\sigma ^A)$ for any pair of cases A and B, so there exist six options of ordering these values:

• 1. $V^A(\sigma ^A) \le V^A(\sigma ^B) \le V^B(\sigma ^B) \le V^B(\sigma ^A)$;

• 2. $V^B(\sigma ^B) \le V^B(\sigma ^A) \le V^A(\sigma ^A) \le V^A(\sigma ^B)$;

• 3. $V^A(\sigma ^A) \le V^B(\sigma ^B) \le V^A(\sigma ^B) \le V^B(\sigma ^A)$;

• 4. $V^B(\sigma ^B) \le V^A(\sigma ^A) \le V^B(\sigma ^A) \le V^A(\sigma ^B)$;

• 5. $V^B(\sigma ^B) \le V^A(\sigma ^A) \le V^A(\sigma ^B) \le V^B(\sigma ^A)$;

• 6. $V^A(\sigma ^A) \le V^B(\sigma ^B) \le V^B(\sigma ^A) \le V^A(\sigma ^B)$;

Here in cases 1–4, we can use lemma 1 to prove that considered difference is less than the right side of inequality (A.3). In this inequality, the same solution is applied to both instances, so this is correct for the values of $V^B(\sigma ^A)$ and $V^B(\sigma ^B)$ within the bounds of a similar form (for example, in the first case it is bounded by $V^A(\sigma ^A)$ and $V^B(\sigma ^A)$).

We prove the same for case 5 in the following way. We use the same approach as in Lemma 1. Firstly, we show that the difference

$$\begin{aligned} V^A(\sigma ^A) - V^B(\sigma ^B) \le \sum \limits _{r\in R} [e_r^A - e_r^B]^+ \sum \limits _{j\in J} W_{jr}. \end{aligned}$$

The changed instance B can provide a better solution only with a reduction of cost. Secondly, we estimate this difference

$$\begin{aligned} V^B(\sigma ^A) - V^A(\sigma ^A) \le \sum \limits _{r\in R} [e_r^A - e_r^B]^-\sum \limits _{j\in J} W_{jr}. \end{aligned}$$

In instance B,the same solution $\sigma ^A$ may provide a worse objective function value, with the difference up to the total reduction of resource amount. These two components are bounded and form the initial difference $V^B(\sigma ^A) - V^B(\sigma ^B)$, in total the upper bound is the same as in inequality (A.4). The same logic can be applied for case 6. $\square$

Lemma 3

Let instances A and B differ only by parameters $L_{rt}$. If we apply the same solution $\sigma$ to the both instances, the upper bound for objective function values difference can be evaluated as follows:

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| \le \rho _L^\sigma (A,B), \end{aligned}$$

(A.5)

where $\rho _L^\sigma (A,B)$ is a particular metric estimation,

$$\begin{aligned} \rho _L^\sigma (A,B)= \max \left\{ \sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^+,\sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^-\right\} . \end{aligned}$$

(A.6)

Proof

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| = |\sum \limits _{r\in R}\sum \limits _{t\in T}e_r^A o_{rt}^A - \sum \limits _{r\in R}\sum \limits _{t\in T}e_r^B o_{rt}^B|, \end{aligned}$$

(A.7)

here $o_{rt} = \max \{0,\sum \limits _{j\in J} c_{jrt} - L_{rt}\}$, and taking into account that costs are equal $e_r^A=e_r^B=e_r$ and $|\max \{a,b\} - \max \{c,d\} |\le \max \{|a-c|,|b-d|\},$

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| \le \sum \limits _{r\in R}\sum \limits _{t\in T} |e_r^A\left( \sum \limits _{j\in J} c_{jrt} - L^A_{rt}\right) - e^B_r\left( \sum \limits _{j\in J} c_{jrt} - L^B_{rt}\right) |. \end{aligned}$$

(A.8)

For identical solutions, we obtain the following result:

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| \le \sum \limits _{r\in R} e_r \sum \limits _{t\in T} |L^A_{rt} - L^B_{rt}|. \end{aligned}$$

(A.9)

As in Lemma 1, we propose a precise upper estimation $\rho _L(A,B)$.

In instance B, several resources are available differently from instance A, each difference $\Delta L_{rt} \ne 0$ leads to a limited possible impact on the value of the objective function.

Suppose that the first difference is positive, i.e. $\Delta L_1 > 0$. Then the objective function difference lies within $[0,\Delta V_1]$, where $\Delta V_1$ is $\Delta L_1$ multiplied by corresponding extra resource usage cost $e_r$. If there is another $\Delta L_2 > 0$, then the impact on the objective is $[0,\Delta V_1 + \Delta V_2]$. If $\Delta L_3 < 0$, then the range left bound is shifted: $[\Delta V_3,\Delta V_1 + \Delta V_2]$).

If we take into account all k differences, aggregated range for the objective function value variation is

$$\begin{aligned} \left[ \sum \limits _{i=1}^k \min \left\{ \Delta V_i,0\right\} ,\sum \limits _{j=1}^k \max \left\{ \Delta V_j,0\right\} \right] , \end{aligned}$$

i.e. here bounds are formed by the sum of all negative and positive changes. More precisely, $\Delta V_x = e_r \Delta L_{x}$ if $\Delta L_x = L^B_{rt} - L^A_{rt}$ (it was applied at period t for resource r). Then we can represent the range of possible differences in the values of the objective function in the following way

$$\begin{aligned} \left[ \sum \limits _{r\in R} e_r \sum \limits _{t\in T} \min \left\{ (L^B_{rt} - L^A_{rt}),0\right\} ,\sum \limits _{r\in R} e_r \sum \limits _{t\in T} \max \left\{ (L^B_{rt} - L^A_{rt}),0\right\} \right] . \end{aligned}$$

We can compare absolute values of these bounds to estimate the absolute value of the difference:

$$\begin{aligned} \rho _L^\sigma (A,B) = \max \left\{ \sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^-,\sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^+\right\} . \end{aligned}$$

$\square$

Lemma 4

Let instances A and B differ only in parameters $L_{rt}$. If we apply optimal solution of instance A $\sigma ^A$ to instance B, then the upper bound for the difference in the value of the objective function can be evaluated as follows:

$$\begin{aligned} V^B(\sigma ^A) - V^B(\sigma ^B) \le \Delta _L^\sigma (A,B) = \sum \limits _{r\in R} e_r \sum \limits _{t\in T} |L^A_{rt} - L^B_{rt}| \end{aligned}$$

(A.10)

Proof

We consider the same cases as for Lemma 2. For cases 1–4, we can again use Lemma 3 to prove that the considered difference is less than the right side of inequality (A.9).

A special case 5 is considered in the same way. Instance B can provide a better solution only with additional amount of resources:

$$\begin{aligned} V^A(\sigma ^A) - V^B(\sigma ^B) \le \sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^+ \end{aligned}$$

For instance B, the same solution $\sigma ^A$ may provide a worse value of the objective function, in this case, the difference can be estimated as follows:

$$\begin{aligned} V^B(\sigma ^A) - V^A(\sigma ^A) \le \sum \limits _{r\in R} e_r \sum \limits _{t\in T} [L^B_{rt} - L^A_{rt}]^- \end{aligned}$$

These two components form the initial difference in the inequality (A.9). $\square$

Lemma 5

Let instances A and B differ only in parameters $p_{min,jr}$ or (and) $p_{max,jr}$. If a solution $\sigma$ is applicable to both instances, there is no change for the objective function value, i.e.

$$\begin{aligned} V^A(\sigma ) = V^B(\sigma ),\;\rho _{p_{min/max}}(A,B) = 0. \end{aligned}$$

Proof

These parameters limit the amount of workload and resource $r\in R$ utilized by job $j\in J$, but do not modify directly the value of the objective function. As it was mentioned above,

$$\begin{aligned} |V^A(\sigma ) - V^B(\sigma )| = |\sum \limits _{r\in R}\sum \limits _{t\in T}o_{rt}^A - \sum \limits _{r\in R}\sum \limits _{t\in T}o_{rt}^B|, \end{aligned}$$

and $o_{rt} = \max \{0,e_r(\sum \limits _{j\in J} c_{jrt} - L_{rt})\}$. If $\sigma$ is applicable to A and B, then

$$\begin{aligned}{} & {} d_{jt}^\sigma p_{min,jr}^A \le c_{jrt}^\sigma \le d_{jt}^\sigma p_{max,jr}^A,\;j\in J,\;r\in R,\;t\in T; \\{} & {} d_{jt}^\sigma p_{min,jr}^B \le c_{jrt}^\sigma \le d_{jt}^\sigma p_{max,jr}^B,\;j\in J,\;r\in R,\;t\in T; \end{aligned}$$

the values of $c_{jrt}^\sigma$ will not be changed, as any other part of $|V^A(\sigma ) - V^B(\sigma )|$. $\square$

Lemma 6

Let instances A and B differ only in parameters $p_{max,jr}$ and/or $p_{min,jr}$. Suppose that an optimal solution $\sigma ^A$ of instance A is applicable to instance B and an optimal solution $\sigma ^B$ of instance B is applicable to instance A. If we apply an optimal solution of instance A i.e. $\sigma ^A$ as a solution to instance B, then we obtain the same value of the objective function.

$$\begin{aligned} \Delta ^{\sigma }_{a,p_{min/max}}(A,B) = V^B(\sigma ^A) - V^B(\sigma ^B) = 0 \end{aligned}$$

(A.11)

Proof

With the condition that both solutions are applicable to both instances, we can directly use Lemma 1, as we can estimate all the components ($\rho _{p,max}(A,B,sigma^A)$, $\rho _{p,max}(A,B,sigma^B)$, as well as the same values for $p_{min,jr}$).

If both solutions are applicable to both instances, it means that it is not necessary to modify solution $\sigma ^A$ if it is applied to instance B to reach the optimal value of the objective function, and the same for $\sigma ^B$ applied to instance A.

We can also show that the difference can be more than zero if either $\sigma ^A$ is not applicable to B, or $\sigma ^B$ is not applicable to A, as a consequence, it is impossible to use Lemmas 1 and 5. For $p_{max,jr}$, if $\Delta ^{p_{max}}(A,B) >0$, it means that solution $\sigma ^A$ applied to instance B must be modified to achieve an optimal solution. If $p_{max,jr}^A < p_{max,jr}^B$, then in some period t we allocate $c_{jrt}^B > p_{max,jr}^A d_{jt}^B$ of resource r to a job j, so a resulting $\sigma ^B$ is not applicable to instance A. If there is a difference $p_{max,jr}^A > p_{max,jr}^B$, then it means that in some period t we have to reduce an allocation of resource r to a job j, as $c_{jrt}^A > p_{max,jr}^B d_{jt}^B$, so $\sigma ^A$ is not applicable to instance B. It is possible to formulate a similar statement for parameters $p_{min,jr}$. $\square$

Lemma 7

Let instances A and B differ only in parameters $p_{max,jr}$ (or $p_{min,jr}$). If a schedule $\pi$ is applicable to both instances, there is an upper bound for the difference in the values of the objective function can be estimated as follows:

$$\begin{aligned} |V^A(\sigma ^A(\pi )) - V^B(\sigma ^B(\pi ))| \le \rho _{p_{max}}^\pi (A,B) \end{aligned}$$

(A.12)

where

$$\begin{aligned} \rho _{p_{max}}^\pi (A,B) = md \max \left\{ \sum \limits _{r\in R} e_r \sum \limits _{j\in J} [p_{max,jr}^A - p_{max,jr}^B]^-,\sum \limits _{r\in R} e_r \sum \limits _{j\in J} [p_{max,jr}^A - p_{max,jr}^B]^+\right\} \nonumber \\ \end{aligned}$$

(A.13)

Proof

If $p_{max,jr}^A < p_{max,jr}^B$ for some $r\in R$ and $j\in J$ in instances A and B, the difference in the values of their objective function will be within the following range:

$$\begin{aligned} {[} e_r (p_{max,jr}^A - p_{max,jr}^B)md,0], \end{aligned}$$

If $p_{max,jr}^A > p_{max,jr}^B$, the range will be

$$\begin{aligned} {[}0,e_r (p_{max,jr}^A - p_{max,jr}^B)md], \end{aligned}$$

Any arbitrary set of fluctuations will form the following range representing an estimation of $\rho _{p,max}^\pi (A,B)$:

$$\begin{aligned} \left[ \sum \limits _{r\in R} e_r md\sum \limits _{j\in J} \left[ p_{max,jr}^A - p_{max,jr}^B\right] ^-,\sum \limits _{r\in R} e_r md \sum \limits _{j\in J} \left[ p_{max,jr}^A - p_{max,jr}^B\right] ^+\right] \end{aligned}$$

$\square$

Lemma 8

Let instances A and B differ only in parameters $p_{max,jr}$. Suppose that an optimal solution $\sigma ^A$ of instance A is applicable to instance B and an optimal solution $\sigma ^B$ of instance B is not applicable to instance A. If we apply the optimal solution of instance A $\sigma ^A$ as a solution to instance B, then the objective function values difference is bounded above by the following expression

$$\begin{aligned} \Delta ^{\sigma }_{n,p_{max}}(A,B) = V^B(\sigma ^A) - V^B(\sigma ^B) \le \sum \limits _{r\in R} e_r md \sum \limits _{j\in J} |p_{max,jr}^A - p_{max,jr}^B | \end{aligned}$$

(A.14)

Proof

We note that in our case any fluctuation in values $p_{max,jr}$ (and/or $p_{min,jr}$) does not impact the objective function, as it was shown in Lemma 5, i.e. $V^A(\sigma ^A) = V^B(\sigma ^A)$. We can use the same approach as in the proof of Lemma 4 to compare $V^A(\sigma ^A)$ and $V^B(\sigma ^B)$. Firstly, the absolute value of the difference has the form:

$$\begin{aligned} |V^A(\sigma ^A) - V^B(\sigma ^B)| \le \sum \limits _{r\in R} e_r \sum \limits _{t\in T} \sum \limits _{j\in J} |c_{jrt}^A - c_{jrt}^B |. \end{aligned}$$

Secondly, taking into account the limits for $c_{jrt}$,

$$\begin{aligned}{} & {} p_{min,jr}^Ad_{jt}^A \le c_{jrt}^A \le p_{max,jr}^A d_{jt}^A \\{} & {} p_{min,jr}^Bd_{jt}^B \le c_{jrt}^B \le p_{max,jr}^B d_{jt}^B, \end{aligned}$$

we can provide an upper estimation

$$\begin{aligned} |V^A(\sigma ^A) - V^B(\sigma ^B)| \le \sum \limits _{r\in R} e_r m \sum \limits _{j\in J} |p_{max,jr}^A - p_{max,jr}^B|d, \end{aligned}$$

as $d_{jt}\in [0,d]$ and there are m periods inside the planning horizon. $\square$

Lemma 9

Let instances A and B differ only by $W_{jr}$. If we apply the same schedule $\pi$ to the both instances, the upper bound for the difference in values of the objective function can be estimated as follows:

$$\begin{aligned} |V^A(\sigma ^A(\pi )) - V^B(\sigma ^B(\pi ))| \le \rho _W^\pi (A,B), \end{aligned}$$

(A.15)

where

$$\begin{aligned} \rho _W^\pi (A,B)= \max \left\{ \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \left[ W^A_{jr} - W^B_{jr}\right] ^+,\sum \limits _{r\in R} e_r \sum \limits _{j\in J} \left[ W^A_{jr} - W^B_{jr}\right] ^-\right\} . \end{aligned}$$

(A.16)

Proof

We can refer to the proof of Lemma 3. In this case, it is also possible to evaluate an upper bound for the difference in the values of the objective function and to consider it as an independent sum of estimations for fluctuations $W^A_{jr} \ne W^B_{jr}$.

Each fluctuation $W^A_{jr} > W^B_{jr}$ may lead to the difference in the values of the objective function within the following range $[W^B_{jr} - W^A_{jr},0]$. An upper bound for an aggregation of all these changes can be represented with the following range

$$\begin{aligned} \left[ \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \min \left\{ (W^B_{jr} - W^A{jr}),0\right\} ,0\right] . \end{aligned}$$

The same approach can be applied for the case where $W^A_{jr} < W^B_{jr}$. The difference in the values of the objective function caused by all fluctuations of $W_{jr}$:

$$\begin{aligned} \left[ 0,\sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^B_{jr} - W^A{jr}),0\right\} \right] . \end{aligned}$$

If we regroup the two previous cases, we obtain the following range:

$$\begin{aligned} \left[ \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \min \left\{ (W^B_{jr} - W^A{jr}),0\right\} ,\sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^B_{jr} - W^A{jr}),0\right\} \right] , \end{aligned}$$

and the following estimation

$$\begin{aligned} \rho _W^\pi (A,B)= \max \left\{ \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^A_{jr} - W^B_{jr}),0\right\} ,\sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^B_{jr} - W^A_{jr}),0\right\} \right\} \end{aligned}$$

$\square$

Lemma 10

Let instances A and B differ only in parameters $W_{jr}$. If we apply the optimal schedule $\pi ^A$ of instance A to instance B, then the upper bound for the difference in the values of the objective function can be estimated as follows:

$$\begin{aligned} V^B(\sigma ^B(\pi ^A)) - V^B(\sigma ^B(\pi ^B)) \le \Delta _W^\pi (A,B)= \sum \limits _{r\in R} e_r \sum \limits _{j\in J} |W^A_{jr} - W^B_{jr}|. \end{aligned}$$

(A.17)

Proof

As in the proof of Lemma 4, we consider the following cases $V^A(\sigma ^A(\pi ^A))$; $V^B(\sigma ^B(\pi ^B))$; $V^A(\sigma ^A(\pi ^B))$; $V^B(\sigma ^B(\pi ^A))$.

For the cases 1–4 and 6, we use Lemma 9, concluding that the considered difference is less than the right side of inequality (A.16), that is less than (A.17).

We prove the same for case 5 in the following way. We use the same approach as in Lemma 9. Firstly, we show that the difference

$$\begin{aligned} V^A(\sigma ^A(\pi ^A)) - V^B(\sigma ^B(\pi ^B)) \le \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^A_{jr} - W^B_{jr}),0\right\} \end{aligned}$$

Instance B can provide a better solution only with additional amount of resource. Secondly, we estimate the difference

$$\begin{aligned} V^B(\sigma ^B(\pi ^A)) - V^A(\sigma ^A(\pi ^A)) \le \sum \limits _{r\in R} e_r \sum \limits _{j\in J} \max \left\{ (W^B_{jr} - W^A_{jr}),0\right\} \end{aligned}$$

In instance B, the same schedule $\pi ^A$ may provide a worse objective function value, with the difference up to the total reduction of the resource amount. These two components are bounded and form the initial difference $V^B(\sigma ^B(\pi ^A)) - V^B(\sigma ^B(\pi ^B))$, therefore, the upper bound is the same as in inequality (A.16). $\square$

Lemma 11

Let instances A and B differ in one of parameters $p_{min,jr}$. $p_{max,jr}$ or $W_{jr}$. Any schedule $\pi$, applicable to instance A, is also applicable to instance B, if B is solvable and:

$$\begin{aligned} d_{min,j}^B \le d_{min,j}^A;\;d_{max,j}^A \le d_{max,j}^B;\;\forall j\in J, \end{aligned}$$

(A.18)

or it can be reformulated in a linear form for the parameters of B:

$$\begin{aligned} W_{jr}^B \le d_{min,j}^A p_{max,jr}^B;\; d_{max,j}^A p_{min,jr}^B \le W_{jr}^B;\;\forall j\in J;\;\forall r\in R. \end{aligned}$$

(A.19)

Proof

From Definition 5, we see that a schedule $\pi$ must guarantee that

$$\begin{aligned} \sum \limits _{t\in T} d_{jt} \in \left[ d_{min,j}^B, d_{max,j}^B\right] ;\;\forall j \in J. \end{aligned}$$

If is is applicable to A, then

$$\begin{aligned} \sum \limits _{t\in T} d_{jt} \in \left[ d_{min,j}^A, d_{max,j}^A\right] ;\;\forall j \in J. \end{aligned}$$

Basically, we can guarantee that $\pi$ is applicable to B if range $[d_{min,j}^A, d_{max,j}^A]$ is fully included in $[d_{min,j}^B, d_{max,j}^B]$, so

$$\begin{aligned} d_{min,j}^B \le d_{min,j}^A;\;d_{max,j}^A \le d_{max,j}^B;\;\forall j\in J. \end{aligned}$$

A linear condition for B is obtained from the definition of $d_{min,j}$ and $d_{max,j}$. If we consider $d_{min,j}^A$ and $d_{max,j}^A$ given and fixed, we rewrite the conditions (27):

$$\begin{aligned} \max \limits _{r\in R} \frac{W_{jr}^B}{p_{max,jr}^B} \le d_{min,j}^A;\;d_{max,j}^A \le \min \limits _{r\in R} \frac{W_{jr}^B}{p_{min,jr}^B};\;\forall j\in J, \end{aligned}$$

and reformulate these conditions without a maximum:

$$\begin{aligned} \frac{W_{jr}^B}{p_{max,jr}^B} \le d_{min,j}^A;\;d_{max,j}^A \le \frac{W_{jr}^B}{p_{min,jr}^B};\;\forall j\in J;\;\forall r\in R. \end{aligned}$$

$\square$

Theorem 1

Let instances A and E differ by parameters $L_{rt}$, $W_{jr}$, $p_{min,jr}$, and $p_{max,jr}$. If we apply a schedule $\pi ^A$ that is optimal for instance A to instance E, the upper bound for difference in the values of the objective function can be estimated as follows:

$$\begin{aligned} V^E(\sigma ^E(\pi ^A)) - V^E(\sigma ^E) \le \Delta ^\pi (A,E); \end{aligned}$$

(A.20)

and

$$\begin{aligned} \Delta ^\pi (A,E) =\Delta _L^\pi (A,E) + \Delta _{p_{min}}^\pi (A,E) + \Delta _{p_{max}}^\pi (A,E) + \Delta _{W}^\pi (A,E). \end{aligned}$$

(A.21)

Proof

It is possible to separate this function:

• B, all parameters equal to instance A except $L_{rt}$, and $L_{rt}^B=L_{rt}^E$;

• C, all parameters equal to instance B except $p_{min,jr}$, $p_{min,jr}^C=p_{min,jr}^E$;

• D, all parameters equal to instance E except $p_{max,jr}$, $p_{max,jr}^D=p_{max,jr}^E$;

We note that instances D and E differ only in parameters $W_{jr}$.

As each expression includes a sum of absolute values, $\Delta ^\pi (A,E)$ has an addictive property:

$$\begin{aligned} \Delta ^\pi (A,E) \le \Delta ^\pi (A,B) + \Delta ^\pi (B,C) + \Delta ^\pi (C,D) + \Delta ^\pi (D,E). \end{aligned}$$

We take into account that $\Delta ^\pi (A,B) = \Delta _L^\pi (A,B)$, $\Delta ^\pi (B,C) =\Delta ^\pi _{p_{min}}(B,C)$, $\Delta ^\pi (C,D) = \Delta ^\pi _{p_{max}}(C,D)$, $\Delta ^\pi (D,E) = \Delta ^\pi _W(D,E)$, so

$$\begin{aligned} V^E(\sigma ^E(\pi ^A)) - V^E(\sigma ^E) \le \Delta _L^\pi (A,B) + \Delta ^\pi _{p_{min}}(B,C) + \Delta ^\pi _{p_{max}}(C,D) + \Delta ^\pi _W(D,E), \end{aligned}$$

and as the parameters of all instances B, C, D are either equal to parameters A or E,

$$\begin{aligned} V^E(\sigma ^E(\pi ^A)) - V^E(\sigma ^E) \le \Delta _L^\pi (A,E) + \Delta _{p_{min}}^\pi (A,E) + \Delta _{p_{max}}^\pi (A,E) + \Delta _{W}^\pi (A,E). \end{aligned}$$

$\square$

Lemma 12

Suppose that instance B is produced from instance A by the following transformation: all job-resource-related parameters are multiplied by a coefficient $k > 0$. We will define it as $kA = B$ meaning that $k L_{rt}^A = L_{rt}^B;\;\forall r \in R;\;\forall t \in T;$ and $k p_{min,jr}^A = p_{min,jr}^B;\; k p_{max,jr}^A = p_{max,jr}^B;\; k W_{jr}^A = W_{jr}^B;\;\forall j\in J;\;\forall r\in R.$

In this case, both instances A and B have the same set of feasible schedules and the set of optimal schedules with scaled solution variables

$$\begin{aligned} k c_{jrt}^A = c_{jrt}^B;\;\forall j\in J;\;\forall r\in R;\; k o_{rt}^A = o_{rt}^B;\;\forall r \in R;\;\forall t \in T. \end{aligned}$$

(A.22)

Thus, objective function values are also scaled:

$$\begin{aligned} V^B(\sigma ^B) = k V^A(\sigma ^A). \end{aligned}$$

(A.23)

Proof

Firstly, this transformation does not change any parameter involved in the definition of a schedule, applicable to an instance (see Def. 5). It does not change precedence relations nor values of minimal and maximal duration. These values equal to a ratio of required workload $W_{jr}$ and a maximal or a minimal amount of allocated resource ($p_{max,jr}$ or $p_{min,jr}$), both multiplied by k. Thus, such a transformed instance is still solvable.

Secondly, we consider the solutions. If schedule $\pi ^A$ with variables $d_{jt}^A$ is optimal for instance A, providing a solution $\sigma ^A(\pi ^A)$ with variables $c_{jrt}^A$, then this schedule is also applicable to instance B. It produces a scaled optimal solution $\sigma ^B(\pi ^A)$ with variables $c_{jrt}^B$. This solution is also optimal, as the solution variables $c_{jrt}$ are defined on a base of a schedule (i.e. variables $d_{jt}$), that are connected by the constraints (2). We can represent these constraints with parameters of instance A:

$$\begin{aligned} \left. \begin{aligned} k p_{min,jr}^A d_{jt}^A = p_{min,jr}^B d_{jt}^B \le c_{jrt}^B \\ c_{jrt}^B \le p_{max,jr}^B d_{jt}^B = k p_{max,jr}^A d_{jt}^A,\; \end{aligned} \right\} \quad \forall j\in J,\;\forall r\in R,\;\forall t\in T; \end{aligned}$$

and by $W_{jr}$ with constraints

$$\begin{aligned} \sum \limits _{t\in T} c_{jrt}^B = W_{jr}^B = k W_{jr}^A,\;\forall j\in J,\;\forall r\in R. \end{aligned}$$

All these linear constraints are scaled for instance B, and it keeps the same ratio between all these parameters. Finally, objective function (3) involves variables: $o_{rt}\in [0,\infty )$

$$\begin{aligned} Minimize\;\sum \limits _{r\in R}\sum \limits _{t\in T} e_r o_{rt}, \end{aligned}$$

defined by constraints (4):

$$\begin{aligned} o_{rt} \ge \sum \limits _{j\in J} c_{jrt} - L_{rt}, \;\forall t\in T,\;\forall r\in R; \end{aligned}$$

where both $c_{jrt}$ and $L_{rt}$ are multiplied by k in instance B:

$$\begin{aligned} \sum \limits _{j\in J} c_{jrt}^B - L_{rt}^B = k \left( \sum \limits _{j\in J} c_{jrt}^A - L_{rt}^A\right) , \;\forall t\in T,\;\forall r\in R. \end{aligned}$$

As we minimize $o_{rt}$, then there is no reason to change neither the structure of the schedule nor the solution in the changed instance B. That is why the solution with variables $c_{jrt}^B$ is optimal, as well as schedule $\pi ^A$ providing it with variables $d_{jt}^A$. Therefore, there exists solution $\sigma ^B$ with variables $c_{jrt}^B = k c_{jrt}^A$ based on the same schedule and it is optimal with the following objective value

$$\begin{aligned} V^B(\sigma ^B) = k V^A(\sigma ^A). \end{aligned}$$

$\square$