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In EarlinessTardiness (E/T) scheduling approach, the JustInTime (JIT) schedule is a schedule with zero earliness and zero tardiness. However, this is an optimal schedule and even notional in some instances where tardiness and earliness are inevitable. However, minimizing the deviation at the upper region (tardiness) and the lower region (earliness) from the JIT schedule is a challenge. This work proposes solutions. Two proposed heuristics; TA1 and TA2 as well as some existing heuristics were explored to solve simulated problems ranging from 5≤n≤400 and the results obtained were benchmarked against the JIT schedule. The results obtained show that one of the heuristics, TA2 yielded JIT schedules for many problem sizes at the lower and upper deviation than other solution methods.
JustInTime (JIT), EarlinessTardiness (E/T) scheduling problem, optimal schedule, deviation, heuristics
There exist global economic meltdown and most production firms are faced with the challenges of optimizing profit. Therefore, minimizing sources of leakages and income losses like overproduction, high inventory cost, waiting and down tool time is a prerequisite. JustInTime (JIT) production has proved to be an essential requirement of worldclass manufacturing concept [1]. This has made researchers to explore different variants of EarlinessTardiness (E/T) scheduling problems to support the realization of JIT environment characterized by zero earliness, zero tardiness and zero inventories. However, JIT schedule is either an optimal schedule which is NP hard or even notional schedule and thus deviation is inevitable. In a due window approach, the due date has three components: The earliest due date, the original due date and the latest due date. The interval between the earliest and latest due date is called the due window [2]. This work proposes solution methods to minimize the deviation from the JIT schedule using the E/T scheduling problem with due window approach.
In a general scheduling system, penalty is usually associated with tardy and late jobs while early jobs are compensated. However, this is not always valid. Akande and Ajisegiri [3] discussed extensively three classes of Earliness – Tardiness scheduling problems and highlighted the variation of associated penalty with early jobs. The class three, as defined by the authors, associated penalty with both early and the tardy jobs. This is the concept of JIT production system. There is a global growing interest in JIT production, because the system ensures that all the jobs are completed at exactly due date or within the due windows and thus zero inventory is achieved [4]. Though several researchers have explored the Earliness Tardiness scheduling problems but literature is sparse in which the problem is used to measure the deviation from the JIT schedule. This is the basis for this work. Nevertheless, Sourd [5] explored a dynamic programming procedure for solving the scheduling problems of minimizing the penalties associated with not delivering on time (JIT) and the idleness cost using the deviation in earliness–tardiness function. Also, authors like [6, 7] considered the problem of minimizing the weighted earliness and tardiness on a single machine. The dynamic variant of E/T scheduling problems was explored by Mazzini and Armentano [8] and the results obtained were used as a benchmark by Oyetunji and Oluleye [9] for two proposed two heuristics named ETA1 and ETA2. In this work, static variant of E/T scheduling problem with the due windows approach is considered and explored to measure the deviation from the JIT schedule. Liman et al; Janiak and Marek; Zhu, et al. [1012] among others are some of researchers that have also explored various functions of due windows for scheduling problems.
3.1 Assumptions and notations
The assumptions made in solving the problem are outlined for clarity as follows:
Some notations are also employed, and they are presented in Table 1.
Table 1. Notations used for solving the problem
I 
Job position $\mathrm{i}=1,2, \ldots, \mathrm{n}$ 
N 
Number of Jobs 
$p_{i}$ 
Processing time of job i 
$C_{i}$ 
Completion time of Job i 
$\mathrm{d}_{\mathrm{j}}$ 
Original Due date of Job i 
$D_{i}^{e}$ 
Earliest due date of job i 
$D_{i}^{L}$ 
Latest due date of job i 
$\mathrm{E}_{\mathrm{I}}$ 
Earliness of Job i 
$\mathrm{T}_{\mathrm{i}}$ 
Tardiness of Job i 
$\mathrm{L}_{\mathrm{i}}$ 
Lateness of Job i 
$N_{J I T}$ 
Number of Just in Time Jobs 
UD 
Upward Deviation 
DD 
Downward Deviation 
DOD 
Degree of Deviation 
NSG 
Deviation value less than 1 (unit) 
SG 
Deviation value greater than 1 (unit) 
SPT 
Shortest Processing Time 
MDD 
Modified Due Date 
FCFS 
First Come First Schedule 
EDD 
Earliest Due Date 
3.2 Problem definition
Consider an automobile servicing firm with only one technician working on a service bay and one service advisor for customers scheduling and appointment. For every job completed after the allocated due window given to the customer, there is always a penalty either in terms of goodwill or reduction in the service charge. Also, for all the jobs completed before the due window, there is an associated cost with parking the car (inventory cost) [13, 14]. Therefore, a system where the completed vehicle is released to the customer at the point of completion will eliminate not only the inventory cost but also the penalty cost associated with lateness or tardiness. Such a system is a typical JIT system. At this point, it is expected that $\mathrm{d}_{\mathrm{i}}=\mathrm{C}_{\mathrm{i}}$.
The total number of jobs completed within the due window is given by
$N_{J I T}=\sum_{i=1}^{n} J I T_{i}$ (1)
where
$J I T_{i}=\left\{\begin{array}{c}1 \text { if } \mathrm{d}_{i}=\mathrm{C}_{i} \\ 0 \text { Otherwise }\end{array}\right.$ (2)
If JIT system is achieved, for $i=1,2, \ldots, n$
$\mathrm{d}_{i}=\mathrm{C}_{i}$ (3)
$\mathbf{n}=N_{J I T}=\sum_{i=1}^{n} J I T_{i}$ (4)
Also, the tardiness of each jobs, i and total tardiness of all the jobs will be given by
$\boldsymbol{T}_{i}=\max \left\{0,\left(\mathrm{C}_{i}\mathrm{d}_{i}\right)\right\}=0$ (5)
$T_{\text {tot }}=\sum_{i=1}^{n} \boldsymbol{T}_{i}=\mathbf{0}$ (6)
Similarly, the earliness of each jobs, i and the total earliness of all the jobs will be given by
$E_{i}=\max \left\{L_{i}, 0\right\}=0$ (7)
$E_{\text {tot }}=\sum_{i=1}^{n} E_{i}=0$ (8)
However, it is not feasible to always attain JIT condition, thus
$\mathbf{n} \gg N_{J I T}$ (9)
$T_{i} \geq 0$ (10)
$T_{\text {tot }} \geq 0$ (11)
$E_{i} \geq 0$ (12)
$E_{\text {tot }} \geq 0$ (13)
However, the target is to minimize these inevitable deviations which are the upward curve deviation associated with the tardiness (jobs completed after the due date) and the downward curve deviation associated with the earliness (job completed earlier than the due dates). Therefore, the deviation in JIT can be described as a piecewise function with the $T_{\text {tot }}$ in the upper region domain and the $E_{\text {tot }}$ in the lower region domain. This condition is illustrated in Figure 1.
Figure 1. Completion time against due date to show deviation from JIT
Therefore, if the total tardiness (upward deviation) and the total earliness (downward deviation) are minimized simultaneously with respect to the equilibrium or JIT values, then the total deviation is also minimized. The two solution methods proposed are hereby discussed.
3.3 Proposed solution methods
The steps of the heuristics are stated as follows.
TA1 ALGORITHM: This algorithm is based on the theorem stated below:
When solving the $\sum_{i=1}^{n} E_{i}+\sum_{i=1}^{n} T_{i}$ for any two jobs say j and K, there exists an optimal sequence for which j appear before k if the following conditions hold:
I. $P_{\mathrm{j}} \leq P_{k}$ (14)
II. $\mathrm{D}_{\mathrm{j}} \leq D_{k}$ (15)
For due windows, this corollary is still valid.
If the Eqns. (14) and (15) are valid then, it can be deduced that: $\mathrm{P}_{\mathrm{j}}+\mathrm{D}_{\mathrm{j}} \leq P_{k}+\mathrm{D}_{\mathrm{k}}$.
Therefore, the steps of the TA1 heuristics are outlined as follows:
STEP 1: Compute the factor time (P+D) for the jobs in job set A.
STEP 2: Arrange the jobs set A in order of increasing Factor Time (P+D) and put the same in Job set C. If there is a tie set, break the tie arbitrarily.
STEP 3: Set i=1 where i is the index number of the job in job set C and update Job Set B with job, i and remove the same job from Job set C. If there is a tie, update Job Set B with the job that has the lowest due date among the tie jobs. Continue until all the jobs have been updated. Job set B is the required sequence.
STEP 5: Compute the earliness and the tardiness of each of the job in the Job Set B.
STEP 8: Stop
TA 2 ALGORITHM
The statements of the TA2 heuristics are outlined as follows:
STEP 1: Arrange the jobs Set A in the order of increasing Factor Time (P + D) and put the same in Job Set C. if there is a tie break the tie arbitrarily.
STEP 2: Arrange Job Set A using the MDD and put same in job Set B.
STEP 3: Compute the earliness and the tardiness of each of the jobs in the Job set B and Job set C.
STEP 4: Compute the FUNCTION $E_{i}+T_{i}$ of each of the jobs in the Job set B and Job set C.
STEP 5: Combine the two schedules by scheduling the job at the same level and with the minimum FUNCTION $E_{i}+T_{i}$ and which has not been previously scheduled.
STEP 6: Remove any jobs that existed more than once. Compute the length of the resultant schedule. Called the schedule Job Set C.
STEP 7: If the length of Job Set C is equal to the length of the job set A. Then Job Set C is the required schedule Then go to step 9. Else go to step 8.
STEP 8: Subtract Job Set C from Job Set A to obtain the jobs that have not been scheduled. Thus, Job Set H = Job Set C – Job Set A.
STEP 9: Arrange job set H at the back of job set C in increasing order of due date. This is called Job set P.
STEP 10: Compute the tardiness and the total tardiness of the optimal sequence schedule.
STEP 11: Stop
3.4 Problem generation for simulation
The utilities of the proposed solution methods were demonstrated by simulated some single processor scheduling problems using the desktop tool module (editor) from MATLAB R2010 programming language. Equations explored by Suer et al. [15]; Akande and Ajisegiri, [16]; Sunday et al. [17]; Bedhief and Dridi [18]; and Joshi and Satpathy [19] were modified to generate the required parameters which include the processing time and the due windows as follows:
$D_{j}=R_{j}+K P_{j}$ (16)
$D_{j}^{L / E}=D_{j} \pm\left(A_{j} \times D_{j}\right)$ (17)
$D_{j}^{e}=D_{j}\left(D_{j} \times A_{j}\right)$ (18)
$D_{j}^{L}=D_{j}+\left(D_{j} \times A_{j}\right)$ (19)
where,
$D_{j}^{e}$: is the earliest due date for job j.
$D_{j}$: is the original due date.
$D_{j}^{L}$: is the latest due date for job j.
Aj: is the flow allowance assigned to job j at time zero. Is set at (20%  40% of $D_{j}$).
In this simulating experiment, the proposed solution methods as well as the existing ones (SPT, EDD, MDD and FCFS) were tested with the three components of the due windows.
The results of the computational experiment are grouped into two classes. Tables 2 – 12 show the results of the total earliness and the tardiness and the Degree of Deviation (DOD) for each of the considered problem sizes for all the solution methods for the three due window components. Table 13, Table 14 and Table 15 show the value of the tardiness for the early due window, original due window, and the latest due window respectively. Table 16, Table 17, and Table 18 show the value of earliness for the early due window, original due window, and the latest due window respectively. However, it should be noted that the value of the earliness and the tardiness for the notional JIT schedule is zero for all the problem sizes. Furthermore, Figure 2, Figure 3, and Figure 4 illustrate the degree of deviation of the total tardiness (Upper deviation) from the notional JIT in the early due window, original due window, and the latest due window respectively, while Figure 5, Figure 6 and Figure 7 show the lower deviation (Total earliness) in the three windows respectively.
Table 2. Results of the total Earliness and Tardiness and the degree of deviation for 5 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
29.65 
2.85 
SG 
NSG 
0.00 
118.00 
JIT 
SG 
0.00 
35.85 
JIT 
SG 
MDD 
28.35 
0.55 
SG 
NSG 
0.00 
8.00 
JIT 
SG 
0.00 
35.85 
JIT 
SG 
TA1 
28.35 
0.55 
SG 
NSG 
0.00 
117.00 
JIT 
SG 
0.00 
35.85 
JIT 
SG 
TA2 
28.35 
0.55 
SG 
NSG 
0.00 
47.00 
JIT 
SG 
0.00 
35.85 
JIT 
SG 
FCFS 
35.35 
0.55 
SG 
NSG 
12.00 
15.00 
SG 
SG 
3.50 
34.35 
SG 
SG 
EDD 
30.35 
0.55 
SG 
NSG 
0.00 
117.00 
JIT 
SG 
0.00 
35.85 
JIT 
SG 
Table 3. Results of the total Earliness and Tardiness and the degree of deviation for 10 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
135.0 
0.96 
SG 
NSG 
9.0 
158.00 
SG 
SG 
82.0 
25.60 
SG 
SG 
MDD 
135.0 
0.95 
SG 
NSG 
29.0 
3.00 
SG 
SG 
67.0 
6.20 
SG 
SG 
TA1 
139.0 
0.95 
SG 
NSG 
0.00 
118.00 
JIT 
SG 
73.0 
8.35 
SG 
SG 
TA2 
135.0 
0.95 
SG 
NSG 
29.0 
3.00 
SG 
SG 
67.0 
6.20 
SG 
SG 
FCFS 
225.0 
6.65 
SG 
SG 
20.0 
32.00 
SG 
SG 
162.0 
42.80 
SG 
SG 
EDD 
139.0 
0.95 
SG 
NSG 
0.00 
104.00 
JIT 
SG 
80.0 
6.20 
SG 
SG 
Table 4. Results of the total Earliness and Tardiness and the degree of deviation for 15 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
350.0 
0.00 
SG 
JIT 
107.0 
107.0 
SG 
SG 
203.0 
14.70 
SG 
SG 
MDD 
350.0 
0.00 
SG 
JIT 
200.0 
7.0 
SG 
SG 
1.97.0 
6.00 
SG 
SG 
TA1 
359.0 
0.00 
SG 
JIT 
105.0 
40.0 
SG 
SG 
222.0 
3.15 
SG 
SG 
TA2 
350.0 
0.00 
SG 
JIT 
118.0 
34.0 
SG 
SG 
197.0 
6.00 
SG 
SG 
FCFS 
543.0 
6.30 
SG 
SG 
196.0 
25.0 
SG 
SG 
401.0 
26.80 
SG 
SG 
EDD 
366.0 
0.00 
SG 
JIT 
118.0 
34.0 
SG 
SG 
244.0 
2.75 
SG 
SG 
Table 5. Results of the total Earliness and Tardiness and the degree of deviation for 20 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 

UD 
LD 

SPT 
8.08e+2 
1.25 
SG 
SG 
2.97e+2 
147.00 
SG 
SG 
3.67e+2 
5.50 
SG 
SG 
MDD 
8.07e+2 
0.00 
SG 
JIT 
5.23e+2 
2.00 
SG 
SG 
3.69e+2 
1.75 
SG 
SG 
TA1 
8.30e+2 
0.00 
SG 
JIT 
3.03e+2 
50.00 
SG 
SG 
3.94e+2 
4.80 
SG 
SG 
TA2 
8.07e+2 
0.00 
SG 
JIT 
4.39e+2 
13.00 
SG 
SG 
3.69e+2 
1.75 
SG 
SG 
FCFS 
11.89e+2 
0.00 
SG 
JIT 
4.49e+2 
28.00 
SG 
SG 
6.83e+2 
12.25 
SG 
SG 
EDD 
8.71e+2 
0.00 
SG 
JIT 
3.47e+2 
41.00 
SG 
SG 
4.07e+2 
1.75 
SG 
SG 
Table 6. Results of the total Earliness and Tardiness and the degree of deviation for 40 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
2.84e+3 
0.00 
SG 
JIT 
2.29e+3 
159.00 
SG 
SG 
1.64e+3 
1.40 
SG 
SG 
MDD 
2.84e+3 
0.00 
SG 
JIT 
2.98e+3 
3.00 
SG 
SG 
1.64e+3 
0.35 
SG 
NSG 
TA1 
2.89e+3 
0.00 
SG 
JIT 
2.64e+3 
32.00 
SG 
SG 
1.70e+3 
0.35 
SG 
NSG 
TA2 
2.84e+3 
0.00 
SG 
JIT 
2.87e+3 
6.00 
SG 
SG 
1.64e+3 
0.35 
SG 
NSG 
FCFS 
4.00e+3 
5.40 
SG 
SG 
3.31e+3 
24.00 
SG 
SG 
2.70e+3 
17.15 
SG 
SG 
EDD 
2.96e+3 
0.00 
SG 
JIT 
2.76e+3 
25.00 
SG 
SG 
1.73e+3 
0.35 
SG 
NSG 
Table 7. Results of the total Earliness and Tardiness and the degree of deviation for 50 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 

UD 
LD 
UD 
LD 

SPT 
3790 
0.95 
SG 
NSG 
3360 
144 
SG 
SG 
3020 
3.05 
SG 
SG 
MDD 
3790 
0.00 
SG 
JIT 
4430 
7.00 
SG 
SG 
3010 
0.35 
SG 
NSG 
TA1 
3910 
0.00 
SG 
JIT 
3990 
29.00 
SG 
SG 
3170 
0.35 
SG 
NSG 
TA2 
3790 
0.00 
SG 
JIT 
4340 
14.00 
SG 
SG 
3010 
0.35 
SG 
NSG 
FCFS 
6120 
0.00 
SG 
JIT 
5340 
29.00 
SG 
SG 
4640 
20.1 
SG 
SG 
EDD 
4050 
0.00 
SG 
JIT 
4300 
26.00 
SG 
SG 
3270 
0.35 
SG 
NSG 
Tables 212 reveal that the two proposed performance measures yielded better results than the selected existing solution methods. This is because the results of TA1 and TA2 yielded a more optimal JIT schedule in the three windows at both the upper deviation (total Tardiness) and the lower deviation (Total earliness) in the three windows components. Also, FCFS yielded the worst results among all the solution methods for each of the problem sizes. However, to measure the performance of the solution over the range of the considered problem sizes (5400), Tables 1318 also shows the values of the total earliness and the tardiness at each of the window’s components. Figure 2  Figure 7 also shows the deviation graphically.
Figure 2 shows that FCFS has the highest upper deviation. The SPT also has a higher deviation especially for the lower number of jobs. As the number of jobs increases, the performance of SPT, EDD, and the two proposed solution methods (TA1 and TA2) coincides.
Table 8. Results of the total Earliness and Tardiness and the degree of deviation for 100 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
LCOF 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
17900 
0.30 
SG 
NSG 
16400 
260.00 
SG 
NSG 
16200 
1.05 
SG 
SG 
MDD 
17900 
0.00 
SG 
JIT 
20200 
1.00 
SG 
NSG 
16204 
0.35 
SG 
NSG 
TA1 
18500 
0.00 
SG 
JIT 
19400 
6.00 
SG 
NSG 
17100 
0.35 
SG 
NSG 
TA2 
17900 
0.00 
SG 
JIT 
19900 
4.00 
SG 
NSG 
16200 
0.35 
SG 
NSG 
FCFS 
25900 
4.50 
SG 
SG 
23300 
17.00 
SG 
NSG 
22600 
23.0 
SG 
SG 
EDD 
19300 
0.00 
SG 
JIT 
20200 
4.00 
SG 
NSG 
17400 
0.35 
SG 
NSG 
Table 9. Results of the total Earliness and Tardiness and the degree of deviation for 150 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
3.819e+4 
0.95 
SG 
NSG 
3.606e+4 
312 
SG 
SG 
3.344e+4 
2.80 
SG 
SG 
MDD 
3.819e+4 
0.00 
SG 
JIT 
4.386e+4 
1.00 
SG 
SG 
3.343e+4 
0.35 
SG 
NSG 
TA1 
3.924e+4 
0.00 
SG 
JIT 
4.176e+4 
1.00 
SG 
SG 
3.500e+4 
0.35 
SG 
NSG 
TA2 
3.819e+4 
0.00 
SG 
JIT 
4.386e+4 
1.00 
SG 
SG 
3.343e+4 
0.35 
SG 
NSG 
FCFS 
5.468e+4 
8.80 
SG 
SG 
5.152e+4 
13.00 
SG 
SG 
5.092e+4 
46.15 
SG 
SG 
EDD 
4.067e+4 
0.00 
SG 
JIT 
4.417e+4 
1.00 
SG 
SG 
3.596e+4 
0.35 
SG 
NSG 
Table 10. Results of total Earliness and Tardiness and the degree of deviation for 200 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
6.679e+4 
0.30 
SG 
NSG 
6.051e+4 
356.00 
SG 
SG 
6.721e+4 
0.35 
SG 
NSG 
MDD 
6.679e+4 
0.00 
SG 
JIT 
7.728e+4 
2.00 
SG 
SG 
6.721e+4 
0.35 
SG 
NSG 
TA1 
6.860e+4 
0.00 
SG 
JIT 
7.335e+4 
15.00 
SG 
SG 
7.095e+4 
0.35 
SG 
NSG 
TA2 
6.679e+4 
0.34 
SG 
NSG 
7.689e+4 
5.00 
SG 
SG 
6.721e+4 
0.35 
SG 
NSG 
FCFS 
9.690e+4 
5.35 
SG 
SG 
9.220e+4 
19.00 
SG 
SG 
9.815e+4 
33.65 
SG 
SG 
EDD 
7.134e+4 
0.00 
SG 
JIT 
7.734e+4 
8.00 
SG 
SG 
7.279e+4 
0.35 
SG 
NSG 
Table 11. Results of total Earliness and Tardiness and the degree of deviation for 300 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
1.661e+5 
0.30 
SG 
NSG 
1.560e+5 
303.00 
SG 
SG 
1.605e+5 
4.80 
SG 
SG 
MDD 
1.661e+5 
0.30 
SG 
NSG 
1.870e+5 
0.00 
SG 
JIT 
1.605e+5 
0.00 
SG 
JIT 
TA1 
1.715e+5 
0.00 
SG 
JIT 
1.812e+5 
0.00 
SG 
JIT 
1.673e+5 
0.35 
SG 
NSG 
TA2 
1.661e+5 
0.00 
SG 
JIT 
1.870e+5 
0.00 
SG 
JIT 
1.605e+5 
0.00 
SG 
JIT 
FCFS 
2.342e+5 
4.45 
SG 
SG 
2.157e+5 
24.00 
SG 
SG 
2.267e+5 
29.50 
SG 
SG 
EDD 
1.779e+5 
0.00 
SG 
JIT 
1.894e+5 
0.00 
SG 
JIT 
1.712e+5 
0.35 
SG 
NSG 
Table 12. Results of total Earliness and Tardiness and the degree of deviation for 400 x 1 problem size

Earliest Due Date 
Original Due Date 
Latest Due Date 

Solution Method 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 
$\sum_{i=1}^{n} T_{i}$ 
$\sum_{i=1}^{n} E_{i}$ 
DOD 

UD 
LD 
UD 
LD 
UD 
LD 

SPT 
2.677e+5 
0.00 
SG 
JIT 
2.708e+5 
304 
SG 
SG 
2.749e+5 
3.05 
SG 
SG 
MDD 
2.677e+5 
0.00 
SG 
JIT 
3.240e+5 
0.00 
SG 
JIT 
2.749e+5 
0.35 
SG 
NSG 
TA1 
2.750e+5 
0.00 
SG 
JIT 
3.161e+5 
1.00 
SG 
SG 
2.876e+5 
0.35 
SG 
NSG 
TA2 
2.677e+5 
0.00 
SG 
JIT 
3.240e+5 
0.00 
SG 
JIT 
2.749e+5 
0.35 
SG 
NSG 
FCFS 
3.846e+5 
2.85 
SG 
SG 
3.826e+5 
10.00 
SG 
SG 
3.876e+5 
28.85 
SG 
SG 
EDD 
2.869e+5 
0.00 
SG 
JIT 
3.302e+5 
1.00 
SG 
SG 
2.946e+5 
0.35 
SG 
NSG 
In the original due date window, as revealed in Figure 3, all the solution methods except FCFS yielded the JIT schedule for 5x1 problem size while only EDD, TA1, and TA2 also yielded JIT schedule for 10x1 problem sizes. However, as the job size increases, the upper deviation increase with SPT has the lowest deviation at 400x1 problem sizes.
In the latest due date window, as revealed in Figure 4, all the solution methods except FCFS yielded the JIT schedule for the 5x1 problem size. However, as the job size increases, the upper deviation increases with SPT, MDD, and TA2 coincides with the lowest deviation at 400x1 problem sizes while FCFS and TA1 has the highest deviation.
The results of lower deviation (Total earliness) were erratic. Though most of the solution method yielded a JIT schedule for most of the problem sizes, the results of FCFS and SPT shows the highest deviation.
Figure 6 reveals that SPT yielded the highest deviation from the JIT schedule for all the problem sizes.
Also, TA1 and EDD show high deviation for problem sizes, n≤50.
Table 13. The Tardiness value for the Earliest due date window for all the solution methods
Problem size 
JIT SCHEDULE 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
29.65 
28.35 
30.35 
35.35 
28.35 
28.35 
10X1 
0 
135 
135 
139 
225 
139 
135 
15 X1 
0 
350 
350 
366 
543 
359 
350 
20X1 
0 
808 
807 
871 
1190 
830 
807 
40x1 
0 
2840 
2840 
2960 
40000 
2890 
2840 
50x1 
0 
3790 
3790 
4050 
6120 
3910 
3790 
100x1 
0 
17900 
17900 
19300 
25900 
18500 
17900 
200x1 
0 
66790 
66790 
71340 
96900 
68600 
66790 
300x1 
0 
166100 
166100 
177900 
234200 
171500 
166100 
400x1 
0 
267700 
267700 
286900 
384600 
275000 
267700 
Figure 2. Plot of Total tardiness against the Problem sizes for the early due date window
Table 14. The Tardiness value for the Original due date window for all the solution methods
Problem size 
JIT SCHEDULE 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
0 
0 
0 
12 
0 
0 
10X1 
0 
9 
29 
0 
20 
0 
0 
15 X1 
0 
107 
200 
118 
196 
118 
105 
20X1 
0 
297 
523 
347 
449 
303 
439 
40x1 
0 
2290 
2980 
2760 
3310 
2640 
2870 
50x1 
0 
3790 
3790 
4050 
6120 
3910 
3790 
100x1 
0 
16400 
20200 
20200 
23300 
19400 
19900 
200x1 
0 
60510 
77280 
77340 
92200 
73350 
76890 
300x1 
0 
156000 
187000 
189400 
215700 
181200 
187000 
400x1 
0 
270800 
324000 
330200 
382600 
316100 
324000 
Figure 3. Plot of Total tardiness against the Problem sizes for the original due date window
Table 15. The Tardiness value for the Latest due date window for all the solution methods
Problem size 
JIT SCHEDULE 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
0 
0 
0 
3.5 
0 
0 
10X1 
0 
82 
67 
80 
162 
73 
67 
15 X1 
0 
203 
197 
244 
401 
222 
197 
20X1 
0 
367 
369 
407 
683 
394 
369 
40 x1 
0 
1640 
1640 
1730 
2700 
1700 
1640 
50x10 
0 
3020 
3010 
3270 
4640 
3170 
3010 
100x1 
0 
16200 
16200 
17400 
22600 
17100 
16200 
150 x1 
0 
33440 
33430 
35960 
50920 
35000 
33430 
200x1 
0 
67210 
67210 
72790 
98150 
150000 
67210 
300x1 
0 
160500 
160500 
171200 
226700 
167300 
160500 
400x1 
0 
274900 
274900 
294600 
387600 
287600 
274900 
Figure 4. Plot of Total tardiness against the Problem sizes for the latest due date window
Table 16. The total Earliness value for Early Due Date Earliness for all the solution methods
Problem size 
JIT SCHEDULE 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
2.85 
0.55 
0.55 
0.55 
0.55 
0.55 
10X1 
0 
0.96 
0.95 
0.95 
6.65 
0.95 
0.95 
15 X1 
0 
0 
0 
0 
6.3 
0 
0 
20X1 
0 
1.25 
0 
0 
0 
0 
0 
40x1 
0 
0 
0 
0 
5.4 
0 
0 
50 x 1 
0 
0.95 
0 
0 
0 
0 
0 
100 x 1 
0 
0.3 
0 
0 
4.5 
0 
0 
150 x 1 
0 
0.95 
0 
0 
8 
0 
0 
200 x 1 
0 
0.3 
0 
0 
5.35 
0 
0.34 
300 x 1 
0 
0.3 
0.3 
0 
4.45 
0 
0 
400 x 1 
0 
0 
0 
0 
2.85 
0 
0 
Figure 5. Plot of Total earliness against the Problem sizes for the early due date window
Table 17. The Earliness value for the Original due date window for all the solution methods
Problem size 
JIT Schedule 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
118 
8.00 
117 
15 
117.0 
47 
10X1 
0 
158.0 
3.0 
104 
32 
118 
3.0 
15 X1 
0 
107 
7.00 
34 
25 
40 
34 
20X1 
0 
147 
2.00 
41 
28 
50 
13 
40 x1 
0 
159 
3.00 
25.00 
24.00 
32.00 
6.00 
50x10 
0 
144.00 
7.00 
26.00 
29.00 
29.00 
14.00 
100x1 
0 
260.00 
1.00 
4.00 
17.00 
6.00 
4.00 
150 x1 
0 
312 
1.00 
1.00 
13.00 
1.00 
1.00 
200x1 
0 
356.00 
2.00 
8.00 
19.00 
15.00 
5.00 
300x1 
0 
303.00 
0.00 
0.00 
24.00 
0.00 
0.00 
400x1 
0 
304 
0.00 
10.00 
1.00 
1.00 
0.00 
Figure 6. Plot of Total earliness against the Problem sizes for the original due date window
Table 18. The total earliness value for the Latest Due Date window for all the solution methods
Problem size 
JIT SCHEDULE 
SPT 
MDD 
EDD 
FCFS 
TA1 
TA2 
5X1 
0 
35.85 
35.85 
35.85 
34.85 
35.85 
35.85 
10X1 
0 
25.6 
6.2 
6.2 
42.8 
8.35 
6.2 
15 X1 
0 
14.7 
6 
2.75 
26.8 
3.15 
6 
20X1 
0 
5.5 
1.75 
1.75 
12.25 
4.8 
1.75 
40 x1 
0 
1.4 
0.35 
0.35 
17.15 
0.35 
0.35 
50x10 
0 
3.05 
0.35 
0.35 
20.1 
0.35 
0.35 
100x1 
0 
1.05 
0.35 
0.35 
23.05 
0.35 
0.35 
150 x1 
0 
2.8 
0.35 
0.35 
46.15 
0.35 
0.35 
200x1 
0 
0.35 
0.35 
0.35 
33.65 
0.35 
0.35 
300x1 
0 
4.8 
0 
0.35 
29.5 
0.35 
0 
400x1 
0 
3.05 
0.35 
0.35 
28.85 
0.35 
0.35 
Figure 7. Plot of Total earliness against the Problem sizes for the latest due date window
Figure 7 reveals that FCFS and SPT yielded the highest deviation from the JIT schedule for all the problem sizes. Also, other solution methods show higher deviation at problem sizes; n$\leq$40. As the problem sizes exceeded this limit, the deviation becomes not significant for all the solution methods except FCFS and SPT.
This work proposed two heuristics for EarlinessTardiness scheduling problems as a means of measuring the deviation of a schedule from the notional JIT Schedule. The results obtained revealed that one of the proposed heuristics TA2 yielded a lower deviation at both the upper and the lower region compared to other proposed heuristics (TA1) and other selected solution methods from the literature.
The work can be explored further by expressing the results of the total earliness and tardiness in composite function using any form of the mathematical expression be it linear, quadratic, or arbitrary equations. However, normalization must be carried for the LCOF using the notional JIT as the benchmark to avoid unbalanced and skewed normalized results.
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