







































 
 

53 
© 2025 by the author; licensee Eastern Centre of Science and Education, USA 

 

Asian Business Research Journal 
Vol. 10, No. 10, 53-60, 2025 
ISSN: 2576-6759 
DOI: 10.55220/2576-6759.622 
© 2025 by the author; licensee Eastern Centre of Science and Education, USA 

 
 

 

 
Optimal Job Selection and Scheduling in Hybrid Manufacturing Systems using 
Linear Programming and Sensitivity Analysis 

 
Maha Hasan Sultan 
 

 
 

Al-Bayan University (Private), Iraq. 

 
Abstract 

This study introduces an interpretable linear model for worker selection and scheduling in hybrid 
manufacturing that considers profitability, resource usage, and energy simultaneously in the same 
objective while respecting capacity, sequencing, and time bucket coupling constraints. By 
assigning tunable weights to selection rewards, use penalties, lateness penalties, and energy costs, 
the approach supports policy tunability and, through an explicit objective decomposition, reveals 
marginal effect of each component on the final plan. Empirical application to operational data 
indicates that such an equilibrium trade-off between value, completion, and delay control is 
possible, with temporal load staying in effective capacity; behavioral indicators within actual 
records also suggest a substantial relationship between delays, energy intensity, and machine 
availability variations. From a management viewpoint, the model offers an reproduceable low-cost 
decision basis well suited for sensitivity analysis, scenario planning in terms of capacity and 
energy policy alternatives, and periodic fine tuning to day-to-day fluctuations. Explainability 
allows integration with learning or metaheuristic elements where higher predictive power and 
scalability are needed while allowing transparent attributions from parameters to outcomes. Such 
established limitations as weight calibration dependency and unavoidable process dynamics 
approximations; yet, the model's expansibility provides for a realistic pathway towards 
incremental real-world data-driven refinement and establishes groundwork for future extensions, 
including coupling with learned estimators and more refined logistical constraints. 

 
Keywords: Cost, Energy, Job, Linear, Profitability, Time. 

 
1. Introduction 

The combination of additive and subtractive processes in Hybrid Manufacturing Systems in the last few years 
has boosted customizing, lowered lot sizes, and enhanced variety in products while making scheduling at the same 
time much more challenging: an attainable schedule must cope with heterogeneous processes (turning and milling 
and additive and drilling) all together, indefinite processing times, time-varying availability windows of machines, 
and persistent mismatches between “scheduled” and “actual” shop-floor timestamps. FMS and JS-FMS research has 
shown static dispatching rules (SPT and EDD and FCFS) suffer with significant plan reality discrepancies in 
dynamic settings, which lead to data-driven “prediction-then-optimization” pipelines and/or metaheuristic hybrid 
learning strategies (Abidi et al., 2020; Meilanitasari & Shin, 2021). Conversely, mixed-integer formulations (MILP 
and MIP), as greatly expressive as they are for loading and selection, routing, and sequence-dependent setups, are 
computationally demanding and less manager-interpretable in industry scale (Abazari et al., 2012; Roshanaei et al., 
2010; Akbaripour et al., 2018). This is precisely where “interpretable linear programming” and “sensitivity 
analysis” pay a strategic dividend: having readily available operational ranges Processing_Time, 
Machine_Availability, and Scheduled and Actual timestamps it is possible to build an open model that includes 
capacity and window limits directly and then, through acceptable ranges on objective coefficients and RHS 
capacities, evaluate “what-if” situations cheaply and quickly (Shapiro, 1993; Monfared & Yang, 2004; Khan et al., 
2021). Industrial experience under real conditions varying from workforce distancing constraints to disconnected 
parallel machines and transport and AGV integration also demonstrates that adding operational realism without a 
sensitivity point of view produces brittle, high-risk decisions (Bazargan-Lari et al., 2022; Saidi-Mehrabad et al., 

2015; Um et al., 2009; Akbaripour et al., 2018) . 
Our question directly falls at this intersection: “Optimal Job Selection and Scheduling in Hybrid Manufacturing 

Systems Using Linear Programming and Sensitivity Analysis,” built on three non-proprietary, readily available 
data pillars processing time (Processing_Time), machine availability (Machine_Availability), and scheduled and 
actual timestamps (Scheduled and Actual) so that (1) a subset of jobs is selected and assigned to machines with 
maximum throughput and utilization or minimum total tardiness, (2) capacity constraints are imposed at the 
machine level within real availability windows, and (3) plan robustness against small parameter perturbations is 
quantified through sensitivity analysis and represented in managerial terms as “allowable increases and decreases” 
for objective weights and capacities. Scientifically, this advancement acts as a bridge between two prevailing camps: 

https://doi.org/10.55220/2576-6759.622


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heavyweight MILP and metaheuristics that are powerful but costly to compute and hard to interpret on the 
decision table side versus static rules or simulation-alone research without an “interpretable bridge” to managerial 
actionability (Byrne & Bakir, 1999; Meilanitasari & Shin, 2021). In our system, objective can be stated to “maximize 
weighted sum of selected jobs” or “minimize total tardiness and incompleteness”; capacity constraints bind the 
aggregate processing time on a machine to its “available time”; window constraints cause Actual to adhere to 
Scheduled (with controllable slack); then sensitivity analysis gives “allowable ranges” on objective coefficients and 
RHS values so managers know how far they can deviate from the priority weights, utilization targets, or shift and 
machine capacities without breaking the optimal basis. The contribution thus is twofold: an implementable 
“baseline LP” for HMS operating over small data domains, and a “robustness map” that distinguishes between safe 
vs. risky parameter moves precisely what decision makers would desire under real world constraints like shift 

changes, periodic faults, or safety considerations . 
 

2. Literature Review 
The flexible setting scheduling literature investigates a few options. At one level, ML and metaheuristics-

driven methods select and predict dispatching rules dynamically and optimize predictive accuracy; e.g., Abidi et al. 
combine weighted feature extraction with a hybrid fuzzy DBN classifier and a lion algorithm variation to propose 
rules in FMS, with accuracy gains from combining metaheuristics and deep learning (Abidi et al., 2020). 
Meilanitasari and Shin’s review highlights that static policies (SPT and EDD and FCFS) create significant gaps 
under JS-FMS dynamics and encourages “prediction then-optimization” with sequence learning to bridge the 
uncertainty and optimal scheduling gaps (Meilanitasari & Shin, 2021). In contrast, well-defined mathematical 
models are used: Abazari et al. propose a hybrid continuous and 0-1 programming model with a GA for machine 
loading to optimize profitability and utilization within capacity, batch size, processing time, tool, and magazine 
constraints (Abazari et al., 2012). Roshanaei et al. formulate JSS with sequence-dependent setup times as a MILP to 
optimize makespan and apply an electromagnetism  like algorithm for the large instances (Roshanaei et al., 2010). 
In cloud manufacturing, Akbaripour et al. develop service selection and scheduling over mixed composition 
structures (sequence and parallel and loop and selective), combine service occupancy and transportation on hybrid 
hub and spoke networks, and demonstrate that adding transportation and availability offers more realistic 
solutions; sensitivity analyses also estimate policy robustness (Akbaripour et al., 2018). Methodologically, the 
ancient underpinning of LP sensitivity to coefficient and RHS changes underlies implemented “what-if” analyses 
(Shapiro, 1993; cf. Monfared & Yang, 2004 on fuzzy scheduling and control sensitivity and parameter tuning) 

The field also leans towards hybrids: combining mathematical models with simulation and metaheuristics to 
achieve high quality, scalable solutions. Examples include neural networks with simulated annealing for stochastic 
job shops (Tavakkoli Moghaddam et al., 2005), hybrid simulation analytical models of multi period, multi product 
planning (Byrne & Bakir, 1999), and joint scheduling maintenance models with multiobjective search (Mishra et al., 
2022; also Tirkolaee et al., 2020 for energy-aware JIT). In FMS and JS-FMS, other papers introduce realism: 
integrated JSS with conflict-free AGV routing (Saidi-Mehrabad et al., 2015), FMS with AGVs and multiobjective 
ES and MONLP (Um et al., 2009), and GA–TOPSIS simulation for operator assignment (Azadeh et al., 2011). At 
the design level, RSM and BWM frameworks make FMS design parameters flexible and optimize performance vs. 
deployment cost (Pasha et al., 2023). Recent RMS and Cloud studies indicate that estimation of actual availability, 
transportation, and reconfigurability significantly alters schedule and planning decisions, and sensitivity and 
ANOVA are key in parameter effect quantification (Imsetif et al., 2025; Yazdani et al., 2022). Multiobjective studies 
during the COVID period include such constraints as social distancing of workers into parallel machines models 
and provides direct impacts on profit and annual scheduling (Bazargan-Lari et al., 2022). In additive processes, 
scheduling non-identical parallel SLM machines with makespan and tardiness objectives and a learning-based 
NSGA-II is the merging of the field towards “explicit model + learning” hybrids (Rohaninejad et al., 2021). 
Generally, the literature shows: (1) MILP and metaheuristics excel with complete constraints but sometimes 
sacrificing interpretability and low cost sensitivity; (2) static rules fall short in dynamic settings, with the need for 
Scheduled and Actual data (Meilanitasari & Shin, 2021); and (3) interpretable LP with complete sensitivity can 
provide a reproducible baseline to HMS decision-making, especially where only usual operational parameters such 
as Processing_Time, Machine_Availability, and Scheduled and Actual are accessible (Shapiro, 1993; Monfared & 
Yang, 2004; Khan et al., 2021). Therefore our gap an LP formulation for “job selection and scheduling” in HMS 
based on public operational fields and reporting allowable ranges on coefficients and capacities addresses two 
requirements directly: analytical transparency for managers and simplicity with real world data for fast, low cost 

deployment . 
 

3. Data and Methodology 
3.1. Study Data 

The data set consists of actual planning and execution data from a hybrid manufacturing system, where each 
row is a production job with a unique identifier and includes the operational and temporal attributes necessary for 
linear modeling and sensitivity analysis. For each operation, operation type (milling, drilling, lathe, additive, or 
grinding) and preassigned machine (Machine_ID) are entered; Processing_Time is the typical job time in base time 
units; Machine_Availability is documented as percent or effective capacity rating, converted to per interval machine 
capacity; Scheduled_Start and Scheduled_End define the planned window, and Actual_Start and Actual_End 
denote the actual timestamps (used for calibration and evaluation). Energy_Consumption per job identifies energy 
intensity per unit time of processing, while Job_Status and Optimization_Category are only used for weighting or 
ex-post evaluation of performance. In order to transform raw data into model inputs, the planning horizon [H] is 

discretized into a uniform time grid with interval length Δ; by representing calendar timestamps as bucket indices, 

each job’s admissible window along the time axis is defined. Per-machine, per-bucket capacity limit cap𝑚,𝑡  is 

calculated from cap𝑚,𝑡 = 𝛥 ⋅
Availability𝑚,𝑡

100
. Processing times 𝑝𝑗  are simply read from Processing_Time following 

unit harmonization to base minute or hour. Actual_* timestamps are not used as constraints for the purpose of 



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keeping the model predictive and deployable; instead, they will be applied once optimization is done in order to 

evaluate and calibrate the penalties for early start and late finish . 
 

3.2. Modeling and Solution Method 
We propose a proactive, time-indexed linear program that simultaneously optimizes the selection of jobs, 

allocation of capacity per machine, and adherence to planned windows. Allowing preemption (that is, splitting a job 
over multiple time buckets) enables a linear, free binary formulation at no increased complexity and yet with only 
temporal stickiness through the introduction of penalties for processing outside the planned window. We then 

define index sets and parameters, decision variables, objective, and constraints . 

Index sets : 

𝐽 =  𝑗𝑜𝑏𝑠. 
M = set of machines . 

T = set of discrete time buckets {1, 2, … , 𝐻}. 

For each j in J, let 𝑚(𝑗) in M be its preassigned machine. For each j in J, let 𝑇on(𝑗)= buckets between 

Scheduled_Start_j and Scheduled_End_j, 𝑇early(𝑗) = buckets preceding Scheduled_Start_j, and 𝑇late(𝑗)= buckets 

following Scheduled_End_j. 
 

3.3. Parameters 
Δ = length of each time bucket. 

𝑝𝑗= typical processing time of job j in units of Δ . 

cap𝑚,𝑡 = capacity of machine m available in bucket t (in units of Δ), obtained from Machine_Availability and 

work calendar. 

𝑒𝑗 = energy rate of job j per unit of processing time. 

𝑤𝑗 = job weight and priority (in simplest case 𝑤𝑗 = 1 or by Optimization_Category) . 

α, β, γ, δ, κ = nonnegative objective weights trading off throughput, lateness, idle capacity, energy, and 

undercompletion . 

 

3.4. Decision Variables 
𝑥𝑗 ∈ [0,1]= job j selection variable (1: completely processed; fractional values: partially processed in the horizon) . 

𝑦𝑗,𝑡
on ≥ 0= amount of processing of job j in bucket t in the scheduled window (𝑡 ∈ 𝑇on(𝑗)) . 

𝑦𝑗,𝑡
early

≥ 0= early processing of job j (𝑡 ∈ 𝑇early(𝑗)) . 

𝑦𝑗,𝑡
late ≥ 0= late processing of job j (𝑡 ∈ 𝑇late(𝑗)). 

𝑜𝑚,𝑡 ≥ 0= overtime for machine m in bucket t (discouraged through a big penalty) . 

idle𝑚,𝑡 ≥ 0= idle capacity of machine m in bucket t (slack to quantify unused capacity) . 

𝑢𝑗 ≥ 0= undercompletion of job j in the horizon (penalized to make it desirable to complete even if 𝑥𝑗  is fractional) . 

 

3.5. Aggregate Helpers 

𝑦𝑗,𝑡 = 𝑦𝑗,𝑡
on + 𝑦𝑗,𝑡

early
+ 𝑦𝑗,𝑡

late, ∀𝑡 ∈ 𝑇 (and for buckets not in the specified subsets, the respective components are zero 

by default) . 

𝐸𝑗 = ∑ 𝑦𝑗,𝑡
early

𝑡∈𝑇early(𝑗)
 (total early processing of job j) . 

𝐿𝑗 = ∑ 𝑦𝑗,𝑡
late 

𝑡∈𝑇late(𝑗)
(total late processing of job j) . 

𝑌𝑗 = ∑ 𝑦𝑗,𝑡
𝑡

 (total allocated processing of job j over the horizon) . 

 

3.6. Objective Function 

,Maximize  j j j m t j j j

j J j J m M t T j J j J

Z w x L idle e Y u    
     

=  −  −  −  −       

The first encourages throughput and job selection; the second inhibits processing with delay; the third reduces 
idle capacity; the fourth imposes a cost for energy usage; and the fifth imposes a cost for undercompletion to make 

the plan reach completion even if 𝑥𝑗 is fractional. 

 

3.7. Constraints 

Job processing balance (flow of each job through time): ∀𝑗 ∈ 𝐽: ∑ 𝑦𝑗,𝑡 + 𝑢𝑗 = 𝑝𝑗𝑥𝑗
𝑡∈𝑇

  Machine capacity per 

bucket (including idle measurement and overtime control): ∀𝑚 ∈ 𝑀, ∀𝑡 ∈ 𝑇: ∑ 𝑦𝑗,𝑡 + idle𝑚,𝑡 ≤
𝑗∈𝐽:𝑚(𝑗)=𝑚

cap𝑚,𝑡 + 𝑜𝑚,𝑡. 
Unavailability of machines (optional hardening of constraint 2):  

, ,,   with  0 :  0m t j tm M t T cap y    = =   

 For each j such that 𝑚(𝑗) = 𝑚 and 𝑜𝑚,𝑡 = 0 and idle𝑚,𝑡 ≥ 0 (this line is a reminder that with zero cap only 
idle can equal cap and y and o are zero.) 
Splitting up processing by temporal region and lateness and earliness definition:  



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∀𝑗 ∈ 𝐽: 𝐸𝑗 = ∑ 𝑦𝑗,𝑡
early

𝑡∈𝑇early(𝑗)

 

∀𝑗 ∈ 𝐽: 𝐿𝑗 = ∑ 𝑦𝑗,𝑡
late

𝑡∈𝑇late(𝑗)

 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇on(𝑗): 𝑦𝑗,𝑡
on ≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇early(𝑗): 𝑦𝑗,𝑡
early

≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇late(𝑗): 𝑦𝑗,𝑡
late ≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇\(𝑇on(𝑗) ∪ 𝑇early(𝑗) ∪ 𝑇late(𝑗)): 𝑦𝑗,𝑡
on = 𝑦𝑗,𝑡

early
= 𝑦𝑗,𝑡

late = 0 

 
 

3.8. Bounds and Nonnegativity 
∀𝑗 ∈ 𝐽: 0 ≤ 𝑥𝑗 ≤ 1 

∀𝑗 ∈ 𝐽: 𝑢𝑗 ≥ 0 

∀𝑚 ∈ 𝑀, ∀𝑡 ∈ 𝑇: idle𝑚,𝑡 ≥ 0 

∀𝑚 ∈ 𝑀, ∀𝑡 ∈ 𝑇: 𝑜𝑚,𝑡 ≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇: 𝑦𝑗,𝑡
on ≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇: 𝑦𝑗,𝑡
early

≥ 0 

∀𝑗 ∈ 𝐽, ∀𝑡 ∈ 𝑇: 𝑦𝑗,𝑡
late ≥ 0 

Policy control of overtime caps: ∀𝑚 ∈ 𝑀, ∀𝑡 ∈ 𝑇: no𝑚,𝑡 ≤ Omax𝑚,𝑡  where Omax𝑚,𝑡 is policy-defined (usually 

zero or a small percentage of cap𝑚,𝑡) . 

Critical jobs (optional hard finish or soft slack): ∀𝑗 ∈ 𝐽critical ⊆ 𝐽: 𝑥𝑗 = 1 ∨ ∑ 𝑦𝑗,𝑡 ≥ 𝑝𝑗𝑥𝑗 − Uslack𝑗
𝑡∈𝑇

 with 

high penalty on Uslack𝑗 in the objective . 

Hard time windows (if strict timing is essential instead of penalties): 

∀𝑗 ∈ 𝐽: 𝑦𝑗,𝑡
early

= 0 for all 𝑡 ∈ 𝑇early(𝑗) 

𝑦𝑗,𝑡
late = 0 for all 𝑡 ∈ 𝑇late(𝑗) 

(in this case, 𝐿𝑗 drops out of the objective and only 𝑦on is admissible.) 

Choosing Δ: smaller Δ increases temporal fidelity but increases model size; standard Δ in the 5–15 minutes 

range. 

Extrapolating Machine_Availability to cap𝑚,𝑡}: if in percentage form, cap𝑚,𝑡 = 𝛥 ⋅
Availability𝑚,𝑡

100
; off-shift 

buckets get cap = 0 . 

Energy: if Energy_Consumption is a per-job total, scale to 𝑒
^

𝑗 =
Energy_Consumption𝑗

𝑝𝑗
 so that 𝛿 ⋅ ∑ 𝑒

^

𝑗𝑦𝑗,𝑡
𝑡

 is 

meaningful . 
Excessive fragmentation avoidance: if contiguity in time matters, one can impose soft penalties on inter bucket 

variance of 𝑦𝑗,𝑡; to maintain linearity, penalizing early and late aggregates typically does the trick . 

Single-machine per job: as Machine_ID is fixed in the data, no assignment constraint is needed; if some 

operations are multi option, generalize to 𝑦𝑗𝑚𝑡 and write ∑ ∑ 𝑦𝑗𝑚𝑡 + 𝑢𝑗 = 𝑝𝑗𝑥𝑗
𝑡

𝑚

 with capacity per (m,t) (still 

linear and preemptive) . 

Sensitivity results: post determine an admissible range on objective coefficients (𝑤𝑗, δ, β) and RHS values 

(cap𝑚,𝑡); this is beyond the formulation and requires post-optimal analysis of the LP solver. 

In the ideal plan, 𝑥𝑗 is the portion of job 𝑗 which was scheduled and finished by the horizon; 𝑦𝑗,𝑡  is the time 

capacity plan on machine 𝑚(𝑗); 𝐿𝑗 is a measure of how much late processing was required to finish the job; idle𝑚,𝑡 

reports unused capacity, acts as a control to fill shifts; and 𝑢𝑗 measures undercompletion so managers could balance 

between serving more orders on Windows and energy consumption. 
 

3.9. Data Analysis 
The discussion begins with scenario behavior and the overall impact on the objective, selected jobs, utilization, 

and costs; followed by objective composition; followed by job-level schedule quality and time bucket occupancy; 

then plan alignment with actual execution; finally, capacity slack. 
From Table 1, increasing alpha raises the objective monotonically while selected_jobs stays at 144 in most 

combinations. Increasing kappa from 50 to 150 slightly lifts avg_utilization; with 𝑘𝑎𝑝𝑝𝑎 = 300 the usage penalty 
total_u pushes total_L upward. This is most severe at alpha=400, where selected_jobs drops to 114. The preferred 

region is alpha∈{700,1000} with 𝑘𝑎𝑝𝑝𝑎 ∈ {50,150}, balancing a high objective, full completion, and controlled 

delay. 
To clarify the objective, Table 2 shows alphasum(wx) as the dominant positive term, and -kappasum(u)  is the 

largest subtraction; -betasum(L) and -deltasum(energyYDelta) also reduce the total but to a lesser extent. This 
accounts for the fact that mid-range kappa performs better than very high kappa in Table 1: usage must be reined 

in, but over-penalizing inflates total_L and wreaks havoc on objective. 
 
 



 
 

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Table 1. Parametric scenarios. 

Alpha Kappa Beta Gamma Delta Cap_Scale Objective Selected_Jobs Total_U Total_L Total_Energy Avg_Utilization 

400 50 10 0.1 0.05 1 47065.92107 144 102.8534885 0.365 555.0900639 0.553998707 

400 150 10 0.1 0.05 1 36932.80242 144 100.7334697 16.875 568.5425474 0.571665531 
400 300 10 0.1 0.05 1 25384.25683 114 48.97521156 134.6281429 521.7602717 0.52236105 
700 50 10 0.1 0.05 1 86245.92107 144 102.8534885 0.365 555.0900639 0.553998707 

700 150 10 0.1 0.05 1 76112.80242 144 100.7334697 16.875 568.5425474 0.571665531 
700 300 10 0.1 0.05 1 61135.30115 144 98.23120246 78.59 588.7622905 0.592517757 
1000 50 10 0.1 0.05 1 125425.9211 144 102.8534885 0.365 555.0900639 0.553998707 

1000 150 10 0.1 0.05 1 115292.8024 144 100.7334697 16.875 568.5425474 0.571665531 
1000 300 10 0.1 0.05 1 100315.3011 144 98.23120246 78.59 588.7622905 0.592517757 

 
 

Table 2. Objective composition. 

Component Value 
alphasum(wx) 130600 
- beta*SUM(L) -245.8333333 
- deltaSUM(energyY*Delta) -28.50065745 
- kappa*SUM(u) -20051.42204 
TOTAL (objective) 110274.244 

 
 



 
 

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Table 3. Job selection summary (sample). 

Job_ID Machine_ID 
Operation_ 
Type 

Processing_ 
Time 

Energy_ 
Consumption 

Scheduled_ 
Start 

Scheduled_ 
End 

Job_ 
Status 

Optimization_ 
Category 

x u L 
Y_ 

sum 
Total_Energy 

J001 M01 Grinding 76 11.42 
3/18/2023 

8:00 
3/18/2023 

9:16 
Completed 

Moderate 
Efficiency 

1 1.2667 0 0 0 

J002 M01 Grinding 79 6.61 
3/18/2023 

8:10 
3/18/2023 

9:29 
Delayed Low Efficiency 1 0.4267 0 1.78 5.8829 

J014 M04 Additive 112 2.01 
3/18/2023 

10:10 
3/18/2023 

12:02 
Completed 

Optimal 
Efficiency 

1 0.0867 5.76 3.56 3.5778 

 
Table 4. Time-bucket schedule Y (sample). 

Job_ID Machine_ID bucket_start bucket_end Y 

J002 M01 3/18/2023 8:00 3/18/2023 8:30 0.89 
J011 M01 3/18/2023 9:30 3/18/2023 10:00 0.89 
J016 M01 3/18/2023 10:30 3/18/2023 11:00 0.89 
J058 M01 3/18/2023 18:00 3/18/2023 18:30 0.242833333 
J060 M01 3/18/2023 18:00 3/18/2023 18:30 0.647166667 

 
Table 5. Hybrid manufacturing categorical (sample). 

Job_ID Machine_ID 
Operation_ 
Type 

Material_ 
Used 

Processing_ 
Time 

Energy_ 
Consumption 

Machine_ 
Availability 

Scheduled_ 
Start 

Scheduled_ 
End 

Actual_ 
Start 

Actual_ 
End 

Job_ 
Status 

Optimization_Category 

J001 M01 Grinding 3.17 76 11.42 96 3/18/2023 8:00 
3/18/2023 
9:16 

3/18/2023 
8:05 

3/18/2023 
9:21 

Completed Moderate Efficiency 

J010 M01 Drilling 2.10 27 3.66 97 3/18/2023 9:30 
3/18/2023 
9:57 

3/18/2023 
9:54 

3/18/2023 
10:21 

Delayed Low Efficiency 

J014 M04 Additive 2.33 112 2.01 95 3/18/2023 10:10 
3/18/2023 
12:02 

3/18/2023 
10:10 

3/18/2023 
12:02 

Completed Optimal Efficiency 

 



 
 

59 
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For quality at the job level, Table 3 (lp selected subj summary) indicates that most rows have x=1 and 
Job_Status=Completed. Values of u and L indicate pressure from machine usage and delay risk. Rows marked 
Optimal or High Efficiency tend to have lower energy with controlled L, whereas Low Efficiency is found with 

higher L or energy . 
To align plan and reality, Table 5 (hybrid manufacturing categorical) shows Scheduled vs Actual times. 

Delayed entries tend to coincide with higher energy or lower machine availability. Together with Table 3, this 
indicates that re-ordering toward lower-energy operations and tighter availability constraints can reduce delay 

incidence. 
To complete the capacity snapshot, Table 6 (idle by time-bucket) shows zero Idle in the sampled intervals; this 

aligns with Y distribution and avg_utilization rates in Table 1. Further gains are expected to arise from reducing 

usage and energy costs as well as avoiding Delayed/Failed instances rather than filling idle holes . 
 

Table 6. Idle by time-bucket (sample). 

Machine_ID bucket_start bucket_end Idle 

M01 3/18/2023 8:00 3/18/2023 8:30 0 
M01 3/18/2023 10:30 3/18/2023 11:00 0 
M01 3/18/2023 14:30 3/18/2023 15:00 0 

 

4. Discussion and Conclusion 
The combined evidence from Table 1 to Table 6 indicates that a transparent linear formulation with weights 

alpha, kappa, beta, gamma, and delta yields an operational balance between objective value, completion, and delay 
control. Within reasonable levels of weights, the objective increases while preserving the set of completions; 
excessive use penalties at high levels degrade performance by raising total_u and, in turn, total_L. This is in 
accordance with Table 2: alphasum(wx) is the main driver, whereas -kappa*sum(u) is the main deduction and, if 
overdone, overstates delays and makes the objective worse. Temporal load in Table 4 shows Y staying within 
effective capacity, as is expected from avg_utilization in Table 1. At the job level, Table 3 reveals that Optimal and 
High Efficiency options generally pair with less energy and controlled delay, and Low Efficiency pairs with more 
usage pressure or energy consumption. Plan-versus-actual agreement in Table 5 confirms that delayed cases often 
overlap with more energy or less availability. Finally, Table 6 shows no major idle pockets, hence improvement 

lever is towards weight tuning and reordering towards lower energy operations. 
Contrary to literature, the results both support and augment three prevalent themes. First, 

learning/metaheuristic models adaptively choose dispatching rules and have reported enhancements in predictive 
accuracy. Our findings via Table 3 and Table 4 show that without learning, a replicable and interpretable baseline 
can indeed be established and served as a foundation for “prediction then optimization” hybrids. Second, well-
defined mathematical models used in the past often favor MILP with sequence-dependent setups and 
transportation, and search heuristics to ensure scalability. Here, planar structure and objective decomposition in 
Table 2 retain explainability and sensitivity at little computational cost, which is a luxury for rapid deployment to 
HMS. Third, energy-aware and availability-aware models drastically change scheduling decisions. The consistency 
across Table 1, Table 3, and Table 5 sends the same message: with energy and capacity weighted properly, both 

delay and consumption are both optimized together . 
The lesson of operation is to keep weights in a well-balanced band so that the target stays high and risk of 

delay is controlled, with this plausible baseline being a stepping stone toward learning or metaheuristic 
optimization. Adaptive retuning of weights against Actual signals and efficiency labels in Table 5 is a plausible next 
step, while using Table 4 to recognize congested time windows and resequence towards lower energy strings. This 

reaffirms compatibility with the “explicit model + learning” hybrids emphasized in the literature . 
An understandable linear model introduced objective benefit, work completion, and delay control into 

equilibrium. The objective breakdown highlighted selection incentives’ key role and extreme sensitivity to use 
penalties. Time-bucket scheduling kept load in efficient capacity, with real execution data correlating delays and 
energy use with availability. The approach presents an open, low cost foundation waiting for incorporation of 

learning and metaheuristic capabilities . 
 

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