Mathematical Model of the Assembly Robot Performance in Automated Manufacturing

Kizito Paul Mubiru; Nalubowa Maureen Ssempijja

doi:doi:10.11648/j.mma.20251003.12

Research Article |

| Peer-Reviewed

Mathematical Model of the Assembly Robot Performance in Automated Manufacturing

Kizito Paul Mubiru^*

, Nalubowa Maureen Ssempijja

Published in Mathematical Modelling and Applications (Volume 10, Issue 3)

Received: 23 September 2025 Accepted: 5 October 2025 Published: 30 October 2025

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Abstract

Robot performance in automated manufacturing supports the competitive edge in manufacturing industries, enhancing customer sustainability and reducing operational costs. The robot cell controller must therefore optimize the operational tasks in the assembly of components or parts of the product to support automated manufacturing strategies for the manufacturing plant. This study proposes a Markov decision process model for robot performance in a steel assembly plant. The performance of the robot is characterized as a Markov chain, and its operational cost matrix represents the expected reward for the Markov decision process problem. This study utilized hourly data for two consecutive weeks. The principal challenge addressed focused on determining the best product assembly option to reduce the assembly expenses of the robot cell. We considered a multi-period planning horizon, where the optimal decision was determined for assembling or not assembling additional products based on the demand and availability of finished assembled products. The model was tested, and the results demonstrated the existence of an optimal state-dependent decision and assembly expenditure of managing the robot cell of the steel assembly plant. As a cost optimization strategy for managing robot cells, improved efficiency and resource utilization for assembled products were realized, supporting automated manufacturing initiatives of the steel assembly plant used in the case study.

Published in	Mathematical Modelling and Applications (Volume 10, Issue 3)
DOI	10.11648/j.mma.20251003.12
Page(s)	49-58
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Automation, Manufacturing, Mathematical Model, Performance, Robot

1. Introduction

1.1. Background

Definitions of automation in an industrial context are “the implementation of processes by automatic means” (ISOTR/TR11065

[1]

, ISO, ISO8373

[2]

). In this context, industrial automation comprises: platforms (usually machines, tools, devices, installations, and systems), autonomy (defined through an organization, process control, automatic control, intelligence, and collaboration), processes (actions, operations, functions), and power sources. The use of advanced manufacturing technologies seems to build firm ground on prospects for technological manufacturing concerns in metal industries, permitting Computer Integrated Manufacturing (CIM) controlled industrial robotics mechanics, Kotha et al

[3]

. Mahbub

[4]

defined automation as a self-regulation process performed using programmable machines to carry out a series of tasks.

The use of industrial robots in metal industries replaces workers in monotonous, repetitive, heavy, hot, and inhospitable working environments, reducing costs, increasing productivity, improving product quality, and eliminating harmful tasks Wallén

[5]

. The main robotic technologies, common tasks, and typical industrial applications explain the wider applications of industrial robots in manufacturing industries today. World Robotics

[6]

, Isak et al.

[7]

further stressed that the use of robots for industrial applications in metal industries is prevalent in the motor vehicle assembly plants, while robot application in metal processing industries is drastically decreasing. Karabegovic et al.

[8]

argue that industrial robot applications in the metal processing industries is due to the automation and modernization of the manufacturing processes. According to Wilson

[9]

, automation is the production of “Automatically controlled operation of an apparatus, process, or system by mechanical or electronic devices that take the place of human labor.”

Before the technological revolution, earlier practices in manufacturing processes were characterized by manual tasks, and later, with the emergence of automated manufacturing processes, Nyori

[10]

. In today’s manufacturing industries, the use of computer-controlled robots to complete manual tasks is increasing, picking up more than ever before, Riben

[11]

. Industrial robots used in steel manufacturing industries today have artificial intelligence embedded with machine learning that powers cognitive competence to inspect bulky items for errors, automate the transportation of work in progress, and avoid safety risks using predictive intelligence. The robot’s capability in accomplishing various manufacturing tasks, Procter et al

[12]

was studied as robot activities meet optimal criteria for planning purposes and robot monitoring to gauge performance ability during design, process, planning, production, and analysis stages. Special interest was taken in validating the technologies and standards relevant to automated robot deployment. Empirical experimentations, in addition, helped to handle the type of speed of the robot Zghair et al

[13]

with additional resource requirements. Aalst et al.

[14]

explained that automating processes addresses the most frequent case types, with less frequent cases not considered due to high costs, which increase when different systems are integrated. With industrial robot use, organizations strengthen the demand for products and innovations, thereby increasing their market share and maintaining customer satisfaction Bader

[15]

. Organizations today are constantly looking to identify manufacturing processes that can be automated using robotic process automation to optimize time-centric and repetitive routine processes, maximize productivity, reduce manufacturing costs by minimizing human error, and increase the agility of manufacturing processes. Leshob, et al.

[16]

. Due to outstanding advantages of high accuracy, high work efficiency, ability to work in a highly hazardous environment, and the ability to bear greater working strength, industrial robots create better conditions for the improvement of productivity, safety, and quality in steel manufacturing industries, Li Guangjun

[17]

. The manufacturing tasks assigned to industrial robots today range from assembly, machining, material handling, packaging, welding, and material transport, among others. PwC,

[18]

1.2. Automation and Industrial Robots

Automation is the replacement of man by machine for the performance of tasks, and it can provide movement, data gathering, and decision-making. Robotic Process Automation (RPA) is the application of technology that is used to configure software robots that capture and interpret existing applications for processing, transactions, manipulating data, and communicating with other software systems Leshob et al.

[16]

RPA uses software-based robots to perform repetitive tasks that require manual labor. Besides, RPA provides a link between traditional process automation and work done by humans through system interrelationships shown in Figure 1 below.

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Figure 1. Robot Process Automation (Aalst, et al. 2018).

RPA endeavours to substitute workers in manufacturing tasks with robots to improve information systems. RPA provides sub-systems that co-exist to substitute manual labor. (Leshob, et al. 2018) explained that manufacturing organizations today are constantly looking to identify processes for automation using RPA to achieve maximum results.

1.3. Application of Industrial Robotic Technologies

The Automotive industry today is the biggest beneficiary of robotic automation technologies. Most industrial robot applications are in the automotive industry. Industrial robotic technology applications in manufacturing industries are expanding far beyond the robot functionality of heavy lifting of materials to handling complex tasks difficult for humans to handle. The applications of robots in the metalworking industry include material handling, machine servicing, manufacturing operations, installation work, and product control tasks.

1.4. Reviews on Robot Performance Modelling

An Open system architecture for CIM, Fang et al

[19]

was presented, where robot performance exhibited a systems approach. The authors found that existing research was insufficient in terms of performance metrics, especially in complex environments. Associated with robot performance is a testing device, Hou

[20]

, that streamlines performance for the industrial robot. The pertinent parameters of temperature and operating speed were considered to facilitate the performance testing of industrial robots. Related studies on how adaptation to fluctuating product variants, Asif et al

[21]

was feasible through program architecture that integrates effective systems and control, and robot abilities in terms of perception showed how empirical results underscored the architecture’s robust perception. The robot cell behavior was studied with a focus on existing programs and simulations in the context of the shop floor, where calculations showed that the fastest task accomplishment yielded profound analysis and validation. Ultimately, the effective performance of robots was complemented with cycle time regions as a core for effective robot performance. Industrial Robot for enhanced precision, Birk et al

[22]

was studied with frequency response using a genetic algorithm. The dynamic characterization was accomplished, and robot dynamics up to 200Hz showed the robot's potential to accomplish dedicated tasks requiring manufacturing precision.

Despite several contributions made by scholars from §1.1 to §1.4, optimality guidelines were lacking in the face of robot performance uncertainty during the product assembly process. The major contributions of this paper to robot performance modeling for automated manufacturing are highlighted as follows:

1) The stochastic matrices characterizing the robot's performance and operational cost were calculated based on existing circumstances of assembly options of a product in the robot cell.

2) The computation procedure calculated the expected operational costs and accumulated operational costs for the assembly decisions of the product.

3) Optimality of assembly options of products under different states of robot performance was attained.

2. Materials and Methods

A discrete-time discrete-state Markov Decision Process (MDP) model was developed with discrete time points t ϵ T = {1, 2,…, E}. At each decision epoch t, the decision maker (ie, robot controller) observed the states of the robot's performance by making observations concerning the hourly number of units assembled by the robot cell. When units on hand exceeded demand, additional units were not assembled, resulting in a decision process. The robot controller instead determined (considering the units required) the superior assembly option to take. The decision continued till the assembly exercise ended for each action the decision maker took. As a result, a quick payoff emerged for the robot’s operational expenditure for the option taken.

The objective, therefore, is to consider the balance between assembling more products, with the associated operational costs, versus not assembling additional units. A formal definition of the core components of the MDP model follows:

2.1. States

The robot performance state i was composed of three state variables: Good (state G), fair (state F), and poor (state P). The states were defined depending on a specified number of tasks accomplished with the speed and accuracy needed

T^{S}

, during the product assembly exercise, with demand

D^{S}

realized at a decision point t of the robot cell, where S ϵ {0,1}, t = 1, 2,…………T.

2.2. Actions

The action space was defined as A= {

a_{0}, a_{1}, \dots \dots a_{k}

} where

a_{i} = 0

represented not assembling and

a_{i} = 1

represented assembling more products. We assumed that if

a_{i} = 0

was chosen, additional units were not assembled when available units exceeded demand, while units had to be assembled whenever demand was higher than on-hand inventory.

2.3. Transition Probabilities

When the robot controller chose an action

s_{t}

ϵ S at decision epoch t when the robot's performance was in state

s_{t}

, the robot's performance state moved to

s_{t + 1}

at t+1 with probability

P_{t}

(

S_{t} / S_{t + 1}, a_{t}) .

We assumed that

P_{t} (S_{t + 1} / S_{t}, a_{t}

P_{t}^{ƛ}

(

c_{t}

a_{t}

P_{t}^{α}

(

α_{t + 1} P_{t}^{α}

(

α_{t + 1}

α_{t}

)(1)

where

P_{t}^{ƛ}

(

c_{t}

a_{t}

) and

P_{t}^{α}

(

α_{t + 1}

α_{t}, a_{t}

) represented the good performance, fair performance and poor performance transition probabilities. This assumption was consistent with our proposition that good state, fair state and poor state were independent, but influenced by the robot controller’s options.. Note that, we assumed that

P_{t}^{ƛ}

(

c_{t + 1}

c_{t}, a_{1}

P_{t}^{ƛ}

(

c_{t + 1}

c_{t}

a_{2}

)>….>

P_{t}^{ƛ}

(

c_{t + 1}

c_{t}, a_{k}

)(2)

where

c_{t + 1}

represented a good state.

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}, a_{1}

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}

a_{t}

)<…<

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}, a_{k}

)(3)

where

ƛ_{t + 1}

represented a fair state than

ƛ_{t}

and

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}, a_{1}

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}

a_{t}

)<……<

P_{t}^{ƛ}

(

ƛ_{t + 1}

ƛ_{t}, a_{k}

)(4)

where

ƛ_{t + 1}

represented a poor state.

2.4. Reward Function

Our model included a reward function

ƛ_{t}

(

s_{t}

a_{t}

) that reflected the utility/disutility of the decision maker as realized performance state

s_{t}

with action

a_{t}

which was taken at decision epoch t. This was defined as

α_{t}

(

s_{t}

a_{t}

\sum_{i = 1}^{N} α_{t, i} (c_{i}

a_{t}

α_{t}

(

α_{t}

a_{t}

)(5)

where

α_{t, i}

(

c_{i}

a_{t}

) represented the immediate reward for a good performance state

c_{i}

and

α_{t}

(

α_{t}

a_{i}

) was the immediate reward for a fair performance state

α_{t}

and a poor performance state.

Hence, the corresponding reward functions were assumed to follow the following inequality for all t.

α_{t}

(

c_{i}, a_{t}

α_{t}

(

c_{i}^{'}

a_{t}

α_{t}

(

α

a_{t}

α_{t}

(

α^{'}

a_{t}

)(6)

2.5. Value Function

Noting that the action required minimization of the expected operational costs over the planning period, the recursive equations for all

s_{t}

ϵ S and t = 1, 2,……. T yields

V_{t}

(

s_{t}

\min_{a_{t ϵA}}

{

α_{t}

(

s_{t}

a_{t}

\sum_{s_{t + 1 ϵS}} P_{t}

(

s_{t + 1}

s_{t}

a_{t}

)

v_{t + 1}

(

s_{t + 1}) / s_{t}, a_{t}

)

v_{t + 1}

(

s_{t + 1})}

(7)

where

v_{t}

(

s_{t}

) represented the minimum expected total reward at the decision epoch t when the robot's performance was in a state

s_{t}

with the boundary condition

V_{α + 1}

(s)=

α_{T + 1}

(s)(8)

2.6. Formulating the Finite-Period Dynamic Programming Problem

Recalling the robot’s performance was considered as a Good state (state G), Fair state (state F), or Poor state (state P), the problem was considered an optimal product assembly decision, and this was modelled as a dynamic programming problem over a finite-period planning horizon. We denoted

g_{n} (i)

as the expected total operational costs accumulated by the robot in robot cell c during the periods n, n+1,…, N, given that the state of the system at the beginning of period n was i є {G, F, P}. The recursive equation relating g_n and g_n+1 became

g_{N}

(i)=

\min_{S}

[

e_{i}^{S}

Q_{iG}^{S} g_{n + 1}

(G)+

Q_{iF}^{S} g_{n + 1}

(F)+

Q_{iP}^{S} g_{n + 1}

(P)](9)

with

g_{N + 1}

(G)=

g_{N + 1}

(F)=

g_{N + 1}

(P)=0(10)

The robot operational costs C^S_ij + g_n+1 (j) resulting from reaching state j ϵ {G, F, P} at the start of period n+1 from state i ϵ {G, F, P} at the start of period n occurred with probability Q^S_ij.

Therefore,

e^{S}

Q^{S}

][

C^{S}

]^TSε[0,1](11)

The corresponding dynamic programming recursive equations were thus obtained

g_{N}

(i)=

\min_{S}

[

e_{i}^{S}

Q_{iG}^{S} g_{n + 1}

(G)+

Q_{iF}^{S} g_{n + 1}

(F)]+

Q_{iP}^{S} g_{n + 1}

(P)](12)

g_{N}

(i)=

\min_{S}

[

e_{i}^{S}

](13)

Assembling products demanded above the available products in the robot cell yielded the robot operational cost matrix

C^{S}

c_{a}

c_{s}

)[

D^{S}

A^{S}

](14)

Otherwise

C^{S}

c_{h}

[

A^{S}

D^{S}

](15)

When available units exceeded demand

Clearly,

C_{ij}^{S} = \{\begin{matrix} (c_{a} + c_{s} + c_{h}) [D_{ij}^{S} - A_{ij}^{S}] if D_{ij}^{S} > A_{ij}^{S} \\ c_{h} [A_{ij}^{S} - D_{ij}^{S}] if D_{ij}^{S} \leq A_{ij}^{S} \end{matrix}

(16)

for: i, j є {G, F, P}, S ϵ {1,0}.

The justification for expressions (13) and (14) was that D^S_ij – A^S_ij units had to be assembled to meet excess demand. Otherwise, assembling was cancelled when demand was less than or equal to available units of assembled product. The following conditions were therefore sufficient to execute the model.

S=1 when c_a > 0 otherwise S=0 when c_a = 0.

c_s > 0 when shortages were allowed; otherwise, c_s=0 when shortages were not allowed.

2.7. Optimization

The robot’s assembly decision and operational costs were optimized for periods 1 and 2.

2.7.1. Optimization - Period 1

Considering the good (state G) performance, the optimal assembly decision was determined as

S = \{\begin{matrix} 1 if e_{G}^{1} < e_{G}^{0} \\ 0 if e_{G}^{1} \geq e_{G}^{0} \end{matrix}

(17)

with expected operational costs

g_{1} (G) = \{\begin{matrix} e_{G}^{1} if S = 1 \\ e_{G}^{0} if S = 0 \end{matrix}

(18)

When robot performance was fair (i.e., in state F), the optimal assembly decision was determined as

S = \{\begin{matrix} 1 if e_{F}^{1} < e_{F}^{0} \\ 0 if e_{F}^{1} \geq e_{F}^{0} \end{matrix}

(19)

with expected operational costs

g_{1} (F) = \{\begin{matrix} e_{F}^{1} if S = 1 \\ e_{F}^{0} if S = 0 \end{matrix}

(20)

When robot performance was poor (i.e., in state P), the optimal assembly decision was determined as

S = \{\begin{matrix} 1 if e_{P}^{1} < e_{P}^{0} \\ 0 if e_{P}^{1} \geq e_{P}^{0} \end{matrix}

(21)

with expected operational costs

g_{1} (P) = \{\begin{matrix} e_{P}^{1} if S = 1 \\ e_{P}^{0} if S = 0 \end{matrix}

(22)

2.7.2. Optimization - Period 2

Reconsider equations (10) and (11) with

a^{S}

denoting the accumulated operational costs of period 1.

a_{i}^{S}

e_{i}^{S}

Q_{iG}^{S}

min[

e_{G}^{1}

e_{G}^{0}

Q_{iF}^{S}

min[

e_{F}^{1}, e_{F}^{0}

Q_{iP}^{S}

min[

e_{P}^{1}, e_{P}^{0}

](23)

a_{i}^{S}

e_{i}^{S}

Q_{iG}^{S} g_{2}

(G)+

Q_{iF}^{S} g_{2}

(F)+

Q_{iP}^{S} g_{2}

(P)(24)

Therefore, for good performance (ie, in state G), the optimal assembly decision during period 2 was determined as

S = \{\begin{matrix} 1 if a_{G}^{1} < a_{G}^{0} \\ 0 if a_{G}^{1} \geq a_{G}^{0} \end{matrix}

(25)

while the associated accumulated operational costs were

g_{2} (G) = \{\begin{matrix} a_{G}^{1} if S = 1 \\ a_{G}^{0} if S = 0 \end{matrix}

(26)

Similarly, when performance was fair (ie, in state F), the optimal assembly decision during period 2 was determined as

S = \{\begin{matrix} 1 if a_{F}^{1} < a_{F}^{0} \\ 0 if a_{F}^{1} \geq a_{F}^{0} \end{matrix}

(27)

In this case, the associated accumulated operational costs were

g_{2} (F) = \{\begin{matrix} a_{F}^{1} if S = 1 \\ a_{F}^{0} if S = 0 \end{matrix}

(28)

When performance was poor (ie. in state P), the optimal assembly decision during period 2 was determined as

S = \{\begin{matrix} 1 if a_{P}^{1} < a_{P}^{0} \\ 0 if a_{P}^{1} \geq a_{P}^{0} \end{matrix}

(29)

With operational expenditure

g_{2} (P) = \{\begin{matrix} a_{P}^{1} if S = 1 \\ a_{P}^{0} if S = 0 \end{matrix}

(30)

2.8. A Case Study: Roofings Uganda Ltd

The model developed was presented using a Case study from Roofings Uganda Ltd, a manufacturing company of steel products. The factory, therefore, required an optimal assembly decision for wheel burrows in the robot cell based on the demand and availability of wheel burrows. Depending on the robot’s ability to assemble wheel burrows in terms of speed, accuracy, and quality, its performance was classified as Good (state G), Fair (state F), or Poor (state P). Decision support was therefore sought in terms of an optimal decision for assembling wheel burrows and the associated operational costs, considering a planning horizon of two days.

2.9. Data Collection

The number of wheel burrows assembled was observed and recorded from the robot cell. The states of performance of the robot under the assembly decisions were considered over fifty days, when the robot’s performance was good (state G), fair (state F), or poor (state P); the data was captured in Tables 1-3.

Table 1. Tasks versus state transitions of robot performance.

Performance States	Assembly Decision (S=1)			Assembly Decision (S=0)
	G	F	P	G	F	P
G	12	8	5	13	7	5
F	8	11	6	8	12	5
P	5	7	13	6	6	13

G: Good performance (state G) F: Fair performance (state F) P: Poor performance (state P)

Table 2. Demand of wheel burrows versus state transitions of robot performance.

Performance States	Assembly Decision (S=1)			Assembly Decision (S=0)
	G	F	P	G	F	P
G	100	60	45	110	60	30
F	60	80	65	60	90	50
P	50	40	110	50	50	100

G: Good performance (state G) F: Fair performance (state F) P: Poor performance (state P)

Table 3. Availability of wheel burrows versus state transitions of robot performance.

Performance States	Assembly Decision (S=1)			Assembly Decision (S=0)
	G	F	P	G	F	P
G	60	80	60	70	40	40
F	80	60	80	70	40	80
P	40	20	60	75	90	45

G: Good performance (state G) F: Fair performance (state F) P: Poor performance (state P)

2.10. Data Analysis

Graphical analysis of the data is presented in the following Figures:

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Figure 2. State-transitions versus robot performance transition probabilities.

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Figure 3. State transitions versus operational costs (in USD).

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Figure 4. Expected operational costs versus states of robot performance.

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Figure 5. Accumulated operational costs versus states of robot performance.

For either assembly decision taken, unit assembly cost (c_a) = 1.20 USD per wheel burrow, unit holding cost (c_h) = 0.80 USD per week, and unit shortage cost (c_s) = 0.32 USD per week.

3. Results

Calculating Parameters

We can illustrate the process of calculating demand transition probabilities and operational costs for matrices

P^{1}

and

C^{1}

respectively.

P_{GF}^{1}

(T_GF^1) / (T_GG^1 + T_GF^1 + T_GP^1)

\frac{8}{12 + 8 + 5}

= 0.320

P_GP^1 = (T_GP^1)/(T_GG^1+ T_GF^1+T_GP^1) =

\frac{5}{12 + 8 + 5}

= 0.200

P_FG^1 = (T_FG^1)/(T_FG^1+ T_FF^1+T_FP^1) =

\frac{8}{8 + 11 + 6}

= 0.320

P_FF^1 = (T_FF^1)/(T_FG^1+ T_FF^1+T_FP^1) =

\frac{11}{8 + 11 + 6}

= 0.440

P_FP^1 = (T_FP^1)/(T_FG^1+ T_FF^1+T_FP^1) =

\frac{6}{8 + 11 + 6}

= 0.240

P_PG^1 = (T_PG^1)/(T_PG^1+ T_PF^1+T_PP^1) =

\frac{5}{5 + 7 + 13}

= 0.200

P_PF^1 = (T_PF^1)/(T_PG^1+ T_PF^1+T_PP^1) =

\frac{13}{5 + 713}

= 0.280

P_PP^1 = (T_PP^1)/(T_PG^1+ T_PF^1+T_PP^1) = 13/(5+7+13) = 0.520

Hence,

P^{1}

(\begin{matrix} P_{GG}^{1} & P_{GF}^{1} & P_{GP}^{1} \\ P_{FG}^{1} & P_{FF}^{1} & P_{FP}^{1} \\ P_{PG}^{1} & P_{PF}^{1} & P_{PP}^{1} \end{matrix})

(\begin{matrix} 0.480 & 0.320 & 0.200 \\ 0.320 & 0.440 & 0.240 \\ 0.200 & 0.280 & 0.520 \end{matrix})

We note that

\sum_{iϵG, F, P} P_{iG}^{1}

P_{iF}^{1} + P_{iP}^{1} = 1

and

P_{ij}

≤ 0 for all i, j ϵ {G, F, P}.

Equations (12), (13) and (14) yield the following operational costs (in USD) for the robot cell given assembly decision 1.

c_{GG}^{1}

= (1.20+0.80+0.32) (100 – 60) = 92.8

c_{GF}^{1}

= (0.80) (80 – 60) = 16.0

c_{GP}^{1}

= (0.80)(60 - 45) =12.0

c_{FG}^{1}

= (0.80)(80 – 60) = 16.0

c_{FP}^{1}

= (1.20+0.80+0.32) (100 – 60) = 46.4

c_{FP}^{1}

= (0.80)(80 – 65) = 12.0

c_{PG}^{1}

= (1.20+0.80+0.32) (50 – 40) = 23.2

c_{PF}^{1}

= (1.20+0.80+0.32) (40 – 20) = 46.4

c_{PF}^{1}

= (1.20+0.80+0.32) (110 – 60) = 116.0

C^{1}

(\begin{matrix} c_{GG}^{1} & c_{GF}^{1} & c_{GP}^{1} \\ c_{FG}^{1} & c_{FF}^{1} & c_{FP}^{1} \\ c_{PG}^{1} & c_{PF}^{1} & c_{PP}^{1} \end{matrix})

(\begin{matrix} 92.8 & 16.0 & 12.0 \\ 16.0 & 46.4 & 12.0 \\ 23.2 & 46.4 & 116.0 \end{matrix})

Assembly decision 0 of the robot cell yielded the following performance matrix and operational cost matrix.

P⁰ =

(\begin{matrix} P_{GG}^{0} & P_{GF}^{0} & P_{GP}^{0} \\ P_{FG}^{0} & P_{FF}^{0} & P_{FP}^{0} \\ P_{PG}^{0} & P_{PF}^{0} & P_{PP}^{0} \end{matrix})

(\begin{matrix} 0.520 & 0.280 & 0.200 \\ 0.320 & 0.480 & 0.200 \\ 0.240 & 0.240 & 0.520 \end{matrix})

C^{0}

(\begin{matrix} c_{GG}^{0} & c_{GF}^{0} & c_{GP}^{0} \\ c_{FG}^{0} & c_{FF}^{0} & c_{FP}^{0} \\ c_{PG}^{0} & c_{PF}^{0} & c_{PP}^{0} \end{matrix})

(\begin{matrix} 92.8 & 46.4 & 8.0 \\ 8.0 & 116 & 24.0 \\ 24.0 & 32.0 & 127.6 \end{matrix})

The expected and accumulated operational costs (in USD) were computed under the states of robot performance: Good (state G), Fair (state F), and Poor (state P), whose results are summarized in Table 4.

Table 4. Expected and Accumulated Operational Costs (in USD) for the robot cell.

State of robot performance (i)	Expected operational costs eSi		Accumulated operational Costs aSi
State of robot performance (i)	Assemble additional units (S=1)	Do not assemble additional units (S=0)	Assemble additional units (S=1)	Do not assemble additional units (S=0)
Good	52.06	62.85	101.73	113.46
Fair	28.42	63.04	76.29	108.93
Poor	77.95	79.79	136.85	139.64

4. Discussion of Results

When robot performance was Good (state G), and noting that 52.06 < 62.85, S=1 represented the optimum assembly option for day 1 and the expected operational expenditure of 52.06 USD for the case of Good performance (state G). Since 28.42<63.04, S=1 represented the optimum assembly option for day 1 with associated expected operational costs of 28.42 USD for the case of Fair performance (state F). Similarly, noting that 77.95<79.79, S=1 was the optimum assembly option for day 1 with expected operational expenditure of 28.42 USD for the case of Poor performance (state P).

When performance was good (state G) and noting that 101.73 < 113.46, S=1 represented the optimum assembly option for day 2 with accumulated operational costs of 101.73 USD for the case of Good performance. Noting that 76.29<108.93, S=1 represented the optimum assembly decision for day 2 with an accumulated operational expenditure of 76.29USD for the case of Fair performance (state F). Similarly, noting that 136.85<139.64, S=1 represented the optimum assembly option for day 2 with an accumulated operational expenditure of 136.85USD for poor performance (state P).

Future Research

Further research is highly sought to examine robot performance and assembly decisions for robot cells under non-stationary transition probabilities of performance. Critical issues can also be considered for model extensions under the infinite period planning horizon. Special interest is also sought in further extending the model by considering robot performance in the context of Continuous-Time Markov Chains (CTMC).

5. Conclusions

Robot performance for automated manufacturing can be modelled using Markov Decision Processes. The optimal assembly decision and operational costs were determined under robot performance uncertainty based on the Markovian states of performance. The decision of whether or not to assemble wheel burrows in the robot cell incorporated solution methods for the Markov decision process algorithm. A case study from Roofing Uganda Ltd showed that the optimal robot assembly decisions and operational costs are state-dependent. Therefore, as an optimization strategy for robot performance in manufacturing plants, Markov decision processes provide promising results.

Acknowledgments

The authors are thankful to Roofings Uganda Limited for the assistance during data collection.

Author Contributions

Kizito Paul Mubiru: Conceptualization, Investigation, Methodology, Validation, Writing – review & editing

Nalubowa Maureen Ssempijja: Formal Analysis, Resources, Supervision, Visualization, Writing – original draft

Funding

The authors received no funding.

Data Availability Statement

The data is available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Mubiru, K. P., Ssempijja, N. M. (2025). Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Mathematical Modelling and Applications, 10(3), 49-58. https://doi.org/10.11648/j.mma.20251003.12

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Mubiru, K. P.; Ssempijja, N. M. Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Math. Model. Appl. 2025, 10(3), 49-58. doi: 10.11648/j.mma.20251003.12

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AMA Style

Mubiru KP, Ssempijja NM. Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Math Model Appl. 2025;10(3):49-58. doi: 10.11648/j.mma.20251003.12

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@article{10.11648/j.mma.20251003.12,
  author = {Kizito Paul Mubiru and Nalubowa Maureen Ssempijja},
  title = {Mathematical Model of the Assembly Robot Performance in Automated Manufacturing
},
  journal = {Mathematical Modelling and Applications},
  volume = {10},
  number = {3},
  pages = {49-58},
  doi = {10.11648/j.mma.20251003.12},
  url = {https://doi.org/10.11648/j.mma.20251003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20251003.12},
  abstract = {Robot performance in automated manufacturing supports the competitive edge in manufacturing industries, enhancing customer sustainability and reducing operational costs. The robot cell controller must therefore optimize the operational tasks in the assembly of components or parts of the product to support automated manufacturing strategies for the manufacturing plant. This study proposes a Markov decision process model for robot performance in a steel assembly plant. The performance of the robot is characterized as a Markov chain, and its operational cost matrix represents the expected reward for the Markov decision process problem. This study utilized hourly data for two consecutive weeks. The principal challenge addressed focused on determining the best product assembly option to reduce the assembly expenses of the robot cell. We considered a multi-period planning horizon, where the optimal decision was determined for assembling or not assembling additional products based on the demand and availability of finished assembled products. The model was tested, and the results demonstrated the existence of an optimal state-dependent decision and assembly expenditure of managing the robot cell of the steel assembly plant. As a cost optimization strategy for managing robot cells, improved efficiency and resource utilization for assembled products were realized, supporting automated manufacturing initiatives of the steel assembly plant used in the case study.
},
 year = {2025}
}

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TY  - JOUR
T1  - Mathematical Model of the Assembly Robot Performance in Automated Manufacturing

AU  - Kizito Paul Mubiru
AU  - Nalubowa Maureen Ssempijja
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.mma.20251003.12
DO  - 10.11648/j.mma.20251003.12
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 49
EP  - 58
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20251003.12
AB  - Robot performance in automated manufacturing supports the competitive edge in manufacturing industries, enhancing customer sustainability and reducing operational costs. The robot cell controller must therefore optimize the operational tasks in the assembly of components or parts of the product to support automated manufacturing strategies for the manufacturing plant. This study proposes a Markov decision process model for robot performance in a steel assembly plant. The performance of the robot is characterized as a Markov chain, and its operational cost matrix represents the expected reward for the Markov decision process problem. This study utilized hourly data for two consecutive weeks. The principal challenge addressed focused on determining the best product assembly option to reduce the assembly expenses of the robot cell. We considered a multi-period planning horizon, where the optimal decision was determined for assembling or not assembling additional products based on the demand and availability of finished assembled products. The model was tested, and the results demonstrated the existence of an optimal state-dependent decision and assembly expenditure of managing the robot cell of the steel assembly plant. As a cost optimization strategy for managing robot cells, improved efficiency and resource utilization for assembled products were realized, supporting automated manufacturing initiatives of the steel assembly plant used in the case study.

VL  - 10
IS  - 3
ER  -

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Author Information

Kizito Paul Mubiru

Mechanical and Production Engineering, Kyambogo University, Kyambogo, Uganda

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http://orcid.org/0009-0007-8289-0324
Nalubowa Maureen Ssempijja

Mechanical and Production Engineering, Kyambogo University, Kyambogo, Uganda

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Plain Text BibTeX RIS

APA Style

Mubiru, K. P., Ssempijja, N. M. (2025). Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Mathematical Modelling and Applications, 10(3), 49-58. https://doi.org/10.11648/j.mma.20251003.12

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ACS Style

Mubiru, K. P.; Ssempijja, N. M. Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Math. Model. Appl. 2025, 10(3), 49-58. doi: 10.11648/j.mma.20251003.12

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AMA Style

Mubiru KP, Ssempijja NM. Mathematical Model of the Assembly Robot Performance in Automated Manufacturing. Math Model Appl. 2025;10(3):49-58. doi: 10.11648/j.mma.20251003.12

Copy | Download

@article{10.11648/j.mma.20251003.12,
  author = {Kizito Paul Mubiru and Nalubowa Maureen Ssempijja},
  title = {Mathematical Model of the Assembly Robot Performance in Automated Manufacturing
},
  journal = {Mathematical Modelling and Applications},
  volume = {10},
  number = {3},
  pages = {49-58},
  doi = {10.11648/j.mma.20251003.12},
  url = {https://doi.org/10.11648/j.mma.20251003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20251003.12},
  abstract = {Robot performance in automated manufacturing supports the competitive edge in manufacturing industries, enhancing customer sustainability and reducing operational costs. The robot cell controller must therefore optimize the operational tasks in the assembly of components or parts of the product to support automated manufacturing strategies for the manufacturing plant. This study proposes a Markov decision process model for robot performance in a steel assembly plant. The performance of the robot is characterized as a Markov chain, and its operational cost matrix represents the expected reward for the Markov decision process problem. This study utilized hourly data for two consecutive weeks. The principal challenge addressed focused on determining the best product assembly option to reduce the assembly expenses of the robot cell. We considered a multi-period planning horizon, where the optimal decision was determined for assembling or not assembling additional products based on the demand and availability of finished assembled products. The model was tested, and the results demonstrated the existence of an optimal state-dependent decision and assembly expenditure of managing the robot cell of the steel assembly plant. As a cost optimization strategy for managing robot cells, improved efficiency and resource utilization for assembled products were realized, supporting automated manufacturing initiatives of the steel assembly plant used in the case study.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Model of the Assembly Robot Performance in Automated Manufacturing

AU  - Kizito Paul Mubiru
AU  - Nalubowa Maureen Ssempijja
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.mma.20251003.12
DO  - 10.11648/j.mma.20251003.12
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 49
EP  - 58
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20251003.12
AB  - Robot performance in automated manufacturing supports the competitive edge in manufacturing industries, enhancing customer sustainability and reducing operational costs. The robot cell controller must therefore optimize the operational tasks in the assembly of components or parts of the product to support automated manufacturing strategies for the manufacturing plant. This study proposes a Markov decision process model for robot performance in a steel assembly plant. The performance of the robot is characterized as a Markov chain, and its operational cost matrix represents the expected reward for the Markov decision process problem. This study utilized hourly data for two consecutive weeks. The principal challenge addressed focused on determining the best product assembly option to reduce the assembly expenses of the robot cell. We considered a multi-period planning horizon, where the optimal decision was determined for assembling or not assembling additional products based on the demand and availability of finished assembled products. The model was tested, and the results demonstrated the existence of an optimal state-dependent decision and assembly expenditure of managing the robot cell of the steel assembly plant. As a cost optimization strategy for managing robot cells, improved efficiency and resource utilization for assembled products were realized, supporting automated manufacturing initiatives of the steel assembly plant used in the case study.

VL  - 10
IS  - 3
ER  -

Copy | Download