Exact solution of the optimal control problem of coordinating a

supplier-manufacturer supply chain in advanced geometric concepts

ATEFEH HASAN-ZADEH

Fouman Faculty of Engineering, College of Engineering, University of Tehran, IRAN

Abstract: Supply chain coordination deals with collaborative efforts of supply chain parties and making globally-

optimal decisions that can improve overall performance and efficiency of the entire supply chain. In many

situations, the problem of supply chain coordination requires formulation of a continuous time optimal control

model, in which optimal solution is identified approximately through numerical estimation. Therefore, in this

paper, a novel approach was presented for optimal control problem by developing a new formulation based on

advanced ingredients of differential and Poisson geometry. Thus, the exact optimal solution of control problem

can be obtained using an analytical methodology that converts the Hamilton-Jacobi-Bellman partial differential

equation (PDE) into a reduced Hamiltonian system. The proposed approach was applied to the problem of

coordinating supplier development programs in a two-echelon supply chain comprising of a single supplier and

a manufacturing firm. For further illustrating applicability and efficiency of the proposed methodology, a

numerical example was also provided. The proposed approach offers unique advantages and can be applied to

find the exact solution of optimal control models in various optimization problems.

Key-Words: Optimal Control Problem, Hamiltonian System, First Integral, Supply Chain Coordination, Supplier

Development

Received: June 12, 2021. Revised: February 20, 2022. Accepted: March 22, 2022. Published: April 28, 2022.

1 Introduction

In recent years, supply chain coordination has

become a major issue in supply chain management

and received much attention from both supply chain

researchers and practitioners [1].

Supply chain coordination implies collaborative

efforts of supply chain members working together

towards mutually-defined goals and activities,

including supplier development, coordination with

suppliers and customers, etc. [2].

It is concerned with making globally-optimal supply

chain decisions that can benefit all supply chain

members, instead of individual decisions [3].

In the recent years, supply chain coordination has

become a major issue in supply chain management

and has received a great deal of attention both from

researchers and practitioners in the field of supply

chain [4].

Supply chain coordination implies collaborative

efforts of supply chain members working together to

achieve the mutually-defined goals and activities

including supplier development, coordination with

suppliers and customers, etc. [5]. It is concerned with

making globally-optimal supply chain decisions that

can be useful for all supply chain members, instead

of individual decisions [6].

Supply chain coordination plays a critical role in

improving the overall performance of supply chain

and the lack of coordination among supply chain

partners may reduce efficiency and result in

undesirable consequences in supply chain operations.

Therefore, the centralized decision-making and

various mechanisms are used by supply chain

partners including revenue sharing, risk sharing,

synchronized operation, etc., to achieve coordination

purposes [7].

In many industries, manufacturing firms develop

strategic, long-term relationships with their suppliers

by implementing and supporting supplier

development programs [8].

The goal is improving performance and

capabilities of the suppliers to meet short- and long-

term supply needs of manufacturing firm, which in

turn results in improving operational performance in

terms of cost, quality, delivery, etc. [9].

Such a strong relationship between manufacturers

and suppliers enhances the overall efficiency and

profitability of both parties and helps to create

sustainable competitive advantage [10].

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

Despite potential benefits of supplier

development programs, they might be unattractive

for suppliers, because suppliers might be reluctant to

modify their internal processes and instead pursue

their own objectives [11]. Since, success of supplier

development program depends on mutual recognition

and aligned objectives, coordination between

supplier and manufacturer is required [8], [12]. Thus,

optimal decision on supplier development is

characterized by a solution for the problem of supply

chain coordination.

Many problems of supply chain coordination

which were mentioned above, involve formulating

and solving a continuous time optimal control model

with an equation of incomplete Hamiltonian system,

in which the exact optimal solution cannot be

obtained, and instead it should be approximately

estimated by numerical analysis (e.g., [8], [13]).

Therefore, in this paper, a novel analytical

solution approach is presented based on differential

and Poisson geometry by reformulating and

converting the original problem to a reduced

Hamiltonian system ([14-19]. The proposed

approach is applied to obtain optimal solution to the

problem of coordinating supplier development in a

two-echelon supply chain comprising of a single

supplier and single manufacturer.

2 Problem of coordinating supplier

development

2.1 Problem description and formulation

We consider the problem of coordinating supplier

development in a two-echelon supply chain as

presented in the study by Proch et al., [8]. Supply

chain comprises of a single supplier and a single

manufacturing firm where manufacturer assembles

components from supplier and sells the final products

to the market. The goal is identifying the optimal

decision of supplier development investment.

A centralized decision-making process is assumed

and supply chain is considered as an integrated

system ,in which all parameters including the optimal

amount of effort invested in supplier development are

simultaneously chosen. This decision-making

process ensures efficiency of the entire system and

opts for the optimum level of supplier development,

i.e., maximizes the total profit of the supply chain.

Variables and parameters for this model are

summarized in Table 1.

Table 1. Parameters and Decision Variables ([8])

Parameters/

Variables

Description

Prohibitive price (e.g.

maximum willingness to pay)

Price elasticity of the

commodity

Manufacturer's unit production

cost

Supply cost per unit charged by

the supplier

Supplier's unit production cost

at the begininig of the contract

period

 

The measurement of the efforts

invested in the supplier

development

The supplier learning rate

 

ln ;

[0,1], 1

c x = c x









Supplier production cost

The supplier fixed profit

margin

 

The effort at time

 



Capicity limit of (resource

availability in terms of time,

man power or budget)

The profit function





 

: 0, ,

J L T RR

the set of measurable functions and the model of

efforts invested in supplier development are defined

by the following problem:

 









 

0, 0 ,

a - c - c x t - r

:= -c u t dt,

subject to x= u;

u : T ,ω

x = x = .





(1)

The centralized collaboration strategy should be

determined such that, the accumulated profit function

(1) is maximized. Using the maximum principle

applied to the optimal control problem (1) with the

Hamiltonian function of

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

 

     

, , ,

H t x u

a - c -c x t - r

-c u t + t u t



(2)

switching time

t

can be obtained by the solution to

   

 

, , 0

Hx u t t = -c + t =







Then, as investigated in a previous study [8],

t

is obtained by numerical analysis from the following

equation:

   

 

m+ m

mc + t a - c - c + t t -T



  

(3)

More details on the above fromulation have been

given in the previous study [8].

2.2. Conversion of the model based on the

proposed methodology

The optimization problem given in Equation (1) is a

common form in many problems of supply chain

coordination. It results in an equation with different

parameters for switching time and the optimal control

function, which can be only evaluated by numerical

estimation. In fact, there exists only one equation

with different parameters (Equation (3)).

The case where the Hamiltonian

Σφάλμα!

Δεν έχει οριστεί σελιδοδείκτης. is linear in

control

is of special interest. Especially, it is a

simple situation to handle when

Σφάλμα! Δεν

έχει οριστεί σελιδοδείκτης. is plotted against

Σφάλμα! Δεν έχει οριστεί σελιδοδείκτης. either

as a positively-or negatively-sloped straight line,

since the optimal control is always to be found at a

boundary of

. Thus, the only task is determining this

boundary. Moreover, this case serves to highlight

how a complex situation in the calculus of variations

has now become easily manageable in optimal

control theory.

This simple approach apparently results in

elimination of some equations of the Hamiltonian

system in the mentioned coordination optimization

problem. For example, because accurate

determination of the capacity limit

 



 

in the problem is not critical to our discussion, it is

exogenously assessed to be feasible to the problem.

However, given the proposed approach, we

consider all the functions and parameters in the

system along with their actual effect. Thus, it will be

possible to incorporate more variables in the

coordination optimization model. This can be

implemented by considering some variables as

multiple functions and then, the Hamiltonian

function as a function of these variables and their

derivatives.

This simple approach apparently results in

elimination of some equations of the Hamiltonian

system in the mentioned coordination optimization

problem. For example, because accurate

determination of the capacity limit

 



 

in the problem is not critical to our discussion, it is

exogenously assessed to be feasible to the problem.

However, given the proposed approach, we

consider all the functions and parameters in the

system along with their actual effect. Thus, it will be

possible to incorporate more variables in the

coordination optimization model. This can be

implemented by considering some variables as

multiple functions and then, the Hamiltonian

function as a function of these variables and their

derivatives.

2.3. Solution method

According to Equation (4), we can rewrite the

corresponding Hamiltonian function (2) as

 

     

, , ,

M SC

H = H x u d

= d p d t - c -c

-c u t + t u t



(4)

with production quantity of

 

M SC

a - c -c

d t = b

and price distribution of

   

 

M SC

p d = p d t = a -bd = a+c +c

c = r+c x

Step 1 (Hamiltonian System): We have the

Hamiltonian system

   

 

, ,

m- m

M SC

- mc x t a - c -c x t

= = -

x b dt

H du

= d a -bd - c - c - bd = -

d dt

H dx H

= u = = -c + = d

dt u

























which can be written as follows

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

     

 

m- m

mc x t a- c - c x t

t= b



(5)

   

d t =c + t



(6)

 

M SC

u t = d -a+bd +c +c +bd 

(7)

Step 2 (First Integrals): According to Equation

(5), we have

 

   

 

2 1 2 1

m- m

M m- m-

m- m-

mc x s a - c - c x s

- ds

- a - c I t - I t

+ I t - I t







(8)

where,

   

I s = x k dk



. Also, according to

Equation (6), we have

 

   

 

SD t

d t = d t - s - c ds

= d t - t - t c - s ds





 



(9)

Then, substituting this into Equation (7) results in

 

2 1 2 1

M m- m-

m- m-

u t = u t

+ -a+c +r I t - I t

+ I t - I t

- a - c - r t - t

+ I t - I t



(10)

Finally, for

( ) 1x t = + t



[0, ]tt





, as

expressed in Equation (8), we conclude that

 

222

c a - c

- + t - + t

+ + t - + t













(11)

In addition, based on Equation (9), we have:

 

   

 

   

 

221 21

4 2 1

m+ m+

d t = d t

c a - c

+ + t t - t

c a - c

- + t - + t

b m+

- + t t - t

+ + t - + t

b m+





















(12)

Finally, Equation (10) results in

 

222 22

m+ m+

= u t

+ -a+c +r + t - + t

+ + t - + t

m+ b

- a - c - r t - t

+ + t - + t















(13)

Step 3 (Reduction): Following Step 2, we have

 

SD HH

-c + p = u



Then, its first integral is

u= p - c

and Equation (4) is reduced to

 

M SC SD SD

H p,x,d

= d p d t -c - c p - c -c + p

2.4. Numerical example and discussion

For further illustrating applicability and superiority

of the proposed methodology, a numerical example

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

is presented. Data for the example are adopted from

the study by Proch et al., [10]. We apply the proposed

approach and the exact solution algorithm presented

in the current research to obtain the results and

compare them with those obtained from numerical

estimation. It helps to evaluate performance and

efficiency of the proposed algorithm and analyze

quality of the obtained solution against a reference

solution. Characteristics of the numerical example

are given in Table 2.

Table 2. Parameter Values for Numerical Analysis

(Adopted from Proch et al. [8])

200

0.01



100000

100

0.1-

For numerical analysis of the problem using the

given parameter values, from Equation (11), we

obtain

 

     

 

c a - c

t = T - + T - + t

+ + T - + t

   











Since

 

t = c





, then we have

 

100 200 70 01

100000 0 1 60 1

0 02

100 02

1 60 1

0 04

( - ) -.

= - + +t

+ + - +t

















resulting in

9.844t=



Substituting the identified value in Equation (11),

we have

     

0.1 0.2

25655 0650000 1 250000 1

t =- - +t - +t



Since

   

 

019348.65

a - c - rc x

d t = = -



then based on Equation (12), we conclude that

   

    

 



0.9

0.8

19348.65 512146.73 9.844

722222.22 8.54 1 155203.71 9.844

312500 6.73 1

d t = - + -t

- - +t - -t

+ - +t

Also, according to Equation (13), we obtain

 

   

 

287500 0 78 1

13888 88 73 1

57 5 9 88 55 55 8 54 1

u t = u t - . - +t

+ . - +t

- . . -t + . . - +t



The approximate value of

t

is equal to 9.212, as

obtained numerically in the study by Proch et al., [8].

However, the analytical solution algorithm

developed herein provides a better answer as it yields

a bigger objective value. The difference between the

results is due to elimination of some equations of the

Hamiltonian system, which is also a prevalent

practice to find the answer to the optimal control

model in coordination optimization problems.

Using our proposed methodology, the value of

switching

t

was obtained as 9.844, which is clearly

better than the result obtained in the study by Proch

et al., [8] for the presented maximization control

problem. In the previous works (e.g., [8], [12-13]),

the optimal solution has been identified by

eliminating some critical equations. Thus, some

important characteristics of the problem should be

overlooked. In fact, an accurate determination of

 



 

and

 

variables has been exogenously

assessed to be feasible or they should be

approximately identified. But in the proposed

method, in actual inspection, we consider the

variables as multiple functions and then, the

Hamiltonian function as a function of these variables

and their derivatives.

3. Conclusions

In this paper, a novel methodology was presented to

find the exact optimal solution of the general

continuous time optimal control problem by

developing a novel reformulation drawing upon

differential and Poisson geometry. For this purpose,

we applied geometric notions about symmetric

groups and first integrals to reduce the order of the

Hamiltonian system. The proposed approach and

solution method was applied to supply chain

coordination problem in a two-echelon supply chain

with the objective of finding the optimal decision of

supplier development investment. We obtained the

exact optimal solution and the optimum switching

time for corresponding coordination problem with a

single supplier and single manufacturing firm.

The main advantage of the proposed methodology

is that it outperforms the numerical estimation

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

approach which is prevalent in solving the optimal

control models in coordination optimization

problems. The proposed methodology converts the

original problem to the system of fully Hamiltonian

equations with equations as equal as variables. It

provides the analytic optimal solution and, thus, it

yields better results than those obtained through

numerical estimation. The proposed approach can be

also successfully applied to solve optimal control

models in other optimization problems.

Acknowledgment

The author of the article considers it necessary to

thank and appreciate the valuable guidance of Dr.

Mohammad-Sadegh Sangari.

References:

[1] Hosseini-Motlagh, S.M., Ebrahimi, S.,

Zirakpourdehkordi, R., “Coordination of dual-

function acquisition price and corporate social

responsibility in a sustainable closed-loop

supply chain”, Journal of Cleaner Production,

Vol. 251, 2020, 119629, DOI:

10.1016/j.jclepro.2019.119629.

[2] Jian, J., Zhang, Y., Jiang, L., Su, J.,

“Coordination of Supply Chains with

Competing Manufacturers considering Fairness

Concerns”, Complexity, Vol. 2020, 2020,

4372603, 15 pages, DOI:

10.1155/2020/4372603.

[3] Hu, H., Zhang, Z., Wu, Q., Han, S.,

“Manufacturer’s customer satisfaction incentive

plan forduopoly retailers with Cournot or

collusion games”, Advances in Production

Engineering & Management, Vol. 15, No. 3,

2020, pp. 345-357.

[4] Hosseini-Motlagh, S.M., Nouri-Harzvili, M.,

Zirakpourdehkordi, R., “Two-level supply chain

quality improvement through a wholesale price

coordination contract on pricing, quality and

services”, International Journal of Industrial

Engineering & Production Research, Vol. 30,

No. 3, 2019, pp. 287-312.

[5] Hosseini-Motlagh, S.M., Nouri-Harzvilia, M.,

Johari, M., Sarker, B.R., “Coordinating

economic incentives, customer service and

pricing decisions in a competitive closed-loop

supply chain”, Journal of Cleaner Production,

Vol. 255, 2020, 120241, DOI:

10.1016/j.jclepro.2020.120241.

[6] Johari, M., Hosseini-Motlagh, S.M.,

“Coordination contract for a competitive

pharmaceutical supply chain considering

corporate social responsibility and pricing

decisions”, RAIRO-Operations Research, Vol.

54, No. 5, 2020, pp. 1515-1535.

[7] Adabi, H., Mashreghi, H., “Coordination and

competition in a duopoly with two-manufacturer

and two-retailer with wholesale-price contract

and demand uncertainty”, International Journal

of Industrial Engineering & Production

Research, Vol. 30, No. 4, 2019, pp. 465-476.

[8] Proch, M., Worthmann, K., Schluchtermann, J.,

“A negotiation–based algorithm to coordinate

supplier development in decentralized supply

chains”, European Journal of Operational

Research, Vol. 256, No. 2, 2017, pp. 412-429.

[9] Dastyar, H., Rippel, D., Freitag, M.,

“Optimization of Supplier Development under

Market Dynamics”, Mathematical Problems in

Engineering, Vol. 2020, 2020, 2912380, DOI:

10.1155/2020/2912380.

[10] Pham, T.H., Doan, T.D.U., “Supply chain

relationship quality, environmental uncertainty,

supply chain performance and financial

performance of high-tech agribusinesses in

Vietnam”, Uncertain Supply Chain

Management, Vol. 8, 2020, pp. 663–674.

[11] Li, S., Kang, M., Haney, M.H., “The effect of

supplier development on outsourcing

performance: the mediating roles of

opportunism and flexibility”, Production

Planning & Control, The Management of

Operations, Vol. 28, No. 6-8, 2017, DOI:

10.1080/09537287.2017.1309711.

[12] Kotzab, H., Darkow, I.L., Bäumler, I., Georgi,

C., “Coordination, cooperation and

collaboration in logistics and supply chains: a

bibliometric analysis”, Production, Vol. 29,

2019, DOI: 10.1590/0103-6513.20180088.

[13] Hsieh, F.S., “Dynamic configuration and

collaborative scheduling in supply chainsbased

on scalable multi-agent architecture”, Journal of

Industrial Engineering, 2018, DOI:

10.1007/s40092-018-0291-5.

[14] Hasan-Zadeh, A., “A note on variational

symmetries of the constrained variational

problems”, Advances in Differential Equations

and Control Processes, Vol. 18, No. 2, 2017, pp.

69-76.

[15] Hasan-Zadeh, A., Mohammadi-khanaposhti,

M., “Inertial manifolds for Navier-Stokes

equations in notions of Lie algebras”, WSEAS

Transactions on Heat and Mass Transfer, Vol.

13, No. 1, 2018, pp. 95-102.

[16] Hasan-Zadeh, A. “Exact inertial manifolds for

dynamical systems”, Advances in Differential

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022

Equations and Control Processes, Vol. 21, No.

1, 2019, pp. 117-122.

[17] Hasan-Zadeh, A. “Geometric Classification of

analytical solutions of Fitzhugh-Nagumo

equation and its generalization as the reaction-

diffusion equation”, Advances in Differential

Equations and Control Processes, Vol. 24, No.

2, 2021, pp. 167-174.

[18] Hasan-Zadeh, “Dynamic optimal control

problems in Hamiltonian and Lagrangian

systems”, Advances in Differential Equations

and Control Processes, Vol. 24, No. 2, 2021, pp.

175-185.

[19] HasanZadeh, “Geometric investigation of

nonlinear reaction-diffusion-convection

equations: Fisher equation and its extensions”,

Advances in Differential Equations and Control

Processes, Vol. 24, No. 2, 2021, pp. 145-151.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.6

Atefeh Hasan-Ζadeh

E-ISSN: 2224-2678

Volume 21, 2022