The optimization of lot sizing with supplier selection problem in multi-echelon defective supply chain network

Abstract

A new problem called lot sizing with supplier selection problem in the multi-product multi-echelon defective supply chain network (MDSCN) is proposed in this study. We explain the problem by a case study. We take the multi-product MDSCN of X enterprise into account. Back and front engine blocks are products of X enterprise. The aim of this study is to identify how many components will be purchased from which supplier while meeting the demands of the customers for these two products. The supply chain (SC) network of X enterprise is formed by mixed-integer linear programming (MILP). The optimization of current SC network of X enterprise is carried out by using Linear, INeractive, Discrete Optimizer (LINDO) program. The customer expectations of X enterprise are met at the highest level, and it gives the opportunity to have the knowledge, which reduces the total cost, of purchasing–production–distribution strategy with this work.

Keywords:

• supply chain management
• multi-echelon defective supply chain network design
• optimization
• lot sizing with supplier selection
• mixed-integer linear programming

1. Introduction

In the most general sense, the purpose of a company is to provide income in the area it serves by supplying the product or service to the consumer. Competitors are increasing, and rivalry is difficult in this area. So in order to gain the competitive superiority, companies must obtain the present sources with high efficiency, high quality and low cost. Companies must offer products that are flexible and are in accordance with changing customer demands, and they must resolve failures in the chain from suppliers to end customers. The aim of supply chain (SC) design is to provide an optimal platform for efficient and effective supply chain management (SCM). This problem is often an important and strategic operations management problem in SCM.

With the rapidly changing process in the world, the high quality and fast service that companies must perform in order to meet the needs of the customers make the companies change. The managers realizing these difficulties have recognized that an integrated approach rather than a traditional approach is needed. The need for managing and following all the activities until transporting goods and services to the markets has given birth to the SC concept. The integration adding value to the customers and the other money/land owners in the basic business processes from the end user to the original supplier producing goods, service and information is called SCM [1]. Supplier selection problem is one of the most important ones in SCM. Companies need to work with different suppliers. Company managers discuss situations where buyers can select the most suitable suppliers and optimize the lot size of each item (product). Erroneous selection of suppliers may lead to operations and finance problems.

After fulfilling the competitive superiority over a long period, there is a great importance for the companies to develop an SC distribution network that focuses on ‘customer satisfaction’ and ‘low cost’. If an item moves more than one echelon before reaching the last customer, this must be in the form of a ‘multi-echelon’ inventory system. In real life, due to various reasons (mechanical failure, human error, etc.), defects occur in the products and/or product components. If these defects are not taken into consideration during the planning process, it will lead to customer dissatisfaction. If there is at least one loss in multi-staged SC network, this SC network is called multi-echelon defective supply chain network (MDSCN) [2–5]. In real life, most of the companies have this kind of SC network. On the contrary, it is accepted in the SC network studies in the literature that there is no defect between the stages. In Figure 1, a sample MDSCN with defect rates is shown.

Figure 1. The MDSCN with defect rates.

The problem we considered in our study is based on lot sizing with supplier selection problem. The problem is related with purchasing the product components from the most appropriate suppliers (supplier selection), at the most economical prices (determining the order size), producing in the most appropriate quantities, storing the produced items in the most appropriate quantities and delivering the products to the customers from the distribution centres (DCs) considering the size of the defect between echelons. This problem is adapted to SC network [2–5]. This new problem is called lot sizing with supplier selection problem in the MDSCN. Literature on this problem assumes that there is a single item. The main contribution of our study is extending the single-item problem to multi-item.

The literature related to this research can be divided into two streams: lot sizing with supplier selection and SC network modelling. There are studies on lot sizing with supplier selection problem in the literature. The first study on lot sizing with supplier selection problem has been done by Basnet and Leung [6]. Basnet and Leung examined multi-supplier, multi-product, multi-period order size scenarios in their studies. The authors explained lot sizing with supplier selection problem and presented a mixed-integer mathematical model in order to solve this problem [6].

Awasthi et al. considered a supplier selection problem for a single manufacturer/retailer that faces a random demand with a limitation on minimum and maximum order sizes. They proposed a heuristic method for their model [7].

Zhang and Zhang [8] approached the supplier selection and purchase problem with fixed selection cost and limitation on minimum and maximum order sizes under stochastic demand. Dahel [9] presents a multi-objective mixed-integer programming (MIP) approach to simultaneously determine the number of suppliers to employ and the order quantities to allocate to these suppliers in a multi-product, multi-supplier competitive sourcing environment.

Goossens et al. [10] discussed the procurement problem of buying multiple items from a set of suppliers considering just the minimization of purchasing cost function so other affecting criteria are not included in their linear mathematical model. Burke et al. [11] analyse the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single buyer.

Crama et al. [12] describe the purchasing decisions faced by a multi-plant company. The suppliers of this company offer discount schedules based on the total quantity of materials purchased.

Aissaoui et al. [13] provide an updated and extensive review focusing on supplier selection and order lot size modelling. The first study on lot sizing with supplier selection problem in the MDSCN has been done by Şenyiğit. He discussed lot sizing in connection with supplier selection problem in the MDSCN. It was accepted that there was a defect only in the supplier's echelon in the SC network. Multi-stage mixed-integer linear programming (MILP) model under material requirement constraint was developed [2].

Şenyiğit and Göleç examined Şenyiğit's problem in the case that the demand was stochastic. In their study, they developed a new heuristic method for lot sizing with supplier selection problem in the MDSCN with stochastic demand. They analysed the performance of heuristic method in four different cases, two different variation coefficients and two different service levels. The supplier capacity was accepted to be infinite in their study [3].

Şenyiğit considered purchasing raw materials costs, production costs, fixed operation costs, transportation costs and lost sale cost in his study similar to his earlier study with Göleç. ProModel simulation software was used to model heuristics and MDSCN system. Şenyiğit extended the work he did with Göleç by finite supplier capacity and a new heuristic. The aim of Şenyiğit's work was to provide a solution to this problem. Additionally, a practice of these heuristics in an X firm in the Turkish furniture industry is presented. Numerical experiments are carried out to show the performance of the proposed heuristics [4].

In his later work, Şenyiğit assumed the MDSCN system to be small-scale model. There were three factories (Izmir, Kayseri and Erzurum), five DCs (Ankara, Adana, Istanbul, Kayseri and Van) and five groups of customers which are in the same city as the DCs on the small scale of MDSCN model. On the medium scale of MDSCN model, it has been assumed that another new factory, DC and customer group with the same values of all parameters have been added to already-existing places in the small scale of MDSCN model. Similarly, it has been assumed that in the large scale of the MDSCN model, two new factories, DCs and customer groups with the same values of all parameters have been added to each of the already-existing ones. The comparison of heuristic performances in three different scales of MDSCN models is done. Numerical experiments are carried out to show the performance of the proposed heuristics. H2 heuristic outperformed the H1 heuristic for three defective SC models [5].

In their study, Moghadam, Afsar and Sohrabi made use of a hybrid intelligent algorithm in the solution of lot sizing with supplier selection problem. Moghadam et al. [14] presented genetic algorithm and hybrid intelligent algorithm using fuzzy artificial neural network in order to plan the material, to determine the best supplier and to guess the demand ratio. Rezaei and Davoodi [15] developed two different multi-object mixed-integer non-linear models for multi-echelon lot size determination problem that has multi-product and multi-supplier in their studies.

In the literature, there are some papers on SC network design. Wang focused on an MDSCN design for choosing the appropriate corporations from a number of potential participators to become involved in the network and to make optimal production–distribution planning decisions. We took the MDSCN system into account like in Wang [16]. Che analysed the defective SC system to discuss its supplier selection, production and distribution as in our article. They developed an optimal mathematical model for both balanced and defective models and adopted particle swarm optimization (PSO) to obtain solutions for both models [17].

Yan et al. proposed a strategic production distribution model for SC design with consideration of Bill of Material (BOM). They showed how these relationships were formulated as logical constraints in an MIP model, thus capturing the role of BOM in the selection of suppliers in the strategic design of an SC [18].

Georgiadis et al. considered a detailed mathematical formulation for the problem of designing SC networks comprising multi-product production facilities with shared production resources, warehouses, DCs and customer zones and operating under time-varying demand uncertainty. They formulated the problem as an MILP problem and solved the global optimality using standard branch-and-bound techniques [19].

Demirtas and Üstün [20] combined analytic network process and multi-objective MILP and proposed an approach for selecting best suppliers and defining the optimum quantities among selected suppliers to maximize the total value of purchasing and minimize the budget and defect rate.

Alfares presented an MILP model for the optimum planning of multi-plant, multi-supplier and multi-grade petrochemical production. A discrete-time model is proposed that determines the optimum mix of petrochemical grades for each plant, the quantity to produce of each selected grade and the optimum production sequence of different grades. The model is applied to real-life data from multigrade polypropylene production in a large petrochemical company as our article [21].

Ertuğrul and Aytaç [22] proposed MILP model considering the variables like designing and planning the delivery networks, supplier, factory, capacity of the store, customer demands, supply, production, storage and transportation cost within SC. The products we have taken into consideration in our study are similar to those in Ertuğrul and Aytaç's study.

The rest of this article is organized as follows: In Section 2, mathematical models presented in the study and notations used are shown. Information about the case study is given in Section 3. In Section 4, numerical results obtained in the study are stated. In Section 5, results and suggestions are presented.

2. MILP model

The following MILP model is presented to determine the optimum lot sizing with supplier selection in the multi-product MDSCN with limited supplier capacities.

The assumptions, notation, variables, objective and limitations of this model are presented below.

Assumptions

1.	The firm has a four-echelon defective SC system that consists of supplier echelon, factories echelon, DCs echelon and customers echelon (multi-echelon). Customer echelons are retail sellers.
2.	Each echelon has a different defect rate.
3.	We dealt with back and front engine blocks as products (multi-product).
4.	The back engine block has eight different components (see Figure 2).
5.	The front engine block has six different components (see Figure 3).
6.	There are 10 main components for products. There are three suppliers for each main component.
7.	All suppliers have finite capacities.
8.	The products are produced in two factories (Istanbul and Kayseri).
9.	There are three DCs (Tekirdağ, Izmir and Adana) for each product in the same cities.
10.	There are three groups of customers (Istanbul, Kayseri and Izmir) for back engine block.
11.	There are four groups of customers (Istanbul, Kayseri, Izmir and Adana) for front engine block.
12.	The total cost includes transportation cost, purchasing cost, production cost and storage costs.
13.	We assumed that the transportation costs did not include the timing element.

Figure 2. The bill of materials of front engine block.

Figure 3. The bill of materials of back engine block.

Indices/sets

a: Supplier indices;	=	A, Total number of suppliers; a∈A.
b: Factory indices;	=	B, Total number of factories; b∈B.
c: Component indices;	=	C, Total number of components; c∈C.
d: DC indices;	=	D, Total number of DCs; d∈D.
e: Customer group indices;	=	E, Total number of customer groups; e∈E.
f: Product indices;	=	F, Total number of products; f∈F.

Parameters

PCL bf :	=	The production capacity limits of factory b for product f.
SCL df :	=	The storage capacity limits of DC d for product f.
CD ef :	=	The demand of customer e for product f.
TC abc :	=	The transportation cost of component c from supplier a to factory b.
PC abc :	=	The purchasing cost of component c from supplier a to factory b.
TC bdf :	=	Unit transportation cost of product f from factory b to DC d.
TC def :	=	Unit transportation cost of product f from DC d to customer e.
PRC bdf :	=	Unit production cost of product f from factory b to DC d.
SC bdf :	=	Unit storage cost of product f from factory b to DC d.
RA cf :	=	Units of component c required to produce one unit of product f according to the product BOM.
CCL ac :	=	The capacity limits of component c of supplier a.
ADR bf :	=	The average defect rate of factory b for product f.
ADR df :	=	The average defect rate of DC d for product f.
ADR ef :	=	The average defect rate of customer e for product f.

Decision variables

AC abc :	=	the amount of component c purchased from supplier a to factory b.
NP bdf :	=	the number of product f sent to DC d from factory b.
NP def :	=	the number of product f sent to customer e from DC d.

The objective function is used to minimize the total relevant cost (TRC) of the MDSCN system and all intermediate variables are the functions of the decision variable. The objective function is net cost minimization.

The objective function and constraints of the model are listed as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Constraint number (1) shows the supplier capacity constraint. According to this constraint, the component amount defined considering the defects sent to factories from the suppliers cannot be on the related component capacity of the supplier. Constraint number (2) determines the production capacity requirement. According to this constraint, total product amount determined by considering the defects sent to DCs from a factory cannot be more than the production capacity of this factory. Constraint number (3) shows the capacity of the DCs. According to this constraint, the amount of the product determined by considering the defects stored at the DCs cannot be more than the capacity of the DCs. Constraint numbers (4) and (5) are the equilibrium constraints. Constraint number (4) states that component amount determined considering the defects sent to that factory from the supplier cannot be more than the amount needed to produce that product in that factory.

Constraint number (5) states that the amount of the product including the defects sent to the DC from a factory cannot be more than the amount of the product sent to the customers from the same DC. Constraint number (6) states that the total of product amount including the defects sent to the customer groups from DCs must be equal to the amount of customer e demand for product f. Constraint number (7) shows that variables are positive.

3. Case study

The aim of this study is to identify how many components will be purchased from which supplier while meeting the demands of the customers for these two products (Back engine block and front engine block). The number of components of the items is not defined in the study since the company did not allow it for strategic reasons. Company X is a real company. All the data are real-life data. In Figure 4, the MDSCN of X enterprise is shown on the geographical map of Turkey. The illustrations of average defect rates and MDSCN for back engine block are given in Figure 5 and for front engine block are given in Figure 6. The average defects and transportation costs are real-life data and were proposed by X firm. The average defect rates of the sent components to the factories from the suppliers are shown in Table 1. The average defect rates of the sent products to the DCs from the factories are shown in Table 2. The average defect rates of the sent products from the DCs to the customers are shown in Table 3. Data used in the MDSCN of the X enterprise are given in Tables 4–13. Unit transportation costs of components from the supplier to the factories are shown in Table 4. The capacities of the suppliers for each component are shown in Table 5.

Figure 4. The illustration of MDSCN geographical location of X enterprise.

Figure 5. The illustration of average defect rates and MDSCN for back engine block.

Figure 6. The illustration of average defect rates and MDSCN for front engine block.

Table 1. The illustration of the average defect rates at factories echelon

	Average defect rates (%)
Factories	Back engine block	Front engine block
Istanbul	8	9
Kayseri	9	7

Table 2. The illustration of the average defect rates at DCs echelon

	Average defect rates (%)
DCs	Back engine block	Front engine block
Tekirdağ	7	8
Izmir	3	4
Adana	5	6
Note: DC, distributive centre.

Table 3. The illustration of the average defect rates at customer echelon

	Average defect rates (%)
Customers	Back engine block	Front engine block
Istanbul	8	7
Kayseri	7	9
Izmir	4	5
Adana	–	2

Table 4. The unit transportation costs of components from suppliers to factories

	Factories
	Istanbul			Kayseri
Suppliers	Istanbul	Kayseri	Izmir	Istanbul	Kayseri	Izmir
Components
Six-corners automaton Type 27	0.20	0.25	0.23	0.20	0.18	0.17
Six-corners automaton Type 22	0.30	0.52	0.23	0.18	0.14	0.17
Six-corners automaton Type 19	0.29	0.25	0.29	0.20	0.19	0.17
Disc automaton Type 10	0.25	0.23	0.24	0.20	0.18	0.19
Sphero-moulting	0.30	0.28	0.30	0.28	0.27	0.25
Aluminium AT 171	0.50	0.49	0.47	0.39	0.35	0.38
Punched metal sheet	0.15	0.13	0.13	0.12	0.11	0.13
Base metal sheet	0.19	0.18	0.19	0.12	0.11	0.13
Rail-Baffle M16	0.15	0.14	0.15	0.09	0.09	0.08
Bolts	0.15	0.13	0.10	0.05	0.05	0.05

Table 5. The capacities of suppliers for each component

	Capacities of suppliers (kg)
	Istanbul	Kayseri	Izmir
Components
Six-corners automaton Type 27	3,000	2,900	3,500
Six-corners automaton Type 22	2,800	2,950	3,000
Six-corners automaton Type 19	1,800	1,700	1,750
Disc automaton Type 10	200	150	180
Sphero-moulting	16,800	17,100	17,000
Aluminium AT 171	21,000	20,000	22,500
Punched metal sheet	1,200	1,250	1,300
Base metal sheet	6,030	6,080	6,100
Rail-Baffle M16	100	1,000	90
Bolts	1,750	1,800	1,790

The unit transportation costs of the products from the factories to the DCs are given in Table 6. The unit purchasing costs of the components from the supplier to the factories are given in Table 7. The unit transportation costs of the products from DCs to the customers are given in Table 8. The unit production costs at the factories are shown in Table 9. The monthly production capacities of the factories and the unit storage costs of the DCs are shown in Tables 10 and 11. In Table 12, the monthly storage capacities of the DCs are given. In Table 13, the monthly demands of the customers are shown.

Table 6. The unit transportation costs of products from factories to DCs

	Back engine block			Front engine block
DCs	Tekirdağ	Izmir	Adana	Tekirdağ	Izmir	Adana
Factories
Istanbul	1.5	3.4	4.5	1.3	3.1	5.4
Kayseri	3.2	2.9	2.3	3.5	3.4	2.1
Note: DC, distributive centre.

Table 7. The unit purchasing costs of components from suppliers to factories

	Factories
	Istanbul			Kayseri
Suppliers	Istanbul	Kayseri	Izmir	Istanbul	Kayseri	Izmir
Components
Six-corners automaton Type 27	1.80	2.25	2.07	1.80	1.62	1.53
Six-corners automaton Type 22	2.70	4.68	2.07	1.62	1.26	1.53
Six-corners automaton Type 19	2.61	2.250	2.61	1.80	1.71	1.53
Disc automaton Type 10	2.25	2.07	2.16	1.80	1.62	1.71
Sphero-moulting	2.70	2.52	2.70	2.52	2.43	2.25
Aluminium AT 171	4.50	4.41	4.23	3.51	3.15	3.42
Punched metal sheet	1.35	1.17	0.90	1.08	0.99	1.17
Base metal sheet	1.71	1.67	1.71	1.08	0.99	1.17
Rail-Baffle M16	1.35	1.26	1.35	0.81	0.80	0.72
Bolts	1.35	1.17	0.90	0.43	0.41	0.45

Table 8. The unit transportation costs of products from DCs to customers

	Back engine block			Front engine block
Customers	Istanbul	Kayseri	Izmir	Istanbul	Kayseri	Izmir	Adana
DCs
Tekirdağ	11	4	10	13	5	12	8
Izmir	13	9	4	12	8	3	5
Adana	12	11	11	14	13	15	3
Note: DC, distributive centre.

Table 9. The unit production costs of products for factories

	Back engine block	Front engine block
Istanbul	4.1	6.7
Kayseri	5.1	6.2

Table 10. The monthly production capacities of products of factories

	Back engine block	Front engine block
Istanbul	2000	3000
Kayseri	2900	2800

Table 11. The unit storing costs of products of DCs

	Back engine block	Front engine block
Tekirdağ	1.0	1.2
Izmir	1.2	1.1
Adana	0.9	1.0
Note: DC, distributive centre.

Table 12. The monthly storing capacities of products for DCs

	Back engine block	Front engine block
Tekirdağ	4000	3900
Izmir	3000	4100
Adana	3500	3700
Note: DC, distributive centre.

Table 13. The monthly demands of customers

	Back engine block	Front engine block
Istanbul	1200	1300
Kayseri	1100	1200
Izmir	1500	1400
Adana	–	1215

4. Numerical results

A series of procedures are needed for a product to reach the consumer. The relations of the companies where the processes change off form the SC network. In this chain, each chain is the customer of the previous link and is the supplier of the next chain.

The purpose is to provide the last customer with the best service, to meet customer satisfaction and to reduce the total cost considering the defects between the echelons. As a matter of course, various amounts of demands for this product may occur. In this study, the aim is to present the information of which DC will be used and how many components will be purchased from which supplier.

Table 14 shows from which supplier and how many components needed for producing the items in optimum solution will be purchased for the factories in Istanbul and Kayseri. The solution is obtained using LINDO package software (LINDO Systems Inc., Chicago, IL, USA). To get more rapid solutions, computer software is often used to solve problems of this kind. LINDO is a kind of software package used specifically to solve mathematical programming problem. LINDO runs fast and easy to input, analyse and solves the mathematical programming problems. Therefore, it is widely used in mathematics, science and industry. LINDO is mainly used for solving linear, non-linear, quadratic and integer programming problems. It can also be used in some non-linear and linear equations for solving algebraic equation and the root. LINDO contains a kind of modelling language and many common mathematical functions for users invoking to establish programming problem. The problem discussed in this article can be solved by LINDO software.

Table 14. The presentation of quantities of components and suppliers

	Factories
	Istanbul			Kayseri
Suppliers	Istanbul	Kayseri	Izmir	Istanbul	Kayseri	Izmir
Components
Six-corners automaton Type 27	687.58	0	0	0	0	1,485.05
Six-corners automaton Type 22	0	0	1,173.88	0	1,509.74	0
Six-corners automaton Type 19	0	383.63	0	0	0	828.57
Disc automaton Type 10	0	63.99	0	0	62.22	0
Sphero-moulting	0	10,558.07	0	0	0	10,153.85
Aluminium AT 171	0	0	6,991.17	0	11,393.85	0
Punched metal sheet	0	0	805.62	490	1,739.99	0
Base metal sheet	0	2,571.69	0	0	3,420.11	0
Rail-Baffle M16	0	233.13	0	0	503.11	90
Bolts	0	0	744.18	0	989.67	0

The model determined the product amount that will be sent to the customers from the DCs considering these defect rates. For example, the average defect rate of back engine block for Izmir customer group is 4% (Table 3). Izmir customer group demanded 1500 back engine blocks. The model determined that 1563 back engine blocks must be sent to the customer from the Izmir DC (Table 13). The total cost of the MDSCN of X enterprise is 309,673.50 Turkish Lira.

As there was no fractional transportation, it turned into integer-valued solution by rounding up. In Table 15, the number of products produced in optimum solution at the factory considering the defects is shown. The number of products sent to the customers from the DCs considering the defect rates on delivery stage is specified and shown in Table 16. The number of products sent from DCs is greater than that of the customer demands because of the defects occurring during the delivery.

Table 15. The presentation of the quantities of the products sent to DCs

	Back engine block			Front engine block
DCs	Tekirdağ	Izmir	Adana	Tekirdağ	Izmir	Adana
Factories
Istanbul	1358	0	0	1434	1511	0
Kayseri	0	1611	1290		1482	1319
Note: DC, distributive centre.

Table 16. The presentation of the number of products sent from DCs to customers

	Back engine block			Front engine block
Customers	Istanbul	Kayseri	Izmir	Istanbul	Kayseri	Izmir	Adana
DCs
Tekirdağ	80	1183	0	0	1319	0	0
Izmir	0	0	1563	1398	0	1474	0
Adana	1225	0	0	0	0	0	1240
Total	1305	1183	1563	1398	1319	1474	1240
Note: DC, distributive centre.

5. Conclusion and future work

In this study, lot sizing with supplier selection problem in MDSCN that has an important place in the globalizing world is discussed. The result is obtained using MILP. In the developed model, a design that considers the material requirements for the last product for MDSCN is presented. The customer demands and the defect rates between the echelons are taken into consideration and what quantity will be purchased from which component and from which supplier are determined.

The customer expectations are met at the highest level, and it gives the opportunity to have the knowledge, which reduces the total cost, of purchasing–production–distribution strategy with this work. This study can be applied to more complex problems that consider all products and customers of the company. In the future works, new SC network designs for identifying the most appropriate number of DCs and factories can be determined. Decision variables are indefinite in real life. Customer demands in the SC network can be stochastic. A heuristic method focused on determining simultaneous lot sizing with supplier selection in MDSCN where the demands are stochastic can be developed as a future study.

Acknowledgements

This study originated from the project supported by the Scientific Research Unit (SRU) of Erciyes University (SRU project number is FBA-09-796).

Notes