Two models of inventory control with supplier selection in case of multiple sourcing: a case of Isfahan Steel Company
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Abstract
Selecting the best suppliers is crucial for a company’s success. Since competition is a determining factor nowadays, reducing cost and increasing quality of products are two key criteria for appropriate supplier selection. In the study, first the inventories of agglomeration plant of Isfahan Steel Company were categorized through VED and ABC methods. Then the models to supply two important kinds of raw materials (inventories) were developed, considering the following items: (1) the optimal consumption composite of the materials, (2) the total cost of logistics, (3) each supplier’s terms and conditions, (4) the buyer’s limitations and (5) the consumption behavior of the buyers. Among diverse developed and tested models—using the company’s actual data within three pervious years—the two new innovative models of mixedinteger nonlinear programming type were found to be most suitable. The results of solving two models by lingo software (based on company’s data in this particular case) were equaled. Comparing the results of the new models to the actual performance of the company revealed 10.9 and 7.1 % reduction in total procurement costs of the company in two consecutive years.
Keywords
Inventory control Supplier selection Multiple sourcing Mathematical modelsIntroduction
Supplier selection is turning to become one of the crucial decisions in operations management area for many companies. Nowadays that competition plays a major role in business, two factors, namely, cost reduction and increase in quality of products, are keys to success of a company. Attaining these two factors is heavily dependent on having appropriate suppliers. Therefore, selecting appropriate suppliers can increase the competitiveness of a business.
The main cost of a product is mostly dependent on the cost of raw material and component parts in most industries (Ghodsypour and O’Brien 2001). Under such a condition the raw material supply and its inventory control can play a key role in the efficiency and effectiveness of a business and have a direct impact on cost reduction, profitability and its flexibility. Regarding supplier selection, there are two general situations:
Single sourcing A situation in which there is no constraint and a single supplier of an item is able to satisfy all requirements of the buyer.
Multiple sourcing In this situation there are many suppliers of a required item, but no single suitable supplier can satisfy all requirements of the buyer. Thus, the buyer must choose “an appropriate set of suitable supplies” to work with (Ghodsypour and O’Brien 1998).
Considering many factors such as variations in price, terms and conditions, quality, quantity, transportation costs and distances, etc. of each supplier, the multiple sourcing situations usually involves taking complex decisions.
While there is a paucity of research that takes into account different aspects of this complex decision situation, only a limited number of mathematical models have been proposed for such decisions. Many of the proposed models consider “net price” as the main factor, a few of them consider “the total costs of logistics”.
The present study investigates the issue of multiple sourcing and proposes mathematical models based on considering factors such as net price, transportation costs, inventory costs and shrinkage problems.
The rest of this paper is organized as follows. In “Background” section literature review is presented. In “The situation” section, the case study is described. The mathematical formulating of problem is presented in “Formulating the models” section. Data collection and parameters are described in “Parameters of model” section. Computational result is presented in “Model runs and results” section and finally, some concluding remarks are given in “Discussion and conclusion” section.
Background
Supplier selection literature may generally be divided into two areas: First, descriptive, survey type approaches and, second, quantitative modeling methods. In the first area, the researches of Dickson (1966) and Weber et al. (1991) should be mentioned as the most comprehensive ones. Dickson has identified and summarized a number of criteria that purchasing managers consider for supplier selection. In his view, the most important criteria are quality, delivery, and the performance history of the supplier. Weber et al. (1991) in a review of 74 articles on supplier selection criteria, found that the most important factor is net price, yet, they suggested that supplier selection is dependent on a multitude of factors with different priorities, depending on the particular purchasing situation.
In the second area, which is more relevant to this article, a few number of fine research attempts should be mentioned here.
The features of other reviewed quantitative researches and our research
Researches  Criterion of comparison  

Mathematical model  Multi attribute decision making (MADM)  Mathematical model and MADM approach  Deterministic model  Uncertaint model  Nonlinearity  Supplier selection and inventory control  Real case  
Our research  ✓  ✓  ✓  ✓  ✓  
Benton (1991)  ✓  ✓  ✓  ✓  
Ghodsypour and O’Brien (1997)  ✓  ✓  
Ghodsypour and O’Brien (1998)  ✓  ✓  
Ghodsypour and O’Brien (2001)  ✓  ✓  ✓  ✓  
Kumar et al. (2004)  ✓  ✓  ✓  
Basnet and Leung (2005)  ✓  ✓  ✓  
Chen et al. (2006)  ✓  ✓  
Amid et al. (2006)  ✓  ✓  
Soukhakian et al. (2007)  ✓  ✓  ✓  ✓  
Rabieh et al. (2008)  ✓  ✓  ✓  ✓  ✓  
Jafarnezhad et al. (2009)  ✓  ✓  
Farzipoor saen (2007)  ✓  ✓  
Ustun and Aktar Demirtas (2008)  ✓  ✓  ✓  ✓  
Wu and Blackhurst (2009)  ✓  ✓  ✓  
Kuo and Lin (2012)  ✓  ✓  ✓  
Mendoza and Ventura (2012)  ✓  ✓  ✓  ✓  
Rao et al. (2013)  ✓  ✓  ✓ 
The situation

ABC analysis, classifies items in terms of annual financial requirement.

VED analysis, classifies items in terms of their functional importance (Vital, Essential, Desirable).

FNS analysis, classifies items in terms of their movement speed (Fast, Normal, Slow; Nair 2002).
Iron concentrate is a supplementary material which is very similar to iron ore in appearance and should be mixed with iron ore in agglomeration process. Since it contains more Fe; its price is much higher than iron ore. However, to obtain a desired and consistent quality of the agglomeration process output, a right percentage of these two materials should be mixed together each time. The needed iron ore and concentrate for agglomeration plant is purchased from five different suppliers, none of which has the sufficient capacity to supply the whole annual requirements. Furthermore, there are some quality variations in their products and each supplier has its own supply characteristics.
The developed models in this study take into account such variations, and are formulated in a way to obtain a right combination of the raw materials in one hand, and minimize the total inventory costs in the other.
Formulating the models
Defining model parameters and variables
Before describing the model, the pertaining parameters and variables are defined as follows:
Decision variables

Q: Ordered quantity to all suppliers in each period.

Q _{ i }: Ordered quantity to ith supplier in each period.
 X _{ i }: Percentage of Q assigned to ith supplier.$$ Y_{i} = \left\{ \begin{aligned} 1\quad {\text{if}}\;X_{i} > 0 \hfill \\ 0\quad {\text{if}}\;X_{i} = 0 \hfill \\ \end{aligned} \right. $$
Parameters

D: Annual iron ore and concentrate demand (in term of tons).

T: The length of each period.

T _{ i }: Part of the period in which the lot of ith supplier (Q _{ i }) is used.

n: Number of suppliers

C _{ i }: Annual capacity of the ith supplier to supply raw material.

C _{ ti }: Transportation cost for ith supplier per unit of raw material.

r: Inventory holding cost rate.

A _{ i }: Ordering cost of ith supplier’s raw material.

P _{ i }: Selling price of ith supplier’s raw material.

h _{ i }: Percentage of moisture in the item of the ith supplier.

\( D^{\prime } \): Speed of material consumption.

\( P^{{^{\prime } }} \): Speed of receiving materials.

SS: Safety stock.

SS_{ i }: Safety stock of the ith supplier’s item.
 $$ \beta_{i} = (P_{i} + C_{ti} )\quad {\text{and}}\quad \alpha_{i} = \left( {1  \frac{{D_{i}^{\prime } }}{{P_{i}^{\prime } }}} \right) $$
The basic assumptions

Constant annual demand (D)

Infinite raw materials storage space

Stable prices over the year

Gradual receiving and consumption of raw materials

Stable safetystock levels

Stockout is not allowed.
Graphical explanation of models
The basic model
In Fig. 1, total order cycle (T) is equal to the sum of order cycles of every supplier (T _{ i }) and at the time one supplier’s inventory is used up, the next supplier’s shipment would arrive in.
In general, this model is applicable to situations where, the quality specification of receiving items from different sources is identical. And no mixing of different items is required.
The new models
Model A
The two supplier of iron concentrate in Model A (Q _{4} and Q _{5}) have to cover the inventory in turn during each ordering cycle (T), while the three suppliers of iron ore deliver their shipments simultaneously.
Model B
In Model B all suppliers send their shipments simultaneously, therefore, they are under less pressure to keep up with a tight delivery schedule.
Comparing with The basic model, at the first glance, one may expect a rise in average inventory in Models A and B, which leads to an increase in total annual carrying costs as a result. But, as we will see later, the inherent flexibility of the new models paves the way for formulating more effective ordering policies. This would prevent such increase in costs to materialize in practice.
Obviously, Models A and B were formulated for different purchasing behaviors. The formulation process of both models is very similar to each other. The slight differences actually are in formulating the carrying cost in objective function and in the quality constraints of the models. So, we skip from presenting such details here, and continue our model formulation only for Model A.
Formulating the objective function
Because of the objective function of this model is formed from inventory related costs such as the purchasing price, transportation costs, carrying and ordering costs, shrinkage cost, it is a minimizing type objective function. The shrinkage cost is mainly related to the evaporation of raw materials moisture during the agglomeration process. Since the iron ore quarries are located in both dray and wet areas of the country, the water content of their stones differ significantly, and should be taken into account as a part of the total annual purchasing cost.
Annual purchasing cost (APC)
Annual transportation cost (ATC)
Annual weight reduction cost (AWRC)
The above formula considers the fact that the evaporated moisture is actually bought and paid for its transportation.
Therefore, it is inferred that the unit cost of ith supplier’s material in agglomeration process is equal to \( (1 + h_{i} )\beta_{i} \). We will apply this formula to compute annual holding (carrying) cost.
Annual holding cost (AHC)
Referring to different behavior of inventory levels in Models A and B, especially in regard with iron concentrate, obviously, the formulation of AHC differs slightly. To save us time, we proceed with formulating AHC for Model A only.
Annual ordering cost (AOC)
The model constraints
The constraints of this model actually pertains to the buyer’s annual demand and quality of receiving materials, on one hand, and the suppliers’ allocable capacity, on the other. In the following section, we present formulation of these constraints, as they were introduced to the model:
Demand constraint
Fe quality constraint
Additionally the number \( 10^{20} \) that represented is a very large number in the model.
Capacity constraint
This constraint stems from the fact that the ith supplier can satisfy only a fraction of the annual buyer’s needs, C _{ i }, each year. Thus: \( X_{i} D \le C_{i} \).
Instead, in above constraint formulas, where ever we have X _{ i }, we can multiply it by Y _{ i }.
Fe quality constraint
Model A formulation
Model B formulation
Parameters of model
Common parameters for two successive years
i  r  α _{ i }  q _{Fei }  h _{ i }  C _{ i } 

1  0.16  0.685  60.19  0.0189  2,021,000 
2  0.16  0.594  61.56  0.0434  142,000 
3  0.16  0.837  60.41  0.0773  84,000 
4  0.16  0.282  67.26  0.0915  3,000,000 
5  0.16  0.084  68.06  0.0832  4,000,000 
Different parameters for two successive years
i  First year  Second year  

A _{ i }  β _{ i }  A _{ i }  β _{ i }  
1  10,485,422  157,682  10,083,210  122,000 
2  10,485,422  170,000  10,083,210  128,000 
3  10,485,422  198,000  10,083,210  167,500 
4  10,485,422  254,000  10,083,210  254,000 
5  10,485,422  215,000  10,083,210  215,000 
Model runs and results
Model A results
First year  Second year  

Objective function (Rials)  286,884,300,000  300,929,000,000 
X _{1}  0.8191556  0.9949174 
X _{2}  0.0681916  0 
X _{3}  0  0 
X _{4}  0  0 
X _{5}  0.1126528  0.005082592 
Y _{1}  1  1 
Y _{2}  1  0 
Y _{3}  0  0 
Y _{4}  0  0 
Y _{5}  1  1 
Q  100,844.4  66,727.33 
Q _{1}  82,640.02  66,388.18 
Q _{2}  6879.469  0 
Q _{3}  0  0 
Q _{4}  0  0 
Q _{5}  11,364.91  339.1478 
Take notice of the fact that the model has chosen three of the suppliers for the first year and only two of them in the second year. Exactly, the same results obtained when running Model B too. Obtaining the same results from Models A and B is rather exceptional, and relates only to this studied situation, and stems from the fact that both models rejected buying iron concentrate from a particular vendor. Models are based on important criteria such as cost, quality and capacity.
Comparing the model results with actual data
Results  First year  Second year 

Total cost of the Model A  286,884,300,000  300,292,000,000 
Total actual cost (Rials)  321,929,098,600  323,290,000,000 
Cost reduction (Rials)  35,044,798,600  22,998,000,000 
% of reduction  10.9  7.1 
Discussion and conclusion
Examining the results presented in Tables 4 and 5, reducing cost by 10.9 and 7.1 %, increasing company’s annual profit, attract any top manager’s attention. One might argue that real world mangers of large processing firms like Isfahan Steel Company keep purchasing from different sources to ensure a continuous and reliable stream of supplies. However, it turned out to be a very expensive way to get such assurances. The responsible managers of the said company had not been aware of the tremendous differences that a wrong suppliers selection decisions could create. Furthermore, one of the real potential values of mathematical modeling is reminding the mangers of alternative ways of doing their daily affairs.
 1.
The models determine the percentages of iron ore and iron concentrate to be mixed in agglomeration process. This is done by considering the minimum Fe contents required for specified output quality of the process, \( q_{\text{afe}} \), and other constraints, and objective function. The models recommended a mix of 88.73472 % iron ore and 11.26528 %, concentrate for the first year and 99.49174 % iron ore, 0.005082592 % concentrate for the second year.
 2.
A sensitivity analysis of the models was assumed by changing capacity and quality constraints’ parameters, the number of placed orders in a year, D/Q. The results of the sensitivity analysis revealed that even in the most pessimistic conditions, the models would result in total costs reduction.
One of the interesting findings of the sensitivity analysis of the model was that if the second iron ore supplier had no capacity limitations, and could supply the Company’s whole annual needs, a substantial costs reduction could happened. Based on this finding, we recommended the management helping that particular supplier to invest in increasing its capacity through a joint venture project.
 3.
The nonlinear assumption of these models makes them closer to the real world managerial problems. Most of the relationships in socioeconomic systems are nonlinear in nature. Yet, Model B is flexible enough to be changed to a linear model simply by assuming D/Q as constant. As a result, the new models are applicable to a fairly large area of operations management special problems.
 4.
The new models are not just a simple inventory model. As we have seen, the models can handle a multicriteria situation comprising cost and quality and help us select appropriate suppliers. They also can present a purchasing schedule to tell us when and how much to buy from each vendor. At the same time, the models can suggest an optimum consumption mix of the materials.
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