A mathematical model for multi product-multi supplier-multi period inventory lot size problem (MMMILP) with supplier selection in supply chain system

Introduction

Supplier selection is defined as the process of finding the right suppliers, at the right price, at the right time, in the right quantities, and with the right quality Ayhan, (2013b). It is recorded that, 70% of total production cost is composed by the purchases of goods and services Ghodsypour & O'Brien, (1998). Hence, selecting the right supplier will result in reducing operational costs, increasing profitability and quality of products, improving competitiveness in the market and responding to customers' demands rapidly Abdollahi, Arvan, & Razmi, 2015; Onut, Kara, & Isık, (2009). In the comparative market, it is necessary having a good production planning and replenishment control through effective inventory management. The single product, multiperiod inventory lot-size problem is one of the most Common and basic problems. The present work considers an environment with multiple products-multiple periods and multiple suppliers. This paper is based on the work of Basnet and Laung (2005) [1] which developed the multiperiod inventory lot size with multiple products and multiple suppliers. Woarawichai, Kuruvit & Vashirawongpinyo (2012) has introduced storage capacity limitation with this model. In this paper a supplier selection with multi-period inventory lot size with multiple products and multiple suppliers under budget and storage capacity constraints and all unit quantity discount is developed using GA. Among various algorithms, GAs is develop by Wang, Yung, and Ip (2001), is most suitable for selecting best supplier combination and Hokey, Gengui, Mitsuo and Zhenyu (2005), suggested that GA is the best population-based heuristic algorithm, capable of generating a group of best solutions at once. Lee et al. (2013) introduced a MIP model and genetic algorithm (GA) to solve the lot sizing problem with multiple suppliers. It incorporates the incremental and all-unit quantity discounts and is applicable to determine the replenishment strategy for a manufacturer for multi periods. But it is only suitable for single items.

The paper is organized as follows. Section 1 describes the introduction and literature review and section 2 introduces assumptions and notations and initial analysis of mathematical model has developed on section 3. In section 4 we define the problem with its mathematical formulation and numerical case study has discussed in section 5. Section 6 introduces solve the numerical case study by Genetic algorithm and LINGO. Finally, concluding remarks are given in Section 7.

Assumption & Notation

In this section, we introduce following the assumptions and notations Assumption 1. Initial inventory of the first period and the inventory at the end of the last period are assumed to be zero. 2. Demand of products in each period is known over a planning horizon. 3. Transportation cost is supplier dependent, but does not depend on the variety and quantity of products involved. 4. Product needs a storage space and available total storage space is limited. 5. Shortage or backordering is not permitted. 6. Holding cost of product per period is product dependent. 7. Budget is fixed for each period. 8. All unit quantity discount is considered.

Q  , otherwise 0. This binary variable indicate that order size at period t is larger than ijlt Q then discounted prices for the ordered products.

Intermediate variable

it I = Inventory of product i, carried over from period t to period t + 1

Quantity discount with price break

As a marketing policy, Suppliers grant discounts to buyers who buy in quantity larger than of minimum acceptable order. We consider all-unit quantity discounts. An all-units quantity discount is a discount given on every unit that is purchased after the purchasing exceeds a given level (breakpoint).

As discussed above, variable specifies the fact that the order size at period t is larger than and therefore results in discounted prices for the ordered quantity and price breaks are defined as:

Initial analysis of Mathematical Model

All demands must be filled in the period in which they occur: shortage or backordering is not allowed.

There is not an order without charging an appropriate transaction cost.

𝑌 𝑗𝑡 (∑ 𝑑 𝑖𝑘

) -𝑋 𝑖𝑗𝑡 ≥ 0 ∀ 𝑖, 𝑗 & 𝑡 Each product has limited capacity

The objective function consists of three parts: the total cost (TC) of purchase cost of the products, transportation cost for the suppliers, and holding cost for remaining inventory in each period in t+1.

is binary variable 0 or 1.

Formulation of Mathematical model

The objective function of the Mathematical model

Numerical case study

We consider a scenario with three products over a planning horizon of five periods whose requirements are as follows: demands of three products over a planning horizon of five periods are given in Table 1. There are three suppliers and their prices and cost, holding cost and storage space are show in Table 2 andTable 3

Solution of Numerical case study

We applied Lingo and GAs approach to solve this numerical case study. We use LINGO 11.0 and MATLAB R2013 software and experiments are conducted on a personal computer equipped with an Intel Core 2 duo 2.00GHz, CPU speeds, and 2 GB of RAM. Lingo is Optimization Modeling Software for Linear, Nonlinear, and Integer Programming. We use this software for solution of numerical case study. We find Lingo result of this case study in the following table. and the demands are from int [10; 200]. A problem size of I; J; T indicates number of suppliers = I, number of products = J, and number of periods = T. Computation time limit is set at 120 minutes. We obtain same result as LINGO and solution time is 0.01 minutes.

Conclusion

In this paper, we present genetic algorithms (GAs) and LINGO applied to the multi-product, multi supplier and multi-period inventory lot-sizing problem with supplier selection budget constraint and all unit quantity discount with maximum storage space for the decision maker in each period is considered. The decision maker needs to determine what products to order in what quantities with which suppliers in which periods. The mathematical model is formulated as a mixed integer programming and illustrated though a numerical case study. The numerical case study is solved with LINGO and the GAs. GAs provides better solution than LINGO that are close to Optimum in a very short time.

References

1. Basnet C, Leung JMY. Inventory lot-sizing with supplier selection. Computers and Operations Research. 2005; 32:1-14.
2. Holland JH. Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor. 1975.
3. Woarawichai, Kuruvit, Vashirawongpinyo. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 2013; 7:2.
4. Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs. AI Series. Springer-Verlag, New York. 1944.
5. Gen M, Cheng R. Genetic Algorithms and Engineering Design. Wiley, New York. 1977.
6. Gen M, Cheng R. Genetic Algorithms and Engineering Optimization. Wiley, New York. 2000.
7. Davis L, The handbook of genetic algorithms, Van Nostrand Reinhold, New York. 1991.
8. Goldberg DE. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA. 1989.
9. Mogador SRM, Afsar A, Sohrabi B. Inventory lot-sizing with supplier selection using hybrid intelligent algorithm. Applied Soft Computing. 2008; 8:1523-1529.
10. Chan SH, Chung W, Wadhwa S. A hybrid genetic algorithm for production and distribution. Omega. 2005; 33:345-355.
11. Sarker R, Newton C. A genetic algorithm for solving economic lot size scheduling problem. Computers and Industrial Engineering. 2002, 42.
12. Inventory Lot-Sizing Problem with Supplier Selection under Storage Space and Budget Constraints. International Journal of Computer Science. 2011; 8:2.
13. Deb K. Multi-Objective Optimization using Evolutionary Algorithms Wiley, Chichester, 2001.
14. Rezaei J, Davoodi M. Genetic algorithm for inventory lot-sizing with supplier selection under fuzzy demand and costs. Advances in Applied Artificial Intelligence. Springer-Verlag Berlin Heidelberg. 4031, 2006.
15. Rezaei J, Davoodi M. A deterministic, multi-item inventory model with supplier selection and imperfect quality. Applied Mathematical Modelling. 2008; 32:2106-2116.