Use of fuzzy demand to obtain optimal order size through Dynamic Programming

Size: px

Start display at page:

Download "Use of fuzzy demand to obtain optimal order size through Dynamic Programming"

Meagan Nichols
5 years ago
Views:

1 Use of fuzzy demand to obtain optimal order size through Dynamic Programming Priyanka Verma a, Aurobindo Parida b, and Neha Uttam c a Assistant Professor, National Institute of Industrial Engineering (NITIE), Mumbai,400087,India b,c PGDIE Students, National Institute of Industrial Engineering (NITIE), Mumbai, ,India Abstract - In today s highly competitive business environment, accuracy of demand forecast is the key that can determine the success or failure of organizations. While availability of accurate demand information is practically impossible, the organizations may adopt the available theoretical knowledge to approach the accurate demand as closely as possible. Demand generally depends on factors like price, lead time, promotional offers, seasonal variations, after-sales service, availability of active competitors and many more. In this paper we investigate the effect of some of these factors to the demand product using fuzzy logic. We identify and fuzzify the input factors to find a fuzzy demand which is more likely to be closer to the real scenario. Demand variability bears a direct effect on a firm s decision, related to the quantity to be ordered and that carried as inventory. While demand of a product may vary at larger amplitudes, the total costs of ordering and inventory are desired to be the minimal. We apply a simple dynamic programming approach given by Wagner-Whitin algorithm to determine the order size, using the defuzzified demand. Keywords - Fuzzy logic; Demand; Dynamic Programming; Order quantity; inventory 1. Introduction and literature review Although the accurate demand forecast is practically impossible, it is desired to have demand forecast as exact as possible. Forecasting of demand is an arduous job, as it depends upon several factors namely price of the product, service level, lead time etc. The combination of fuzzy models and dynamic programming can be used to minimize costs of ordering and inventory, and to define major factors on which demand depends. The fuzzy mathematical programming is effective in planning for supply chain production. A lot of work has been done in the field of dynamic programming and fuzzy modeling separately. A fuzzy programming based formulation was given for the problem of dynamic capacitated location routing (Nadizadeh and Hosseini, 2014, ). A model is provided for fuzzy optimization if market demand is unknown (Mula, Peidro, Poler, 2006, 1-16). Fuzzy based models can be used in the field of warehouse and inventory. The problems of reorder point inventory in random fuzzy environment were studied and the solutions were suggested, taking unpredictable lead time into account (Wang, 2011, ). In procurement field also, fuzzy approach has been used for formulating a model in scheduling the procurement (Dixit, Srivastava, Chaudhuri, 2014). Retailers place their order to wholesalers in lot basis to meet the demands of the customers. During lead time, these demands are random and fuzzy in nature as they depend upon certain other variable factors (Rong, Mahapatra, Maiti, 2008, ). The lead time of the inventory is supposed to be a control variable in many works. Lead time is having broader aspect; it is the combination of time periods in making of an order, travelling, dispatching, and setting up (Tersine, 1982). There are cases where not only the demand of customers is variable but also number and their arrival is also variable. Models related to inventory are well thought-out, considering the fuzzy elements and taking sales time as deciding factor (Vijayan and Kumaran, 2009, ). Dynamic programming is a powerful tool to deal with multistage decision-making problems (Kacprzyk and Esogbue, 1996, 31-45). The dynamic lot-sizing problem was first analyzed in 1958 (Wagner and Whitin, 1958, 89-96). Since then, uncapacitated and capacitated dynamic lot-sizing problem is considered in literature in various forms. Classical dynamic lot-sizing problem is simplified by taking into account manufacture capacity constraints and time for manufacturing (Hwang, Jaruphongsa, Cetinkaya and Lee, 2010, ).The capacitated dynamic lot sizing problem is well addressed in closed-loop supply chain (Pan, Tang and Liu, 2009, ). The dynamic, capacitated and deterministic work in the field of lot sizing and scheduling is compiled (Drexl and Kimms, 1997, ). For

2 stochastic, non-stationary demand and the dynamic lot-size problem, an algorithm was presented for determining the optimal solution over the complete planning prospect. Wagner Whitin algorithm provides a tool for such type of problems (Vargas, 2009, ). Production planning problems include lot sizing problems with setups. It is often expensive to manufacture a product in each period because of these setups. If product is manufactured in larger quantities, then it results in inventory pile up and hence high inventory holding costs are incurred. Demand for any product can be affected by variety of factors. A lot of work has been done to identify these factors which include cost, seasonality, promotional offers, service level and many more. Demand of customers must be satisfied by minimizing different costs (Brahimi, 2006, 1-16). For random demand and the non-stationary costs, the dynamic lot sizing problem was formulated (Sox, 1997, ). For certainties of supply, demand and process, a model was proposed for fuzzy mathematical programming, which considers the planning of supply chain (Peidro,Mula,Poler,Verdegay, 2009, ). Smaller lead time is desirable when customer s demand is certain (Glock, 2012, 37-44). There is an upward trend in the demand of the customers due to competitive prices (Zare and Nadizadeh, 2013, 75-84). Quantity and volume discounts affect the price-sensitive demand very much. Demand for any product emerges at the retailer s end. A distribution channel, having a single-vendor and single-retailer was considered to study the effect of both the quantity and volume discounts on demand (Viswanathan and Wang, 2003, ). Seasonality is also having considerable effect on customer demand for many products (Ehrenthal, Honhon and Woensel, 2014, ). The effect of price and inventory age on demand is studied for maximizing profits. The quantity to be ordered and replenishment period are the important criteria for optimization (Avinadav, Herbon and Spiege, 2014, ). Joint economic lot size problems can also be addressed by fuzzy mathematical programming. It solves the problem from the end of both seller and buyer by providing an efficient algorithm (Lam and Wong, 1996, ). Sometimes suppliers reduce the price temporarily for operational reasons. This promotion is planned for a certain period of time (Abad, 2003, 63-74). In this paper, there are 6 sections. Section 1 gives the introduction of the paper. In section 2, theoretical approach is discussed in brief. Section 3 consists of description of model for fuzzification of demand. Section 4 describes the problem and methodology. Section 5 contains applications of the developed model. Section 6 concludes the paper with future scope. 2. Theoretical approach In this paper, we have considered the merging of two approaches to give the solutions of a real life problem. These approaches are: Fuzzy logic approach Fuzzy logic is different from the crisp logic which takes absolute values either 0 or 1, as it can take any value in the range of 0-1. Fuzzy set was the first introduced in 1965 (Zadeh, 1965, ). Fuzzy logic is the extension of fuzzy set theory and it has gained much popularity since it can address the problems related to imprecision and uncertainty, depending on the cognitive human processes. It connects the input with the output by the use of IF-THEN rules. Fuzzy logic inference system takes input in fuzzified form and gives output in the fuzzified form which can further be defuzzified to get an absolute value, which is the input for Wagner-Whitin algorithm in this paper. Dynamic programming and Wagner-Whitin algorithm approach The problems related to sequential and multi variable problems can be solved by dynamic programming. Wagner- Whitin approach can be used to decide the lot size when demand is deterministic, change with time and all the assumptions related to economic order quantity system are true. The starting point in Wagner-Whitin method is first week which continues to N th week. If optimal lot size is desired to obtain, then there are two ways: First, inventory carried from period t to t+1 is zero or second, quantity to be produced in t+1 period is zero. Depending on the demand, the production is carried out only in that week, in which holding inventory is zero. Initiating from the first week, there are the N number of production options for N number of demands which can be d 1, d 1 +d 2, d 1 +d 2 +d 3, d 1 +d 2 +d 3 +d 4 to d 1 +d 2 +d 3 +d 4 +.+d N (d 1, d 2, d 3,..d N are the demands for first, second, third Nth week). For example, if there are 4 weeks and production is possible in all the 4 weeks, then to satisfy the demand of first week, production is possible only in first week. For satisfying the demand of second week, there are two options: Produce the total quantity of product equivalent to the demands of both first and second week in the first

3 week itself or produce individual quantities equivalent to the demands of both weeks in the first week and in the second week separately and same is true for third and fourth week also. Selection of the option depends upon the minimum cost. The performance of Wagner-Whitin approach can be compared to the performance of economic order quantity, periodic order quantity and lot for lot methods. 3. Model for fuzzifying the demand In fuzzy logic, a membership function shows the degree of truth and can be defined for any fuzzy set Z as :Z [0,1]; Z is the universe of discourse. Membership functions can be triangular, normal or gaussian, trapezoidal, sigmoidal and many others. We have assumed the membership functions of all the inputs and outputs used in this paper as gaussian. The membership function for gaussian distribution can be written as: i(z) = (1/σ i (2π)e (Z i Z i )/2σ 2 i ; i= 1,2,3,4 n (1) Where σ & Z are standard deviation and mean of range for which membership function is defined. Both vary with inputs such as price, service level, lead time, promotional offer and seasonal factor as well as for output which is demand. Also, demand is the function of price, service level, lead time, promotional offer, after sale service, and seasonal factor, which is shown below: D i = f (P i, SL i, LT i, PO i, SF i ) (2) Where P = Price, SL = service level, LT = Lead time, PO = Promotional offer, SF = Seasonal factor and D = Demand. The model uses the following equation to defuzzify the demand. The equation for centroid method can be shown as: Z= z z zdz z dz The assumptions related to the fuzzy logic approach, considered in this paper are: 1. All the fuzzy function is assumed to follow Gaussian distribution. 2. Centroid method is considered for defuzzification of the demand. 3. If two membership functions share a common region, the minimum of two is taken for developing the model. 4. In applications, demand depends only upon the number of input factors considered. 5. Fuzzy logic based model for the estimation of demand is developed using Mamdani system. (3) 4. Description of the problem and methodology The estimation of demand is cumbersome and requires a lot of efforts. If demand is constant for a period of time, then determination of lot size to be ordered by the company is easy. But in most of the industrial lot sizing problems, demand fluctuates with time. Also, the demand depends upon number of factors; it is required to have information of those factors. There is a need to have a model which can accommodate the uncertain conditions of demand or factors affecting the demand. Furthermore, if demand is estimated, then also planning of lot size for production is complex in nature. Additional methods are required to proceed further once the demand is projected. The methodology used for the development of fuzzy model and application of Wagner-Whitin method to determine the lot size is shown in the fig.1. The inputs (price change, promotional offer, seasonal factor, service level and lead time) are fuzzified according to the rules, to get a fuzzy demand, which is defuzzified by centroid method for the further process. The rules for the development of fuzzy model are taken as IF-THEN type: IF A is healthy and B is light THEN C can run, where A is food of C, B is weight of C and C is a person. The defuzzified demand is taken as input for Wagner-Whitin method to obtain the optimum lot size. The various costs (procurement, production, carrying and set-up costs) are assumed for the Wagner-Whitin dynamic programming.

4 5. Applications: Figure 1. Methodology for obtaining fuzzified demand and optimal lot size In order to illustrate the application of the above proposed model, following cases in real life scenario are shown: Case Example 1: In 1 st example, the industry is B2B, and an automobile part is the product. Input factors which were considered for estimation of fuzzy demand were lead time & seasonal factor. Rules were defined on the basis of experience of people working in the same industry. The fuzzy inputs and output of developed fuzzy model are shown in fig.2. Figure 2. Fuzzy model of B2B case Defuzzification of the demand is done through centroid method; fig.3 shows the demand after defuzzification and fig. 4 shows the three dimensional representation of demand function.

Figure 3. Output of defuzzified demand (Matlab output) Figure 4.

Whitin algorithm. All the assumptions of Wagner-Whitin algorithm are considered.

Discrete demand was estimated by keeping lead time constant as 7 days and by varying seasonal factor over the period of 10 months, using fuzzy model.

5 Figure 3. Output of defuzzified demand (Matlab output) Figure 4. 3D graphical output of demand function (Matlab output) The estimated defuzzified demand was further used for the calculation of lot size for production by using Wagner- Whitin algorithm. All the assumptions of Wagner-Whitin algorithm are considered. It was also assumed that cost of production, set-up cost and carrying cost remains constant throughout planning horizon. Discrete demand was estimated by keeping lead time constant as 7 days and by varying seasonal factor over the period of 10 months, using fuzzy model. Set-up cost & holding cost were assumed to be $100 and $1 respectively. Total production cost (Set-up cost + Holding cost) for optimal lot size of production was obtained by dynamic programming and compared to other methods, which is shown in fig.5. The Wagner-Whitin method saves cost by 25%, 8% and 6% from economic order quantity, lot for lot and periodic order quantity methods respectively. Figure 5. Total production cost for case 1

Case Example 2: In 2 nd example, the industry is B2C, and the product is health drink.

lead time, service level and promotional offer.

Fuzzy model of B2C case In this case also, defuzzification of demand is done by centroid method.

6 Case Example 2: In 2 nd example, the industry is B2C, and the product is health drink. Input factors, considered for estimation of fuzzy demand were change in price, seasonal factor, lead time, service level and promotional offer. The fuzzy model of five input factors and fuzzy demand is shown in fig.7. Figure7. Fuzzy model of B2C case In this case also, defuzzification of demand is done by centroid method. The defuzzified demand along with rules is given in fig.8. Figure 8. Output of defuzzified demand (Matlab output) Figure 9. 3D graphical output of demand function (Matlab output)

7 Defuzzified demand was required for calculation of lot size for minimum production cost. It was calculated over the period of 12 months by varying seasonal factor and keeping the other factors constant in that duration. Cost of set-up & carrying were assumed to be $150 & $25 respectively. With discrete demand and assumed cost, Wagner-Whitin cost was calculated. The comparison of cost of production obtained from different methods for the estimation of lot size is shown in fig.10. In this case, the Wagner-Whitin method saves cost by 71%, 11% and 10% from economic order quantity, lot for lot and periodic order quantity methods respectively. Figure 10. Total production cost for case 2 6. Conclusion and future scope The Wagner-Whitin dynamic programming is the effective model for estimation of optimum lot size for production. But the demand, on which it depends and estimates lot size, further depends upon various factors (seasonal factor, promotional offer, change in price etc). The developed fuzzy model can improve the estimation of demand (or reduce error in forecasting). Combination of both, that is, the estimation of optimum lot size by incorporating fuzzy model and Wagner-Whitin algorithm dynamic programming will improve planning (or reduce waste). The fuzzy model developed can be used for similar types of problems to a great extent. In this paper all the factors were assumed to follow gaussian distribution. Nature of distribution of each factor can be different and may affect the result. Also the cost of production, set-up, and carrying cost vary in real life, which can be further included to make it realistic. References Abad P.L., Optimal price and lot size when the supplier offers a temporary price reduction over an interval. Computers & Operations Research, 30, 63 74, Avinadav, T., A. Herbon, U. Spiegel, Optimal ordering and pricing policy for demand functions that are separable into price and inventory age. Int. J.ProductionEconomics, 155, , Brahimi et. al., Single item lot sizing problems, European Journal of Operational Research, 168, 1 16, Dixit, Vijaya, Rajiv K. Srivastava, Atanu Chaudhuri, Procurement scheduling for complex projects with fuzzy activity durations and lead times. Computers & Industrial Engineering, 2014,

8 Drexl, A., and A. Kimms, Lot sizing and scheduling - Survey and extensions. European Journal of Operational Research, 99, , Ehrenthal, J. C. F., D. Honhon,,T. Van Woensel, Demand seasonality in retail inventory management. European Journal of Operational Research, 238, , Glock, C. H. Lead time reduction strategies in a single-vendor single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. Int. J. Production Economics, 136, Hwang, H. C., W. Jaruphongsa, S. Çetinkaya, C.Y. Lee. Capacitated dynamic lot-sizing problem with delivery/production time windows. Operations Research Letters, 38, , Kacprzyk, Janusz, and Augustine O. Esogbue, Fuzzy dynamic programming: Main developments and applications. Fuzzy Sets and Systems, 81, 31-45, Lam, S. Monika, and Danny S. Wong, A fuzzy mathematical model for the joint economic lot size problem with multiple price breaks. European Journal of Operational Research, 95, , Mula, J., D. Peidro, R. Poler, The effectiveness of a fuzzy mathematical programming approach for supply chain production planning with fuzzy demand. Int. J. Production Economics, 128, , Nadizadeh, Ali, and Hasan Hosseini, Nasab, Solving the dynamic capacitated location-routing problem with fuzzy demands by hybrid heuristic algorithm. European Journal of Operational Research 238, , Pan, Zhendong., Jiafu Tang, Ou Liu, Capacitated dynamic lot sizing problems in closed-loop supply chain. European Journal of Operational Research, 198, , Peidro, David, Josefa Mula, Raul Poler, Jose-Luis Verdegay, Fuzzy optimization for supply chain planning under supply, demand and process uncertainties. Fuzzy Sets and Systems 160, , Rong, M., N.K. Mahapatra, M. Maiti, A multi-objective wholesaler retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number. Applied Mathematical Modelling, 32, , Sox, Charles. R., Dynamic lot sizing with random demand and non-stationary costs. Operations Research Letters, 20, , Tersine, R.J. Principles of Inventory and Materials Management, North-Holland, New York Vargas, Vicente, An optimal solution for the stochastic version of the Wagner Whitin dynamic lot-size model. European Journal of Operational Research, 198, , Vijayan, T., and M. Kumaran, M, Fuzzy economic order time models with random demand. International Journal of Approximate Reasoning, 50, , Viswanathan, S., and Qinan Wang, Discount pricing decisions in distribution channels with price-sensitive demand. European Journal of Operational Research, 149, , Wagner, Harvey M., and Thomson M. Whitin, Dynamic version of the economic lot size model. Management Science, 5 (1), 89 96, Wang, Xiaobin, Continuous review inventory model with variable lead time in a fuzzy random environment. Expert Systems with Applications, 38, , Zadeh, L. Fuzzy sets. Information and Control, 8, , Zare, Mehrjerdi Yahia, and Ali Nadizadeh, Using greedy clustering method to solve capacitated location-routing problem with fuzzy demands. European Journal of Operational Research, 229(1), 75 84, 2013.

Sustainable sequencing of N jobs on one machine: a fuzzy approach

44 Int. J. Services and Operations Management, Vol. 15, No. 1, 2013 Sustainable sequencing of N jobs on one machine: a fuzzy approach Sanjoy Kumar Paul Department of Industrial and Production Engineering,