An Inventory Model for Deteriorating Commodity under Stock Dependent Selling Rate

1. An Inventory Model for Deteriorating Commodity under Stock Dependent Selling Rate by: Wahyudi SUTOPO1,2 & Senator NUR BAHAGIA1 1) Dept. of Industrial Eng., Bandung Institute of Technology, Bandung, INDONESIA 2) Dept. of Industrial Eng., University of Sebelas Maret, Surakarta, INDONESIA. Day 2, Thursday, December 4th, 2008 D2S2, R6, P3, 11.30 - 11.50 WITA

2. Some commodities, like fresh foods, vegetables, fruits and cooking spices, will be perishable or spoiled when the lifetime is over. http://blog.americanfeast.com/images/Dallas%20Farmers%20Market%2031.jpg Research Topic / The deteriorating Commodity The deteriorating commodity is defined as a commodity with decay or loss of quality marginal value that results in the decreasing usefulness from the original condition (Nahmias, 1982; Dave, 1991, and Raafat, 1991). • Firms selling goods whose quality level deteriorates over time often face difficult decisions when unsold inventory remains. • Since the leftover commodity is often perceived to be of lower quality than the new commodity, carrying it over offers the firm a second selling opportunity and also reduced selling price. • In this case, firms should choose optimal strategy as trade of order quantity from supplier and selling price down policy as impact of deterioration rate. How should “A GROSSER” manage order (Q & T) of “Deteriorating Commodity”? 01

3. Literature Review / Research Gaps Shah & Jha (1991) have started to developing inventory model with maximize profit as decision criterion,however the model do not be related to the impact of deterioration rate Aggoun et al (2001) have conducted to integrate among deterioration with inventory level in one particular market environment which is stochastic. However, they do not considering cost or price varying yet as the impact of deterioration rate. We will develop an inventory model for deteriorating commodity under stock dependent selling rate and considering selling price. The performance criterion of this model is to maximize profit by simultaneously determining the selling price. 02

4.   The Length of cycle Time The Selling Rate USER “A GROSSER” SUPPLIER Q (unit) Qs Deteriorated Overview of System Relevant  They should obtain an inventory policy to maximize profit. The problem addressed in this paper concerns with the decision of “the optimal replenishment time” for ordering an EOQ to a supplier. Assumptions: • The item is a single commodity. • Each commodity will decay of quality according to the original condition with randomize characteristics; • All stock outs (shortages) are lost and not recovered, • The excess stocks is expired and no value. Qs t1 Shortage T t2 03

5. Selling price, P, • The function of deterioration rates, • The function of demand rates, • The cost structure of the model : • (a) a fixed ordering cost,A, • (b) a holding cost, h, • (c) a shortage cost,Ck, • (d) a cost/unit, p, How we will develop a proposed-model? Total sales revenue (TR) –Total of Inventory cost (TC) [price x total demand] - [purchase cost (Ob) + ordering cost (Op) + holding cost (Os) +shortage cost (Ok)+ excess cost (Ou) ] Max. profit of sales /order T, the length of the cycle time 04

6. Model’s Components We first analyze the model of Aggoun el al (2001) as reference and then develop a new stochastic inventory model based on: periodic review system (P Model), Aggoun et al (2001), and Shah & Jha (1991). The mathematical model formulation is developed with the following steps: (I). total of inventory cost [(1). total of purchase cost and ordering cost (2). total holding cost + (3). total shortage cost and total excess cost ] (II). The inventory level [ (1). Calculate the level of inventory & (2). Calculate the total holding cost and total shortage cost ] (III). Calculate the total demand (IV). Calculate the Total profit 05

7. The probability of shortage and the mean number short can readily be calculated for the multiple reorder point policy. Since Z is a random variable of demand distribution function under the assumption that f(Z) is normal distribution, we have the total quantity shortage is , , and The Mathematical Model (1) (I). total of inventory cost (II). The inventory level “ (1). Calculate the level of inventory From Padmanabhan & Vrat (1995), the basic model with varying rate of deterioration that describes this model is given by: The solution of equation “the level of inventory”, using Linier First Order Equation Theorem (see Appendix A1), for the boundary condition I(t) = 0, is … (2) Calculate the total holding cost and total shortage cost : The probability of shortage and the mean number short can readily be calculated for the multiple reorder point policy. Since Z is a random variable of demand distribution function under the assumption that f(Z) is normal distribution, we have the total quantity shortage is To calculate the total holding cost and total shortage cost, we require to be determined how many is the total quantity shortage items. 06

8. The Mathematical Model (2) (III). Calculate the total demand The total demand, where the demand rate depends on the change in inventory level and parameter of stock dependent selling, we obtained by solve this equation: (IV). Calculate the Total profit Total profit is equal to total revenue less than total cost. Total revenue is obtained by multiplication demand per length of the cycle time to selling price. Total Revenue Total Cost 07

9. We can easily extend to the more function using three objective variables, and expressed by equation (15): Using convexity rule, we can obtained three objective variables: The necessary conditions and the sufficient condition for equation maximizing are: The Objectives Function (under 3 objective variables) 08

10. From equation (15),first we can derivative the equation to , with implies: Hence, from equation (18), we can defined the probability of stock out, , as Second, we can derivative the equation (15) to then we get to simplify following relation then Third, we can derivative the equation (15) to . The optimum value of T can be obtained by solving the equation (15). Let and Obtained three objective variables then, we can define t1 using logarithmic rule, as The derivative of the latter, according the product-difference rule, after some simplification (see Appendix A3), is 09

11. The optimum value of T * can be obtained from expression (20). The value of is always negative to satisfy the sufficient condition for maximizing. The sufficient condition for maximum value of profit is: For T > 0, expression (26) always negative. Step 1. Start with assumption that t1=T; then find expected TD by solving equation (13).Compute based on Wilson’s model. Step 2. Compute the probability of stock out, , by solving equation (19). Step 3. Compute Qs by assume that demand is normally distributed, . Step 4. Compute the total profit, , by solving equation (14). Step 5. Go to step 1. To make iteration by increasing T, substitute and compute by performing Step 2 and Step 3. If > , let then compute . If < STOP iteration with then go to Step 6. Step 6. To make iteration by decreasing T, substitute and compute by performing Step 2 and Step 3. If > , let then compute . If < STOP iteration. Hence, the uniqueness of the optimum replenishment policy, T , can be provided by choosing the best . Determine T* An evidence of T* will be maximizing value of profit: Modifying the Hadley-Within algorithm to define T* : The optimum value of T* can be obtained by solving equation (25). It requires considerably more computation than a non recursive procedure. As a consequence, we suggest slightly modifying the Hadley-Within algorithm as follows: 10

12. Determine T* - Numerical Example Find the optimum replenishment policy, here we have parameters as follows: a=10; b=0.05; A=25,000; h=500; cu=10,000;  =0.1; P=10,000; and e=2.71828. Compute the optimum replenishment policy by the proposed algorithm above: (1). Compute T0 and TD: Based on the proposed algorithm above, after some simplification, then we get: Next Step Find α, Zα, TR, Qs and TP: 11

13. Determine T* - Numerical Example Find the optimum replenishment policy, here we have parameters as follows: a=10; b=0.05; A=25,000; h=500; cu=10,000;  =0.1; P=10,000; and e=2.71828. 2). Find α, Zα, TR, Qs and TP0: • Next steps: • Go to step 1. To make iteration by increasing T, • To make iteration by decreasing T, Modifying the Hadley-Within algorithm to define T* (Step 1 s.d. Step 4 ) 12

14. Determine T* - Numerical Example Find the optimum replenishment policy, here we have parameters as follows: a=10; b=0.05; A=25,000; h=500; cu=10,000;  =0.1; P=10,000; and e=2.71828. Table 2. The optimum replenishment policy T* = 0.33 and Tp= 33,920,423. In our comparison with reference model, it is clear that this model obtains the length of the cycle time then we calculate the quantity of commodity to be ordered by grosser. 13

15. Conclusion An Inventory Model for Deteriorating Commodity underStock Dependent Selling Rate was derived by this research. In particular, the optimal replenishment time was derived. Moreover, the numerical examples were shown to evidence the usefulness of the proposed model. Firms or Grosser can maximize the profit through determine the optimal replenishment time. In reality, many of the inventory systems dealing with foods items, vegetables, and meats can be tacked by the present model; in which the optimal replenishment time can measured per day, per weeks, etc. due to the commodity’s characteristics. Besides that, the unit in stock can measured per kg, per ton, per stock keeping units (SKU) and etc. For further research, this model could be extended to other characteristics of deteriorations problems in grosser, in examples with consider capacitated stored,treatment cost to pursuing decay of quality, and transaction scheme to supplier. 14

16. Thank You (Q & A) Presenter: Wahyudi SUTOPO E-mail: wahyudisutopo@students.itb.ac.id and sutopo@uns.ac.id