PME I J http://polipapers.upv.es/index.php/IJPME International Journal of Production Management and Engineering https://doi.org/10.4995/ijpme.2019.10745 Received 2018-10-13 Accepted: 2019-01-20 Using system simulation to search for the optimal multi-ordering policy for perishable goods Huang, Y.C.a1, Chang, X.Y.a2, and Ding, Y.A.a3 aDepartment of Industrial Management, National Pingtung University of Science and Technology. No.1, Xuefu Rd., Neipu Township, Pingtung County 912, Taiwan (R.O.C.) a1 ychuang@mail.npust.edu.tw, a2 pink4141114@gmail.com, a3 yian0213@gmail.com Abstract: This paper explores the possibility that perishable goods can be ordered several times in a single period after considering the cost of Marginal contribution, Marginal loss, Shortage, and Purchasing under stochastic demand. In order to determine the optimal ordering quantity to improve the traditional newsvendor and maximize the total expected profits, and then sensitivity analysis is taken to realize the influence of the parameters on total expected profits and decision variables respectively. In addition, this paper designed a multi-order computerized system with Monte Carlo method to solve the optimal solution under stochastic demand. Based on numerical examples, this paper verified the feasibility and efficiency of the proposed model. Finally, several specific conclusions are drawn for practical applications and future studies. Key words: Perishable goods, Single-period, Multi-ordering, Newsvendor model, Monte Carlo method. 1. Introduction There are many goods which are shorter period than the durable commodities in reality. As time goes by, the value of the goods will rapidly decline. This type of goods is very common in our life such as newspapers, magazines, fresh food, and milk, and so on. Before the start of the sales cycle, decision maker often needs to determine how many the goods to be ordered for the entire cycle, and no more ordering before the expiry date. This type of goods will be discussed in newsvendor model. In addition, these products are called as perishable goods or seasonal goods according to their characteristics. There were many kinds of research on newsvendor problems in academic community; they discussed the inventory method, demand situations, single or twice orders and so on. In the past literature, several scholars have discussed the second order in a single period. If the first ordering quantity is sold out, there has time to the end of the period, then determine the second order should be taken or not, and proved that in some cases the expected profit of order twice is higher than order once. However, past literature did not discuss the single-period and multi-order situations, although the expiry period of perishable goods is very short, but if only ordered once before the sales cycle, and do not consider the situation that all the perishable goods was sold out before the To cite this article: Huang, Y.C., Chang, X.Y., Ding, Y.A. (2019). Using system simulation to search for the optimal multi-ordering policy for perishable goods. International Journal of Production Management and Engineering, 7(1), 49-62. https://doi.org/10.4995/ijpme.2019.10745 Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International 49 http://creativecommons.org/licenses/by-nc-nd/4.0/ expiry period, then it may be not an optimal ordering strategy, eventually. This paper is to improve the traditional newsvendor model, and to explore whether the perishable goods should be ordered more than one time in a single period, to achieve the goal of maximizing the total expected profit. The aim of this paper is to determine whether a perishable commodity should be ordered more than once in order to maximize the total expected profit. The purposes of the paper are as follows: 1. To establish a stochastic model under single- period and multi-ordering. 2. Proposed an optimal ordering strategy for single- period and multi-ordering. 3. Proved the total expected profit of multi-ordering is better than the single order under stochastic demand. 2. Literature review Perishable goods were ordered in the case of uncertain demand to meet the needs of the sales cycle. Therefore, the order should be carefully determined when ordering. There were many kinds of research on newsvendor problems which discussed the inventory method, demand situations, single, and twice orders. This paper will discuss the optimal ordering strategy for perishable commodities under single period. 2.1. Order once in a single period Dian (1990) derived an algorithm to determine a sequence of supply quantities which minimizes total costs of over- and undersupply in the most adverse demand conditions. Fujiwara et al. (1997) considered the problem of ordering and issuing policies arising in controlling finite-life-time fresh- meat-carcass inventories in supermarkets. They developed a mathematical model describing actual operations and then simplify the sub-product run out period so that optimal ordering and issuing policies were easily established. The newsvendor problem is also called Single-Period Problem (SPP). Khouja (1999) built taxonomy of the SPP literature and delineated the contribution of the different SPP extensions. Khouja (2000) extended the SPP to the case in which demand was price- dependent and multiple discounts with prices under the control of the newsvendor were used to sell excess inventory. They developed two algorithms for determining the optimal number of discounts under fixed discounting cost for a given ordering quantity and realization of demand. Chun (2003) assumed that the customer’s demand was represented as a negative binomial distribution, and determined the optimal product price based on the demand rate, buyers’ preferences, and length of the sales period. For the case where the seller can divide the sales period into several short periods, finally proposed a multi-period pricing model. Dye and Ouyang (2005) extended Padmanabhan and Vrat’s model (1995) by proposing a time- proportional backlogging rate to make the theory more applicable in practice. Alfares and Elmorra (2005) extended the analysis of the distribution-free newsvendor problem to the case when shortage cost was taken into consideration. A model was presented for determining both an optimal ordering quantity and a lower bound on the profit under the worst possible distribution of the demand. Chen and Chen (2009) presented a newsvendor model with a simple reservation arrangement by introducing the willingness rate, represented as the function of the discount rate, into the models. And mathematical models were developed, and the solution procedure was derived for determining the optimal discount rate and the optimal ordering quantity. In addition, some scholars put forward that the idea of demand forecast updated, which focus only on the trade-off between exact requirements and additional costs, and often assuming that the supplier’s capabilities were unrestricted, but in real life is not the case. Zheng et al. (2016) investigated an extension of the newsvendor model with demand forecast updating under supply constraints. In studying the manufacturer-related effects, two supply modes are investigated: supply mode A, which has a limited ordering time scale, and supply mode B, which has a decreasing maximum ordering quantity. A comparison of the different supply scenarios demonstrated the negative effects of increased purchasing cost and ordering time and quantity restrictions when demand forecast updating implemented. 2.2. Order twice in a single period Gallego and Moon (1993) extended the analysis to the recourse case, where there was a second purchasing Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 50 http://creativecommons.org/licenses/by-nc-nd/4.0/ opportunity; to the fixed ordering cost case, where a fixed cost was charged for placing an order; to the case of random yields; and to the multi-item case, where multiple items compete for a scarce resource. Azoury and Miller (1984) used the concept of flexibility it was anticipated that the quantity ordered under the non-Bayesian policy would be greater than or equal to that under a Bayesian policy. This result was established for the n-period non depletive inventory model. Lau and Lau (1998) considered the very common situation in which a single-period newsvendor type product may be ordered twice during a period. They extended the basic model to consider a non-negligible set-up cost for the second order; it served as an illustration of how one might want to extend their basic two-order model to handle a large number of different combinations of additional factors such as the second-order’s delivery delay time and price differential. Chung and James (2001) extended the classic newsvendor problem by introducing reactive production. Production occurs in two stages, an anticipatory stage and a reactive stage. Their model reduces to a single-period model with piecewise- linear convex costs. They obtain an analogue of the well-known critical fractile formula of the classic newsvendor model. Pando et al. (2013) presented of the newsvendor problem where an emergency lot can be ordered to provide for a certain fraction of shortage. This fraction was described by a general backorder rate function which is non-increasing with respect to the unsatisfied demand. An exponential distribution for the demand during the selling season was assumed. An expression was obtained in a closed form for the optimal lot size and the maximum expected profit. 2.3. Literature review In this paper, we explored the single-period and multi-order strategy for perishable goods. The relevant literature was summarized and shown in Table 1. 3. Construction of the mathematical model This paper proposed the concept of single-period and multi-order strategy for perishable goods, then developed the total expected profits model to determine the optimal ordering quantity and quantity of order. Furthermore, we will prove that the multi- order is superior to the single-order for perishable goods. We will introduce the simulation method and program flow chart in Section 3.7. 3.1. The assumptions of this paper 1. The model assumes no lead time. Each ordering must pay the same ordering cost. If the goods sold out in this period, then the subsequent ordering quantity can be delivered before the start of next period. 2. The demand is a random variable. The marginal contribution, marginal loss, shortage cost, salvage value, and delivery costs are all known and fixed. 3. The sales quantity of each period can be known by the POS system, and the distribution of demand can be reasonably estimated by historical data and goodness-of-fit test. 4. Do not consider the quantity discount and restrictions of storage space. 3.2. Definitions of symbols i: The period, i = 1,2,3…n n: The number of time intervals in expired period j: The jth ordering Xi: The demand quantity of i th time interval (Xi is a random variable) Yj: The total demand from j th ordering to the end of sales cycle (Yj is a random variable). Y Xj i i K n j = = | Coj: Ordering cost of j th ordering CP: Purchase cost per unit of perishable goods Price: Price per unit of perishable goods S: Salvage value per unit of perishable goods CS: Shortage cost per unit of perishable goods MP: Marginal contribution, MP=Price–CP, where Price >CP ML: Marginal loss, ML=CP–S , where CP > S Qj: The ordering quantity of j th ordering (Qj is a decision variable) Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Using system simulation to search for the optimal multi-ordering policy for perishable goods 51 http://creativecommons.org/licenses/by-nc-nd/4.0/ fj ( yj): Probability density function of Yj Fj ( yj): Cumulative distribution function of Yj Kj : The ordering time point of j th ordering Mπj (Qj): Marginal profit under ordering quantity Qj and jth ordering MRj (Qj): Marginal revenue under ordering quantity Qj and j th ordering MCj (Qj): Marginal cost under ordering quantity Qj and jth ordering Tπj (Qj): Total expected profit under ordering quantity Qj and j th ordering Tπ1 (Q1): Total expected profit under ordering quan- tity Qj and 1 st ordering TπM (Q1,Q2,…,QJ): Sum of total expected profit under ordering quantity (Q1,Q2,…,QJ) After symbols definition, the concept of multiple orders in single period for perishable goods can be shown in Figure 1. The expiry period can be divided into n, and X1,X2,X3,…,Xn respectively represents the demand quantity at period 1, 2, 3…n. Only the 1st ordering time point is sure, the other ordering time points K1,K2,… are uncertain. If the demand of the entire cycle can be satisfied by the first ordering quantity, then K2 will not happen. If the initial ordering quantity cannot satisfy the demand of the entire cycle, and reordering has a positive profit, then 2nd ordering will be taken and the time point is K2. The others are reasoned by analogy. Figure 1. The schematic of single-period and multi-order structure for perishable goods. Based on the symbol definition and Figure 1, the demand of Yj is Y Xj i i K n j = = | 3.3. Ordering strategy This section describes the mathematical model of the ordering strategy. 3.3.1. Ordering strategy Assuming that the demand of ith period is Xi, the first ordering quantity is Q1, and the total demand of whole period is Y1, so Y Xi i n 1 1 = = | The total expected profit Tπ1 (Q1) under single order strategy is shown in Equation (1): T Q y MP Q y ML f y dy Q MP y Q C f y dy Co Q SQ 1 1 1 1 10 1 1 1 1 1 1 1 1 1 1 ∞ 1 1 $ $ $ $ $ $ r = - - + - - - ^ ^ ^ ^ ^h h h h h 6 6 @ @ # # (1) Table 1. The comparison between literature and this paper. Project Author Shortage cost Salvage value Order cost Total expected profit maximization Twice orders Multi- orders System Simulation Azoury and Miller (1984)   Dian (1990)   Gallego and Moon (1993)    Fujiwara et al. (1997)   Lau, H. and H. Lau (1998)  Khouja (2000)    Chung and James (2001)   Dye and Ouyang (2005)  Pando et al. (2013)      This paper        Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 52 http://creativecommons.org/licenses/by-nc-nd/4.0/ Tπ1 (Q1) can be taken a first order derivative with respect to Q1 and set the result be equal to zero to obtain the optimal ordering quantity Q1 that maximizes the total expected profit, as shown in Equation (2): Q T Q ML f y dy MP C f y dy F Q MP ML C MP C 0 ∂ ∂ Q s Q s s 1 1 1 0 1 1 1 1 1 1 1 1 ∞ 1 1 ( $ $ r = + + = = + + + ^ ^ ^ ^ ^h h h h h# # (2) Q F MP ML C MP C s s 1 1 1` = + + +- c m (3) After finding out the optimal ordering quantity and if Tπ1 (Q1) < 0, it means the expected profit is negative, then the decision maker will not make an order to purchase the perishable goods; Conversely, if Tπ1 (Q1) ≥ 0, it means the expected profit is positive, then the decision maker will make an order to purchase the perishable goods with the optimal ordering quantity (Q1). The second order derivative of the total expected profit Tπ1 (Q1) with respect to Q1 to verify whether the Tπ1 (Q1) is a concave function of Q1: Q Q T Q Q ML f y dy MP C f y dy ML f Q MP C f Q ML MP C f Q 0 ∂ ∂ ∂ ∂ ∂ ∂ < Q sQ s s 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 ∞1 1 $ $ $ $ $ r = + + = - - + = - + +^ e f ^ ^ ^ ^ ^ ^ ^ ^ h h o h h h h h h hp# # (4) From equation (4) know that Q T Q 0∂ ∂ <2 2 1 1r ^ h , so the Tπ1 (Q1) is the concave function of Q1, Therefore, Q F MP ML C MP C s s 1 1 1= + + +- c m is an optimal ordering quantity, and can make Tπ1 (Q1) have a maximum value. 3.3.2. The construction of multi-order in single period problem The multi-order means that the decision maker may deliver one or more orders during the expiry period, each ordering quantity can be denoted by Q1,Q2,…,QJ, respectively. The total expected profit is expressed by TπM (Q1,Q2,…,QJ), so we have , , ,T Q Q Q T Q M J j j j J 1 2 1 gr r= = ^ ^h h| The total expected profit of jth order, Tπj (Qj), can be expressed as shown in Equation(5): T Q y MP Q y ML f y dy Q MP y Q C f y dy Co j j j j j Q j j j j j j SQ j j j j 0 ∞ j j $ $ $ $ $ $ r = - - + - - - ^ ^ ^ ^ ^h h h h h 7 7 A A # # (5) where ~ , , , , , . Y F y and Y X Q F MP ML C MP C j J1 2 j j j j ii K n j j s s1 j ` g = = + + + = = - ^ c h m | (6) |J Max j T Q 0 >j jr= ^ h# - 3.3.3. Compare single-order and multi-order Multi-order in single period will occur when the first ordering quantity was sold out and second order before the end of the sales cycle is still profitable. It can be inferred that the total expected profit of multi- order will be greater than the single-order, it means TπM (Q1,Q2,…,QJ)≥ Tπ1 (Q1). The proof was shown in Proposition 1. Proposition 1. TπM (Q1,Q2,…,QJ)≥ Tπ1 (Q1) Proof: , , ,T Q Q Q T Q M J j j j J 1 2 1 a gr r= = ^ ^h h| and , , , ,T Q j J0 1 2 >j j 6 gr =^ h , , ,T Q Q Q T Q T Q T Q > M J j j j J 1 2 1 1 2 1 1 gr r r r = + = ^ ^ ^ ^ h h h h | so TπM (Q1,Q2,…,QJ)≥ Tπ1 (Q1) Q.E.D. (7) 3.3.4. Without considering the shortage cost When we do not consider the shortage cost, the total expected profit of perishable goods in 1st ordering Tπ1 (Q1) was shown in Equation (8): T Q y MP Q y ML f y dy Q MP f y dy Co Q Q 1 1 1 1 10 1 1 1 1 1 1 1 1 ∞ 1 1 $ $ $ $ $ r = - - + -^ ^ ^ ^h h h h6 6 @ @# # (8) Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Using system simulation to search for the optimal multi-ordering policy for perishable goods 53 http://creativecommons.org/licenses/by-nc-nd/4.0/ Based on first order condition (so called FOC), we have Equation (9) and (10) as follows: Q T Q ML f y dy MP C f y dy F Q MP ML MP 0 ∂ ∂ Q sQ 1 1 1 0 1 1 1 1 1 1 1 1 ∞ 1 1 ( $ $ r = + + = = + ^ ^ ^ ^ ^h h h h h# # (9) Q F MP ML MP 1 1 1` = + - a k (10) When we consider the shortage cost, the optimal ordering quantity Q1 is F MP ML C MP C s s 1 1 + + +- c m ; whereas, when we do not consider the shortage cost, the optimal ordering quantity Q1 is F MP ML MP 1 1 + - a k . If CS=0, then MP ML C MP C MP ML MP s s + + + = + , so F MP ML C MP C F MP ML MP s s 1 1 1 1 + + + = + - -c am k . If CS > 0, then MP ML C MP C MP ML MP> s s + + + + , so .F MP ML C MP C F MP ML MP≥ s s 1 1 1 1 + + + + - -c am k It means when the shortage cost exists, the optimal ordering quantity will increase. 3.4. Goodness-of-fit test This paper collected sales data, and based on the historical data at different periods to take the goodness- of-fit test to estimate the demand distribution and its population parameters. The Kolmogorov-Smirnov test (K-S test) is a goodness-of-fit test. The test is a nonparametric statistical method to test the sampling data whether follows a specific theoretical distribution, such as uniform distribution, normal distribution, exponential distribution and so on. The testing steps are as follows: Step 1: building a hypothesis Suppose that the actual distribution function of random variable X is F(x), and the specific theoretical distribution function is given as F0(x). The hypothesis of this test is: 1. Null hypothesis H0: X ~ F0(x) 2. Alternative hypothesis H1: ~X0 (H1 is the supplementary set of H0 ) Step 2: calculating the testing statistic Let x1,x2,…,xn be a set of random sample taken from the population distribution F0(x), and let F(x) be the actual distribution function, the testing statistic ,D Max F x F x x 0 6= -^ ^h h , the testing statistic D is the maximum absolute difference between the actual distribution function F(x) and the specific theoretical distribution function F0(x). Step 3: rejection region If D>dα, then reject H0, where dα is a critical value of D. After goodness-of-fit test to estimate the demand distribution of and then construct the mathematical model to search for the optimal ordering strategy. 3.5. The additive property of distributions Assuming that the demand distribution for each period can be estimated from past sales data through by goodness-of-fit test, and then we need to discuss whether the distribution has the property of additive. Let, Y X j ii K n j = =| where ~ ,X F x and X i i i i^ h ╨ ,X i j ≠j 6 And ( ) E Y E X E X j ii K n ii K n ii K n j j j n = = = = = = ^ ah k| | | (11) V Y V X V X j ii K n ii K n ii K n 2 j j j v = = = = = = ^ ^ ah h k| | | (12) If Xi follows normal distribution, it can be denoted as ~ ,X Ni i i 2n v_ i , and Y Xj ii K n j = =| ,then ~ ,Y N j ii K ii K nn 2 j j n v= =a k|| (13) The common distributions are summarized in Table 2 to justify their additive property. 3.5.1. The discussion on ordering quantity Under the premise of additive property or Xii K n j= | follows the central limit theorem, and if MP, ML and Cs are known, then Qj ≥ Qj+1, it means Q1 ≥ Q2 ≥… ≥QJ. The proof is shown in Proposition 2. Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 54 http://creativecommons.org/licenses/by-nc-nd/4.0/ Proposition 2. : If MP, ML and Cs are known and fixed, then Q1 ≥ Q2 ≥…≥ QJ . Proof: Given Y Xj ii K n j = =| and Xi ≥0 so Y Y Y≥ ≥ ≥ ≥ ≥ ≥ J Y Y Y1 2 J1 2(g gn n n and V Y V Y V Y≥ ≥ ≥ J1 2 g^ ^ ^h h h it has , and, as shown in Figure 2: Therefore QJ≥ QJ+1. By the same way, we can prove that Q1 ≥ Q2 ≥…≥ QJ. Q.E.D. Figure 2. Schematic of F C F C≥j j1 11- +-^ ^h h . 3.5.2. The discussion on total expected profit in each period If Xi is additive, and MP, ML and Cs are known, and Co1=Co2=…=CoJ, then Tπ1(Q1) ≥ Tπ2(Q2) ≥ …≥ TπJ(QJ) ≥ 0. The proof is shown in Proposition 3. Proposition 3 : If , MP, ML and Cs are known and Co1=Co2=…=CoJ, then Tπ1(Q1) ≥ Tπ2(Q2) ≥ …≥ TπJ(QJ) ≥ 0. Proof: F yj ja ^ h is an increasing function of yj, and if y1>y2, then Fj(y1)≥Fj(y2). If Tπj(Qj) ≥ 0, then Mπj(Qj) ≥ 0, where MRj(Qj) - MCj(Qj) so MRj(Qj) ≥ MCj(Qj), and MRj(Qj) =MP·P(Yj≥Qj); MCj(Qj)= ML·P(Yj < Qj) MP P Y Q ML P Y Q≥ ≥ j j j j1- - % % +_ _ _ _ii ii7 A . In other words, P Y y P Y y> > >j j j j1 % % +_ _i i . So T Q T Q≥j j j j1 1r r + +^ ^h h . Q.E.D. 3.6. Sensitivity analysis The sensitivity analysis is taken to realize the influ- ences of the system parameters on total expected profit are shown as follows. 1. The influence of marginal contribution (MP) on total expected profit (Tπj(Qj)) has a same changing direction. MP T Q y f y dy Q f y dy 0 ∂ ∂ > j j j j j j Q j j j jQ0 ∞j j $ $ r = + ^ ^ ^ h h h# # (14) 2 The influence of marginal loss (ML) on total expected profit (Tπj (Qj)) has an opposite changing direction. ML T Q Q y f y dy 0∂ ∂ < j j j j j j j Q 0 j $ r = - - ^ ^ ^ h h h# (15) 3 The influence of shortage cost (Cs) on total expected profit (Tπj (Qj)) has an opposite changing direction. C T Q y Q f y dy 0∂ ∂ < s j j j j j j jQ ∞ j $ r = - - ^ ^ ^ h h h# (16) 4 The influence of delivery cost (Coj) on total expected profit (Tπj (Qj)) has an opposite changing direction. Co T Q 1 0∂ ∂ < j j jr = - ^ h (17) 3.7. System simulation The Monte Carlo simulation will be applied and introduced as follows. 3.7.1. Monte Carlo simulation Monte Carlo simulation is a simulation; it can generate random numbers that follow a specific probability distribution. Based on the random numbers and given mathematical model to find out the optimal solution that maximizes the total expected profit or minimizes the total expected cost. In this study, the Monte Carlo method was applied to simulate the demand of each period. After collecting the past sales data and building the demand distribution of each period by goodness-of-fit test, using a random number generator to create a random number between 0 and 1, and let it denote FX(X). Applying the inverse function of FX(X) to find out the value of random variable that follows a specific distribution. The steps of Monte Carlo simulation are as follows: Step 1: Collecting historical sales data. Step 2: Using goodness-of-fit test to estimate the population’s parameters and demand distribution of each period. Step 3: Using random number generator to create a random number (U) between 0 and 1, and U~Uniform(0,1). Step 4: Finding the cumulative distribution function of the demand distribution (F(x)). Step 5: Let U= F(X) Step 6: X= F-1(U). Step 7: Repeat step 4 to 6 until the required random numbers are satisfied. 3.7.2. The relationship between system simulation and uniform distribution A random variable U is generated, and U~U(0,1), then let FX(X)=U, therefore X= FX -1(U). If X~FX , where FX is a cumulative distribution function (c.d.f ) of X. In other words, a random number U can be obtained from the random number generator, where U has a uniform distribution between 0 and 1, and then given X~FX and let FX(X)=U can be used to obtain a mapping value of random variable (X). The proof is shown in Property4. Property 4. : Given X~FX and . .X C R V ! (Continuous Random Variable), let FX(X)=U, where U~U(0,1), then X= FX -1(U). Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 56 http://creativecommons.org/licenses/by-nc-nd/4.0/ Proof: Let U~U(0,1), then F u P U u dt u1≤ U u 0 $= = =^ ^h h # , ,u 0 1! 6 @ If FX(X)=U, then F u P U u P F X u P F F X F u P X F u F F u u ≤ ≤ ≤ ≤ U X X X X X X X 1 1 1 1 $ = = = = = = - - - - ^ ^ ^ ^ ^ ^ ^ h h h h h h h 6 7 7 7 A A @ A And . Q.E.D. P X x P F U x P F F U F x P U F x F F x F x ≤ ≤ ≤ ≤ X X X X X U X X 1 1 = = = = = - - ^ ^ ^ ^ ^ ^ ^ h h h h h h h 7 7 6 6 7 @ @ A A A Since FX is a non-decreasing function of X, it means if a > b, then FX(a) ≥ FX(b), and if . .X C R V ! , then FX(a) > FX(b). 3.8. The flow chart of the proposed system simulation This paper uses the Visual Basic software to develop a multi-ordering computerized system; the system flow chart is shown in Figure 3. 4. Example analysis This chapter will base on the statistical analysis described as above to search for the optimal ordering strategy under single-period and multi-ordering situations. At first, describes the problem and then put the data into simulation system to find out the optimal ordering quantity and total expected profit, then analysis and discuss the simulation results. Finally, sensitivity analysis is carried out to verify the feasibility and correctness of the proposed model. 4.1. Example description Suppose there is a convenience store sells monthly magazine, and its price is $ 120 at cost $ 60. If it is not sold after the end of the sales cycle, it will only be worth $ 1 sold to the recycling dealer. Considering the shortage cost is equal to the marginal contribution of the magazine, and each ordering and delivery cost is 50, and assuming that no lead time, when the expected profit of each ordering is 0 will also carry out an order to satisfy the customer’s need. The sales period of the magazine is 30 days and divided into 3 periods, so each period is 10 days. We collected sales data over the past years and took the goodness- of-fit test to estimate the demand distribution of each period. We found that the demand distribution of each period is a normal distribution, which is Xi~N(µi,σi 2). The influences of the mean and variance of three periods on the number of orders, the optimal ordering quantity and the total expected profit is discussed. Therefore, the mean and variance are classified as large, medium and small, respectively. The large, medium and small of mean were 10, 20 and 30; the large, medium and small of standard deviation were ,3 1 6 1 9 1, and i i in n n . Therefore, there are 729 ((3×3)3) combinations. The random demand (Xi) for period i is calculated by the Monte Carlo simulation method. Each experiment is repeated 1000 times. The model proposed in this paper does not limit the ordering quantity, as long as Tπj (Qj)≥0, the order will be delivered. In this example, there are three possible ordering time points: the first time point is at the beginning of the sales period to meet the demand of entire period; The second time point is at the beginning of period 2 when the magazine was sold out in period 1, and reorder to meet the needs of period 2 and 3; The third time point is at the beginning of period 3 when the magazine was sold out in period 2, and then reorder to meet the need of time3. The purpose of this paper is to decide the optimal multi-ordering policy under stochastic demand to maximize the total expected profit. 4.2. General situation Putting the values of MP, ML and Cs into Equation (6) to find out the optimum ordering quantity Qj, and calculate the total expected profit Tπj (Qj) by Equation (5). If Tπj (Qj)≥0, then takes an order and the ordering quantity is Qj; If Tπj (Qj)<0, then do not take an order. 4.2.1. Analysis of single data We now randomly select the combination No. 8 which ordering twice in a sales period (it has three periods) to explain. The mean demand of the period 1 is 30 and its standard deviation is 10, the mean demand of period 2 is 30 and its standard deviation Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Using system simulation to search for the optimal multi-ordering policy for perishable goods 57 http://creativecommons.org/licenses/by-nc-nd/4.0/ is 10 and the mean demand of time interval 3 is 10 and its standard deviation is 1.67. The data is shown in Table 3. According to Table 3, it can be found that there happened 145 times of twice ordering in 1000 experiments. If order occurs twice, the second ordering quantity will be less than the first ordering quantity (11 < 76). Therefore, the Proposition 2 was verified. When the second order occurs in combination No. 8, the final total expected profit will be greater than which is only order once (4493.97 > 3116.16), so the Proposition 1 is verified. 4.2.2. Analysis of order twice data According to the simulation results where each combination was performed 1000 times. We show partial results of order twice in Table 4. According to Table 4, it can be found that if order occurs twice, the second ordering quantity will be less than the first ordering quantity (Q2 < Q1). Therefore, the Proposition 2 is verified. In addition the final total expected profit of order twice will be greater than which is only order once (TπM (Q1,Q2) > Tπ1 (Q1)), so the Proposition 1 is verified. In general, If Q1 is less than or close to (µ1+3σ1), then it has the opportunity to order twice and the time of second order is at the end of period 1; When Q1 is less than or close to µ1+(µ2+3σ2) or close to (µ1+3σ1)+µ2, it has the opportunity to order twice and the time of second order is at the end of the period 2. It was known from the examples that ordering twice is likely to occur in (µ1 ≥ µ2 ≥ µ3) or (µ2 ≥ µ1 and µ2 ≥ µ3) condition. Figure 3. System flow chart. Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 58 http://creativecommons.org/licenses/by-nc-nd/4.0/ 4.3. Shortage cost Under the other parameters are fixed, we will discuss the magnitude of shortage cost that influences the optimal ordering strategy and total expected profit, simultaneously. There are three kinds of situations need to consider: (1) Thinking of the shortage cost is as the opportunity cost, it means that Cs=MP; (2) Thinking of the shortage cost is as the opportunity cost plus customer run off cost, it means that Cs > MP; (3) Thinking of the shortage costs is as fictitious loss, it means that Cs=0. Applying the simulation system developed in this paper, the results are obtained and shown in Figure 4. According to Figure 4, it can be found that the ordering quantity will be increased when shortage cost rises. Those results just verify the inference in section 3.3.4. 4.4. Order three times’ conditions Based on Section 3.3.4, we knew that the ordering quantity of considering the shortage cost is greater than the one of do not consider. Under do not consider the shortage cost, it can be found that ordering more than one time would occur in some particular combinations, and those results also proved that multi-ordering policy for perishable goods in expiry period (which can be divided into several periods) is worthy. The combinations of order three times are shown in Table 4. According to we found that the situation of order three times is likely to occur only once in 1000 Table 4 times random simulations under some specific combinations. Usually it occurs at the mean and variation of period 1 are large, and the mean of period 3 is small. From Table 4 we can find the ordering quantity is decreasing each time, it means Q1 > Q2 > Q3. Therefore, the Proposition 2 was verified. In addition, the total expected profit is shown in Figure 5. Reorder conditions are based on Tπj (Qj)≥0. Therefore, that can be known the total expected profits will increase when the order number is rising, so the Proposition 1 was verified. When we do not consider shortage cost (it means Cs=0) and then execute 1000 times simulations for each combination. It can be found that when Table 3. Total expected profit and ordering quantity of combination No. 8. Number Time 1 Time 2 Time 3 J=1 J=2 µ1 σ1 µ2 σ2 µ3 σ3 total times of ordering Tπ1(Q1) total average Q1 total times of ordering TπM(Q1 ,Q2) total average Q1 +Q2 8 30 10 30 10 10 1.7 855 3,116.16 76 145 4,493.97 87 Cs=0 Cs=MP Cs>MP NO.246 Order once Total order quantity average 80 86 87 NO.246 Order twice Total order quantity average 110 117 119 NO.333 Order once Total order quantity average 60 65 66 NO.333 Order twice Total order quantity average 70 75 77 80 86 87 110 117 119 60 65 66 70 75 77 55 85 115Total ordering quantity Figure 4. The effects of various shortage costs on ordering quantity. Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Using system simulation to search for the optimal multi-ordering policy for perishable goods 59 http://creativecommons.org/licenses/by-nc-nd/4.0/ (µ1 ≥ µ2 ≥ µ3) and (µ1, σ1, σ2) are very large, furthermore, Q1 is less than or close to (µ1+3σ1) and Q2 is less than or close to (µ2+3σ2), order three times situations will be happened. 4.5. Sensitivity analysis The influences of the system parameters on total expected profit are shown as follows. According to Table 5, it can be observed that Tπj (Qj) will increase when MP is rising. It showed that MP and Tπj (Qj) has a positive correlation. Therefore, Equation (14) was verified. It can be observed that Tπj (Qj) will decrease when ML is rising. It showed that ML and Tπj (Qj) has a negative correlation. Therefore, Equation (15) was verified. It can be observed that Tπj (Qj) will decrease when Cs is rising. It showed that Cs and Tπj (Qj) has a negative correlation. Therefore, Equation (16) was verified. It can be observed that Tπj (Qj) will decrease when Coj is rising. It showed that Coj and Tπj (Qj) has a negative correlation. Therefore, Equation (17) was verified. 0,0 1.500,0 3.000,0 4.500,0 6.000,0 7.500,0 7 8 34 35 36 61 62 Order once in thousand times Total expected profit averages 3.257,5 3.250,2 2.806,1 2.804 2.830,7 2.326,4 2.332,1 Order twice in thousand times Total expected profit averages 4.541,9 4.631,7 3.965,9 4.016,4 4.050,3 3.413,8 3.495,7 Order thrice in thousand times Total expected profit averages 6.693 7.050 5.493 5.850 5.731 4.650 4.650 Total expected profit Figure 5. Total expected profit under three times ordering. Table 4. Total expected profit and ordering quantity for order twice and three times. No. i=1 i=2 i=3 J=1 J=2 J=3 µ1 σ1 µ2 σ2 µ3 σ3 total times of ordering Tπ1(Q1) total average Q1 average total times of ordering TπM(Q1,Q2) total average Q1+Q2 average total times of ordering TπM(Q1,Q2,Q3) total average Q1+Q2+Q3 average 7 30 10 30 10 10 3.3 870 3,092.94 76 130 4,466.06 76+11.3 - - - 49 30 10 20 2.2 20 6.7 981 3,375.84 75 19 4,891.47 75+23 - - - 89 30 5 30 10 10 1.7 899 3,396.94 75 101 4,588.55 75+11 - - - 99 30 5 30 5 10 1.1 961 3,692.56 73 39 4,683.49 73+10 - - - 116 30 5 20 6.7 10 1.7 940 2,999.72 64 60 4,008.30 64+11 - - - 134 30 5 20 2.2 10 1.7 993 3,194.83 63 7 3,896.00 63+11 - - - 269 20 6.7 30 3.3 10 1.7 951 3,061.81 63 49 4,046.29 63+11 - - - 656 10 1.1 30 10 10 1.7 919 2,304.68 55 81 3,481.16 55+11 - - - 7 30 10 30 10 10 3.3 734 3,257.5 70 265 4,541.9 70+10.1 1 6,693 70+40+10 8 30 10 30 10 10 1.7 743 3,250.2 70 256 4,631.7 70+10 1 7,050 70+40+10 34 30 10 20 6.7 10 3.3 796 2,806.1 60 203 3,965.9 60+10.3 1 5,493 60+30+10 35 30 10 20 6.7 10 1.7 797 2,804 60 202 4,016.4 60+10.1 1 5,850 60+30+10 36 30 10 20 6.7 10 1.1 791 2,830.7 60 208 4,050.3 60+10.1 1 5,731 60+30+10 61 30 10 10 3.3 10 3.3 806 2,326.4 50 193 3,413.8 50+11.3 1 4,650 50+20+10 62 30 10 10 3.3 10 1.7 811 2,332.1 50 188 3,495.7 50+11.4 1 4,650 50+20+10 Int. J. Prod. Manag. Eng. (2019) 7(1), 49-62 Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Huang, Y.C., Chang, X.Y. and Ding, Y.A. 60 http://creativecommons.org/licenses/by-nc-nd/4.0/ 5. Conclusions This paper establishes a single-period and multi- ordering mathematical model to revise the traditional newsvendor model and based on numerical examples to verify its feasibility and profitability. The purpose of this paper is to modify the traditional newsvendor model from single-order to multi- order to maximize the total expected profit. With consideration of marginal contribution, marginal loss, and shortage cost, the total expected profit for multiple orders will be better than for single order, and the amount of each order placed under multiple orders and its corresponding expected profit will gradually decrease. Based on numerical examples, the perishable goods will be ordered three times only in few cases. 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