考慮商品數量折扣之聯合補貨問題 Consider quantity discounts for joint replenishment problem 研究生 : 王聖文 指導教授 : 楊能舒 教授

Reporting process Introduction Background and Motivation Research of objective Research Process Literature Quantity discounts Particle swarm optimization Joint replenishment problem Research Method Mathematical model Solving process Experiment and Analysis Instance calculus Analysis of results Conclusion Programming approach Particle swarm optimization Research Plan Discount percentage Numbers of item

The joint replenishment objective adjusts to the replenishment cycle between different products to avoid additional ordering costs. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Joint replenishment problem

Quantity discounts IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Order quantity Price

Consider quantity discounts Establish heuristic method to solve the joint replenishment problem considering quantity discounts Consider quantity discounts for joint replenishment problem IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Research Motivation and objective Related literature Joint replenishment problem Particle swarm optimization Establish heuristic method Experiment parameters set Analysis of results IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Consider quantity discounts for joint replenishment problem Research Process

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan No.AuthorYearMethodResult The relationship between this study 1 J.Kenny, R.C. Eberhart 1995 Propose particle swarm optimization and explain the feature and concept of this algorithm. Particle swarm algorithms consider a few parameters and fast convergence of the solution space, suitable for solving large problems. Use the feature of particle swarm optimization for solving joint replenishment problem. 2 Goyal, S.K.1973 Propose the method to find out the upper and lower bounds of the optimal ordering cycle. By changes in cycle multiplier derived upper and lower bounds of the optimal ordering cycle. Use the method proposed by Goyal to set the upper and lower bound of basic cycle. 3 Goyal, S.K.1974 Propose the method to find out upper and lower bounds of the optimal ordering cycle and give an example. Changes in cycle multiplier will affect the total cost. Similar to Goyal(1973), there is example for reference.

No.AuthorYearMethodResult The relationship between this study 4 Silver, E.A.1976 Derivate cycle multiplier formula and set upper and lower bounds of cycle multiplier also proposed sorting indicators. Silver proposed a simple method can get a good solution performance. Use the method proposed by Silver to set the cycle multiplier lower and upper bound as well as to determine the basic ordering cycle. 5 Shi, Y.,Eberhart, R Propose weight added to the particle swarm optimization. By adjusting the weights to change the solution space of global search or local search. Consider the weight can increase the accuracy of search solutions. 6 I.K.MOON,S.K.GOYAL,B.C.C HA 2008 Using genetic algorithms try to solve consider quantity discounts for joint replenishment problem.. Provide a method to solve consider quantity discounts for joint replenishment. Mentioned joint replenishment model for considering quantity discounts. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan No.AuthorYearMethodResult The relationship between this study 7 Kaspi, M. & Rosenblatt, M.J The extension of Silver (1976). Find out the relationship of cycle multiplier and frequency. Another method for solving joint replenishment problem. 8 Kaspi, M. & Rosenblatt, M.J The extension of Kaspi, M. & Rosenblatt, M.J. (1983). Add frequency upper and lower bounds, and the same amount range. Proposed the RAND algorithm. Another method for solving joint replenishment problem. 9 Goyal, S.K. & Belton, A.S Another indicator to modify the method proposed by Silver (1976) Claimed to use the indicators to get a better solution. Increase the selection of indicators 。

Problem description Consider quantity discounts joint replenishment problem for single supplier to multi-retailers. Objective is minimize the total cost. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Joint replenishment problem (Not consider quantity discounts ) Joint replenishment problem (Consider quantity discounts ) Programmin g approach Analysis of results Heuristic method Find the optimal replenishment strategies Single item replenishment problem (Consider quantity discounts ) Research steps IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Compare Programming approach and heuristic methods

Mathematical Symbol Description Di: Demand for items hi: Items i per unit holding cost ratio S: Major ordering cost si: Minor ordering costs Ci: Unit price of item i ki: Integer number that determines the replenishment schedule of item i T: Basic cycle TC: Total cost IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Mathematical model with quantity discounts Single items replenishment problem Joint replenishment problem IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve joint replenishment problem.(Not consider quantity discounts) S=4000 Item123 si1000 hi0.2 Di Ci500 IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve joint replenishment problem.(Not consider quantity discounts) Objective function : T 、 ki 、 yij are decision variables ki at least one of 1 (basic cycle) The other items cycles is the integer multiple of the basic cycle. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve joint replenishment problem.(Not consider quantity discounts) basic cycle T (k1,k2,k3)(1,2,5) TC IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve Single item replenishment problem (Consider quantity discounts ) D= h=0.2 S=100 Ordering quantityUnit price Q< ≦ Q< Qi ≧ IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve Single item replenishment problem (Consider quantity discounts ) Originally objective function : changed to yj is binary ， Indicates whether to use a discounted price j ， j=1,2,3 Qj 、 yj are decision variables ， IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve Single item replenishment problem (Consider quantity discounts ) The result of programming approach solving is the minimum TC for $ 354,520 occurred when the order quantity Q is IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Programming approach to solve Single item replenishment problem (Consider quantity discounts )

Programming approach to solve joint replenishment problem.(Consider quantity discounts) S=4000 Item123 sisi 1000 hihi 0.2 DiDi IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

C1C2C3 Qi< ≦ Qi< Qi ≧ Quantity discount table IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Programming approach to solve joint replenishment problem.(Consider quantity discounts)

Originally objective function : changed to yij is binary. Denote the items i whether use discounted prices j. i=1,2,3 ， j=1,2,3 T 、 ki 、 yij are decision variables. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Programming approach to solve joint replenishment problem.(Consider quantity discounts) Basic cycle T (k1,k2,k3)(1,9,4) (C1,C2,C3)(440,460,500) TC IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Analysis of results Programming approach example shows that it is feasible to solve small quantity discounts problem. But large quantity discounts problem programming approach can not be solved.  Find another new algorithm for solving IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Optimal solution (k 1,…,k i,T,V 1,…,V i ) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Particle swarm optimization

Mathematical Symbol Description: X id ： Position of the particle i on d-th iteration. V id ： Speed of the particle i on d-th iteration. P id ： The best position of the particle i in d iterations. P gd ： The best position of all particle i in d iterations. C j ： Learning coefficient. ω ： Weight. ω max ： Weight maximum. ω min ： Weight minimum. R j ： Independent random variable. The range is [0, 1]. V max ： The maximum allowable speed when the particle update. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan

Decision variables are k i and basic cycle Tk i and basic cycle T set to particle position Use particle swarm optimization to minimize the total cost. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

Particle velocity update formula ： Particle position update formula ：  Solving Particle speed limit ： IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

According to the method proposed by Goyal (1973 & 1974) set the upper and lower bounds of k i and basic cycle T. According to the method proposed by Silver(1976) to find out k 1 T is the basic cycle.,(ki=1) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

Step1 ： Initialization Randomly generated particles and randomly assigned to the initial position and speed. k 1 =1 ， k 2 、 k 3 upper bound are 、 ,and ≦ T ≦ 。 (k1,k2,k3)Basic cycle T(v1(k2),v2(k3),v3(T)) (1,2,3)0.1090(0.5412,2.4325,3.2516) (1,4,2)0.0706( , ,3.5421) (1,3,9)0.0957(3.7685, ,1.0010) (1,1,10)0.1010(2.5612,0.0025, ) (1,2,5)0.0752( ,2.6570, ) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

Step2 ： Evaluation Evaluate each particle function value.  Randomly generated values ​​ of k, T and v into the objective formula. (k1,k2,k3)Basic cycle T(v1(k2),v2(k3),v3(T))Fitness value (TC) (1,2,3)0.1090(0.5412,2.4325,3.2516) (1,4,2)0.0706( , ,3.5421) (1,3,9)0.0957(3.7685, ,1.0010) (1,1,10)0.1010(2.5612,0.0025, ) (1,2,5)0.0752( ,2.6570, ) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

Step3 ： Update Pid According to the value obtained by Step2 update so far the best position of each particle. Step4 ： Update Pgd According to the value obtained by Step3 update so far the best position of group of particle. The fitness value of each particle after iteration. Update so far the fitness value of each particle. the best position of each particle. (k1,k2,k3,T) (1,2,3,0.1090) (1,4,2,0.0706) (1,3,9,0.0957) (1,1,10,0.1010) (1,2,5,0.0752) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

Step5 ： Randomly generated R 1, R 2 and update X id, V id. Obtained P id and P gd from Step3,Step4 then update particles speed and position. (k1,k2,k3)Basic cycle T(v1(k2),v2(k3),v3(T))(R1,R2) (1,1,11)0.1091( ,4,2.9174)(0.1057,0.5678) (1,1,11)0.1091(-4,4,3.2312)(0.9512,0.7123) (1,4,6)0.1091(3.7685, ,1.0010)(0.6717,0.0045) (1,3,10)0.0408(2.3051,0.0023, )(0.2130,0.7951) (1,1,11)0.0408( ,4, )(0.3016,0.4510) IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

(k1,k2,k3)Basic cycle TFitness value (TC) (1,1,11) (1,1,11) (1,4,6) (1,3,10) (1,1,11) Step6 ： Repeatedly step2 to step5, and stop when it reaches the termination conditions. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan After repeated iteration, the total cost TC will gradually close to the optimal solution. Programming approach Heuristic method Basic cycle T (k1,k2,k3)(1,9,4)(1,2,5) (C1,C2,C3)(440,460,500)(440,480,500) TC Analysis of results

Compared with Programming approach, the calculate time of heuristic method is shorter than Programming approach significantly. And the convergence speed of heuristic method also faster than Programming approach, several iteration that can be get a good solution. Heuristic method is suitable to apply in large joint replenishment problem. IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Conclusion

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Heuristic method solve the example proposed By Goyal(2008) Solving performance ? Research plan

IntroductionResearch of objective Research Process LiteratureResearch Method -Instance calculus Research Method -Analysis of results ConclusionResearch plan Programmi ng approach Heuristic method Small proble m The optimal solution can be obtained. The number of calculation is less and good solution can be obtained. Large proble m Can not be solved. ? Conditions set Numbers of item 520 Discount percentage 10%30% …… … Research plan

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