考慮商品數量折扣之聯合補貨問題 Consider quantity discounts for joint replenishment problem

1. 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

2. Reporting process

3. Joint replenishment problem • The joint replenishment objective • adjusts to the replenishment cycle between different products to avoid additional ordering costs.

4. Quantity discounts Price Order quantity

5. Consider quantity discounts for joint replenishment problem Consider quantity discounts Establish heuristic method to solve the joint replenishment problem considering quantity discounts

6. Research Process • Research Motivation and objective • Related literature Joint replenishmentproblem Consider quantity discounts for joint replenishmentproblem • Particle swarm optimization Establish heuristic method Experiment parameters set • Analysis of results

10. Problemdescription • Consider quantity discounts joint replenishment problem for single supplier to multi-retailers. • Objective is minimize the total cost.

11. Research steps Joint replenishment problem (Not consider quantity discounts ) Single item replenishment problem (Consider quantity discounts ) • Programming approach Analysis of results Joint replenishment problem (Consider quantity discounts ) • Compare Programming approachand heuristic methods Find the optimal replenishment strategies Heuristic method

12. 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

13. Mathematical model with quantity discounts Single items replenishment problem Joint replenishment problem

14. Programming approachtosolve joint replenishment problem.(Not consider quantity discounts) • S=4000

15. Programming approachtosolve joint replenishment problem.(Not consider quantity discounts) • Objective function : • T、ki、yij are decision variables • kiat least one of 1 (basic cycle) • The other items cycles is the integer multiple of the basic cycle.

17. Programming approachtosolve Single item replenishment problem (Consider quantity discounts ) • D=120000 • h=0.2 • S=100

18. Programming approachtosolve Single item replenishment problem (Consider quantity discounts ) • Originally objective function : • changed to • yjisbinary， Indicates whether to use a discounted price j ， j=1,2,3 • Qj、yj are decision variables • ，

19. Programming approachtosolve Single item replenishment problem (Consider quantity discounts ) • The result of programming approach solving isthe minimum TC for $ 354,520 occurred when the order quantity Q is 10000.

20. Programming approachtosolve Single item replenishment problem (Consider quantity discounts )

21. Programming approachtosolve joint replenishment problem.(Consider quantity discounts) • S=4000

22. Programming approachtosolve joint replenishment problem.(Consider quantity discounts) Quantity discount table

23. Programming approachtosolve joint replenishment problem.(Consider quantity discounts) • Originally objective function : • changed to • yij isbinary. Denote the items i whether use discounted prices j. i=1,2,3 ，j=1,2,3 • T、 ki、 yij are decision variables.

25. 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

26. Particle swarm optimization (k1,…,ki,T,V1,…,Vi) (k1,…,ki,T,V1,…,Vi) (k1,…,ki,T,V1,…,Vi) Optimal solution (k1,…,ki,T,V1,…,Vi) (k1,…,ki,T,V1,…,Vi)

27. Mathematical Symbol Description: • Xid： Position of the particle i on d-th iteration. • Vid：Speed of the particle i on d-th iteration. • Pid：The best position of the particle i in d iterations. • Pgd： The best position of all particle i in d iterations. • Cj： Learning coefficient. • ω： Weight. • ωmax： Weight maximum. • ωmin： Weight minimum. • Rj： Independent random variable. The range is [0, 1]. • Vmax：The maximum allowable speed when the particle update.

28. Heuristic method

29. Heuristic method Particle velocity update formula ： Particle position update formula ： Particle speed limit ： Solving

30. Heuristic method • According to the method proposed by Goyal (1973 & 1974) set the upper and lower bounds of ki and basic cycle T. • According to the method proposed by Silver(1976) to find out k1T is the basic cycle. ,(ki=1)

31. Heuristic method • Step1： Initialization Randomly generated particles and randomly assigned to the initial position and speed. k1=1，k2、k3upper bound are 3.2969 、10.5409,and 0.0408 ≦T ≦0.1091 。

32. Heuristic method • Step2：Evaluation Evaluate each particle function value.  Randomly generated values ​​of k, T and v into the objective formula.

33. 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.

34. Heuristic method • Step5：Randomly generated R1, R2 and update Xid, Vid. Obtained Pid and Pgd from Step3,Step4 then update particles speed and position.

35. Heuristic method Step6： Repeatedly step2 to step5, and stop when it reaches the termination conditions.

36. Analysis of results • After repeated iteration, the total cost TC will gradually close to the optimal solution.

37. Conclusion • 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.

38. Research plan

39. Research plan

40. Timetable

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