Inventory model of a deteriorating item with price and credit linked fuzzy demand : A fuzzy differential equation approach

Abstract

This paper deals with an inventory model for a deteriorating item. Here the demand is considered as fuzzy in nature which depends on unit selling price as well as credit period offered by the retailer. Here wholesaler/producer offers a delay period of payment to its retailer to capture the market. At the same time retailer also offers a fixed credit period to its customers to boost the demand. Due to impreciseness of demand, the model is formulated using fuzzy differential equation (FDE). Fuzzy-Riemann integration method is followed to find α-cuts of fuzzy inventory costs and fuzzy average profit. The goal is to find the optimal cycle length, unit selling price and credit period offered by retailer to maximize the average profit. Combining the features of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), a hybrid algorithm named Interval Compared Hybrid Particle Swarm GA (ICHPSGA) is developed and used to find marketing decision for the retailer. Different ranking methods of intervals are used in this algorithm to find fitness of a solution. The model is also solved using Fuzzy Genetic Algorithm (FGA), Multi-Objective GA (MOGA) and results are compared with those obtained using the proposed algorithm (ICHPSGA). Moreover, several non-linear test functions are also tested with the present developed algorithm, conventional MOGA and FGA. Numerical experiments are performed to illustrate the model and some sensitivity analyses have been made. For statistical support, analysis of variance (ANOVA) is performed with the sample of runs for the test functions using the presented algorithm.

Appendix A

1.1 List of test functions (TF)

• TF-1:  (Taken from Bessaou and Siarry [5]): $SH(x_{1},x_{2})=\sum \limits _{j=1}^{5}j\times cos[(j+1)\times x_{1}+j]\times \sum \limits _{j=1}^{5}j\times cos[(j+1)\times x_{2}+j], -10\leq x_{1}, x_{2}\leq 10.$This problem has 760 local minima and 18 global minima. At global minima $(x_{1},x_{2})$, $SH(x_{1},x_{2})= -186.7309$.

• TF-2:  TF-2:  (Taken from Bessaou and Siarry [5]): $MZ(x_{1},x_{2},...,x_{n})=-\sum \limits _{i=1}^{n}sin(x_{i}).[sin(i.(x_{i})^{2}/\pi )]^{2m}, -\pi \leq x_{1},x_{2},...,$ $x_{n} \leq \pi , \text {where} m=10.$For $n=2$, it has one global minima at $(x_{1},x_{2})=(2.25,1.57)$ and $MZ(2.25,1.57)=-1.80$.

• TF-3:  (Taken from Bessaou and Siarry [5]): $RC(x_{1},x_{2})=\left \{x_{2}-[5/(4\times \pi ^{2})].x_{1}^{2}+(5/\pi )\times x_{1}-6\right \}^{2}+10\times \{1-[1/(8\pi )]\}\times cos(x_{1})+10,-5\leq x_{1}\leq 10, 0\leq x_{2}\leq 15.$ This problem has three global minima at $(x_{1},x_{2}){}={}(-\pi ,12.275),$ $(\pi ,2.275),(9.42478,2.475)$ and $RC(x_{1},x_{2})=0.397887$ at any one of these minima.

• TF-4:  (Taken from Michalewicz [37]): Minimize $F(x_{1},x_{2}){}={}(x_{1}-2)^{2}+(x_{2}-1)^{2}$, such that $-x_{1}^{2}+x_{2}\geq 0, x_{1}+x_{2}\leq 2, -5\leq x_{1}, x_{2}\leq 5$, It has one global minima at $(x_{1},x_{2})$ =(1,1), and $F(1,1)=1$.

• TF-5:  (Taken from Michalewicz [37]) Minimize $F(x_{1},x_{2})=100\left (x_{2}-x_{1}^{2}\right )^{2}+(x_{1}-1)^{2}$, such that $x_{1}+x_{2}^{2}\geq 0, x_{1}^{2}+x_{2}\geq 0, -0.5\leq x_{1}\leq 0.5, -1.0\leq x_{2}\leq 1.0$It has one global minima at $(x_{1},x_{2})=(0.5,0.25)$, and $F(0.5,0.25)=0.25$.

• TF-6:  (Taken from Bessaou and Siarry [5]) Minimize $F_{n}(x_{1},x_{2},...,x_{n})=\sum \limits _{i=1}^{n-1}\left [100\times \left (x_{j}^{2}-x_{j+1} \right )^{2}+(1-x_{j})^{2}\right ], -1\leq x_{1},x_{2},...,x_{n}\leq 5$, Three functions $F_{2}, F_{3}, F_{4}$ are used which are denoted by 6A, 6B and 6C respectively. This problems has one global minima at $(x_{1},x_{2},...,x_{n})=(1,1,...,1)$ and $F_{n}(1,1,...,n)=0$.

• TF-7:  (Taken from Bessaou and Siarry [5]): $F2(x_{1},x_{2})=100\times \left (x_{2}^{2}-x_{1}\right )+(1-x_{1}), -2.048\leq x_{1}, x_{2} \leq 2.048.$It has one minima at $(x_{1},x_{2})=(2.048,0)$ and $F2(2.048,0)=-205.8480$.

• TF-8:  (Taken from Bessaou and Siarry [5]): $ES(x_{1},x_{2})=-cos(x_{1})\times cos(x_{2})\times exp\{-[(x_{1}-\pi )^{2}+(x_{2}-\pi )^{2}]\}, -10\leq x_{1}, x_{2} \leq 10.$This test function has one global minima at $(x_{1},x_{2})=(\pi ,\pi )$ and $ES(\pi ,\pi )=-1$.

• TF-9:  (Taken from Bessaou and Siarry [5]): $Z_{n}(x_{1},x_{2},...,x_{n})=\left (\sum \limits _{j=1}^{n}x_{j}^{2}\right )+\left (\sum \limits _{j=1}^{n}0.5j\times x_{j}\right )^{2}+\left (\sum \limits _{j=1}^{n}0.5j\times x_{j}\right )^{4},-5\leq x_{1},x_{2},...,x_{n} \leq 5.$It has one global minima at $(x_{1},x_{2},...,x_{n}){}={} (0,0,...,0)$ and $Z_{n}(0,0,...,0)=0.$

• TF-10:  (Taken from Bessaou and Siarry [5]): $BH(x_{1},x_{2})=x_{1}^{2}+2\times x_{2}^{2}-0.3\times cos(3\pi \times x_{1})\times cos(4\pi \times x_{2})+0.3, -5\leq x_{1},x_{2} \leq 5.$This problem has one global minima at $(x_{1},x_{2}){}={}(0,0)$ and $BH (0,0){}={}0.$

• TF-11:  (Taken from Bessaou and Siarry [5]): $DJ(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}, -5.12\leq x_{1},x_{2},x_{3} \leq 5.12.$It has one global minima at $(x_{1},x_{2},x_{3})=(0,0,0)$ and $DJ(0,0,0)=0$.

The proposed algorithm PSGA (as the test functions are not interval valued objectives, PSGA is used instead of ICHPSGA) is run for the above test functions using different seeds of random number generators for 50 times each and number of wins of finding optimal solutions for each function are noted in Table 9. If optimal solution is found in a run of the heuristic algorithm using a seed, we say that the algorithm wins. Number of wins for different test functions due to another two heuristics MOGA and FGA are also listed in Table 9. It is seen that number of wins is highest in the case of PSGA for all most all the test functions.

Table 9 Results of test functions following different approaches