Multi-echelon Supply Chain Model for Deteriorating Products in a Fuzzy Deterioration Environment

Figures & data

Figure 1. The buyer's inventory level versus time.

Figure 2. The supplier's inventory level versus time.

Figure 3. Triangular fuzzy number $\tilde{d}$.

Figure 4. Triangular fuzzy number.

Figure 5. Optimality of the algorithm for the given example. (a) Effects of D on average total cost (b) Effects of P on average total cost; (c) Effects of H_s on average total cost (d) Effects of H_b on average total cost; (e) Effects of C on average total cost (f) Effects of A on average total cost; (g) Effects of F on average total cost (h) Effects of C_d on average total cost.

Table 1. Solution procedure of the model.

${\mathrm{\Delta }}_1$	${\mathrm{\Delta }}_2$	$N$	$q$	$\widehat{TC}(.)$
0.1	0.1	1	481.0	18,243
		2	279.5	17,110
		3	202.0	16,856
		4	157.7	16,980
		5	131.2	17,140
		6	112.0	17,345
0.1	0.2	1	464.4	18,745
		2	267.1	17,592
		3	192.4	17,303
		4	152.7	17,462
		5	124.2	17,629
		6	109.6	17,833
0.2	0.1	1	500.8	17,718
		2	285.2	16,627
		3	204.4	16,371
		4	161.5	16,500
		5	134.0	16,643
		6	123.0	16,840

Figure 6. Percentage changes of parameters on the average total cost.

Table 2. Effects of $D$ on optimal solution.

% of change in $D$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	201.6	4	21,495
	(0.1, 0.2)	192.0	4	22,049
	(0.2, 0.1)	210.3	4	20,920
	RZ = 2.3%		${\tilde{R}}_Z$=2.4%
+25	(0.1, 0.1)	179.2	4	19,299
	(0.1, 0.2)	178.3	4	19,815
	(0.2, 0.1)	192.9	4	18,764
	RZ = 2.4%		${\tilde{R}}_Z=2.5\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,303
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	160.7	3	13,199
	(0.1, 0.2)	152.8	3	13,586
	(0.2, 0.1)	165.1	3	12,796
	RZ = 2.7%		${\tilde{R}}_Z=2.9\mathrm{\%}$
−50	(0.1, 0.1)	200.5	2	11,079
	(0.1, 0.2)	193.1	2	11,415
	(0.2, 0.1)	208.7	2	10,730
	RZ = 2.8%		${\tilde{R}}_Z=2.9\mathrm{\%}$

Table 3. Effects of $P$ on optimal solution.

% of change in $P$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	198.2	3	17,033
	(0.1, 0.2)	190.5	3	17,506
	(0.2, 0.1)	205.3	3	16,541
	RZ = 2.2%		${\tilde{R}}_Z$=2.6%
+25	(0.1, 0.1)	199.1	3	16,962
	(0.1, 0.2)	192.6	3	17,433
	(0.2, 0.1)	209.4	3	16,474
	RZ = 2.5%		${\tilde{R}}_Z=2.6\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	161.6	4	17,380
	(0.1, 0.2)	155.5	4	17,173
	(0.2, 0.1)	170.6	4	16,153
	RZ = 2.7%		${\tilde{R}}_Z$=2.8%
−50	(0.1, 0.1)	149.9	5	15,956
	(0.1, 0.2)	144.3	5	16,388
	(0.2, 0.1)	156.1	5	15,507
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%

Table 4. Effects of $H_S$ on optimal solution.

% of change in $H_s$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	193.4	3	17,380
	(0.1, 0.2)	186.7	3	17,827
	(0.2, 0.1)	200.8	3	16,916
	RZ = 2.4%		${\tilde{R}}_Z$=2.5%
+25	(0.1, 0.1)	198.5	3	17,120
	(0.1, 0.2)	190.4	3	17,577
	(0.2, 0.1)	204.5	3	16,647
	RZ = 2.5%		${\tilde{R}}_Z=2.6\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	208.6	3	16,585
	(0.1, 0.2)	199.6	3	17,065
	(0.2, 0.1)	215.8	3	16,089
	RZ = 2.6%		${\tilde{R}}_Z$=2.9%
−50	(0.1, 0.1)	168.5	4	16,293
	(0.1, 0.2)	161.4	4	16,789
	(0.2, 0.1)	176.2	4	15,777
	RZ = 2.8%		${\tilde{R}}_Z$=3.0%

Table 5. Effects of $H_B$ on optimal solution.

% of change in $H_B$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	155.0	4	17,196
	(0.1, 0.2)	152.4	4	17,656
	(0.2, 0.1)	162.3	4	16,719
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
+25	(0.1, 0.1)	199.0	3	17,033
	(0.1, 0.2)	191.5	3	17,492
	(0.2, 0.1)	208.1	3	16,555
	RZ = 2.5%		${\tilde{R}}_Z=2.6\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	204.0	3	16,676
	(0.1, 0.2)	198.2	3	17,149
	(0.2, 0.1)	215.6	3	16,184
	RZ = 2.6%		${\tilde{R}}_Z$=2.7%
−50	(0.1, 0.1)	208.3	3	16,494
	(0.1, 0.2)	201.0	3	16,973
	(0.2, 0.1)	217.6	3	15,993
	RZ = 2.4%		${\tilde{R}}_Z$=2.8%

Table 6. Effects of $C$ on optimal solution.

% of change in $C$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	185.0	4	18,958
	(0.1, 0.2)	180.1	4	19,507
	(0.2, 0.1)	196.2	4	18,388
	RZ = 2.7%		${\tilde{R}}_Z$=2.8%
+25	(0.1, 0.1)	174.5	4	17,977
	(0.1, 0.2)	212.6	3	18,487
	(0.2, 0.1)	181.2	4	17,446
	RZ = 2.6%		${\tilde{R}}_Z=2.8\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	181.0	3	15,621
	(0.1, 0.2)	172.2	3	16,039
	(0.2, 0.1)	184.6	3	15,186
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−50	(0.1, 0.1)	213.1	2	14,159
	(0.1, 0.2)	204.2	2	14,519
	(0.2, 0.1)	222.7	2	13,785
	RZ = 2.3%		${\tilde{R}}_Z$=2.4%

Table 7. Effects of $A$ on optimal solution.

% of change in A	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	204.5	3	16,953
	(0.1, 0.2)	197.0	3	17,423
	(0.2, 0.1)	211.2	3	16,465
	RZ = 2.6%		${\tilde{R}}_Z$=2.7%
+25	(0.1, 0.1)	202.8	3	16,904
	(0.1, 0.2)	196.1	3	17,373
	(0.2, 0.1)	211.3	3	16,418
	RZ = 2.6%		${\tilde{R}}_Z=2.6\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	200.1	3	16,807
	(0.1, 0.2)	194.6	3	17,271
	(0.2, 0.1)	209.6	3	16,324
	RZ = 2.6%		${\tilde{R}}_Z$=2.6%
−50	(0.1, 0.1)	199.7	3	16,758
	(0.1, 0.2)	193.8	3	17,220
	(0.2, 0.1)	208.7	3	16,278
	RZ = 2.6%		${\tilde{R}}_Z$=2.7%

Table 8. Effects of $C_d$ on optimal solution.

% of change in $C_d$	$({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$	$q^{\ast }$	$N^{\ast }$	$\widehat{TC}(.)$
+50	(0.1, 0.1)	173.4	3	18,802
	(0.1, 0.2)	167.2	3	19,400
	(0.2, 0.1)	180.1	3	18,177
	RZ = 3.0%		${\tilde{R}}_Z$=3.1%
+25	(0.1, 0.1)	185.1	3	17,865
	(0.1, 0.2)	178.7	3	18,400
	(0.2, 0.1)	193.4	3	17,306
	RZ = 2.8%		${\tilde{R}}_Z=2.9\mathrm{\%}$
0	(0.1, 0.1)	202.0	3	16,856
	(0.1, 0.2)	192.4	3	17,087
	(0.2, 0.1)	204.4	3	16,371
	RZ = 2.5%		${\tilde{R}}_Z$=2.6%
−25	(0.1, 0.1)	222.6	3	15,754
	(0.1, 0.2)	212.1	3	16,141
	(0.2, 0.1)	231.6	3	15,353
	RZ = 2.3%		${\tilde{R}}_Z$=2.4%
−50	(0.1, 0.1)	251.3	3	14,528
	(0.1, 0.2)	242.7	3	14,822
	(0.2, 0.1)	259.1	3	14,225
	RZ = 1.8%		${\tilde{R}}_Z$=1.9%