PARTICLE SWARM OPTIMIZATION-BASED ALGORITHM FOR BILEVEL JOINT PRICING AND LOT-SIZING DECISIONS IN A SUPPLY CHAIN

Figures & data

FIGURE 1 A two-echelon hierarchical system.

Table

The manufacturer's decision variables
β:	manufacturer's number of orders in the planning horizon
p m :	unit wholesale price of the product
The retailer's decision variables
α:	number of retailer's lot size per manufacturer's lot size
Q :	retailer's lot size
p r :	unit retail price of the product
Other related parameters are as follows
T:	weekly length of the planning horizon
D:	weekly demand rate D = b − a * p r
h m :	manufacturer's weekly holding cost rate
h r :	retailer's weekly holding cost rate
O m :	manufacturer's ordering cost per order
O r :	retailer's ordering cost per order
p s :	manufacturer's unit purchase cost
T c :	unit transportation cost
M c :	unit manufacturing or procurement cost for the manufacturer
∏ m :	manufacturer's net profit in the planning horizon
∏ r :	retailer's net profit in the planning horizon

TABLE 1 Best and Average Results Based on BPSO

Test problems	Best results		Average results
	f 1	f 2	f 1	f 2
Problem 1	97.7092	−47.8544	88.0164	−45.8822
Problem 2	10.9967	−10.9967	10.9471	−10.9471
Problem 3	15.7157	−3.9211	15.6553	−3.9343
Problem 4	29.7327	−3.0834	29.0413	−3.2731

TABLE 2 Comparisons of the Best Results Based on Different Algorithms

Test problems		BPSO		GA		PSO
		Best solutions	Best values	Best solutions	Best values	Best solutions	Best values
Problem 1	f 1	x = (13.7128,11.3775)	97.7092	x = (17.4582,10.9055)	85.0551	x = (17.4535,10.9070)	85.0700
	f 2		−47.8544		−50.16937		−50.1745
Problem 2	f 1	x = (15.7590,10.9967)	10.9967	x = (15.9984,10.9968)	10.9968	x = (15.9999,10.9998)	10.9998
	f 2		−10.9967		−10.9968		−10.9998
Problem 3	f 1	x = (3.9524,3.9211)	15.7157	x = (3.9994,3.9974)	15.9917	x = (4,4)	16
	f 2		−3.9211		−3.94636		−4
Problem 4	f 1	x = (0.1932,0.3920) y = (0.5424,0.6753,0.4443)	29.7327	x = (0.000,0.898) y = (0.000,0.599,0.399)	29.1480	x = (0.0004,0.8996) y = (0,0.5995,0.3993)	29.1788
	f 2		−3.0834		−3.1930		−3.1977

TABLE 3 Comparisons of the Average Results Based on Different Algorithms

Test problems		Average values
		BPSO	GA	PSO
Problem 1	f 1	88.0164	84.657812	84.85119
	f 2	−45.8822	−50.03023	−50.0781
Problem 2	f 1	10.9471	10.8082	10.9961
	f 2	−10.9471	−10.8082	−10.9961
Problem 3	f 1	15.6553	15.82567	15.98811
	f 2	−3.9343	−3.946363	−3.996343
Problem 4	f 1	29.0413	21.52948	24.81256
	f 2	−3.2731	−3.39072	−3.1977

TABLE 4 Comparisons of the Best Results Based on Different Algorithms

Test problems		Best results
		BPSO	HPSOBLP	TRM	Original
Problem 5	f 1	−15.2023	−14.7578	−12.68	−12.68
	f 2	1.0346	0.2067	−1.016	−1.016
Problem 6	f 1	66.9516	88.7757	88.79	88.79
	f 2	−15.8347	−0.7698	−0.77	−0.77
Problem 7	f 1	2.0191	2.0000	2	2
	f 2	23.9841	24.0190	24.02	24.02
Problem 8	f 1	1.0376	2.7039	2.75	2.75
	f 2	−0.5164	0.5602	0.57	0.57

TABLE 5 Different Results Based on Different Groups of Coefficients (k ^* = 2)

Coefficients of demand function	α	β	p m	k	p r	Q	∏ m	∏ r
a = 2, b = 600	4	4	9.8824	1.9686	19.4545	1824	119290	276100
a = 4, b = 600	4	2	9.4028	1.9422	18.2612	3425	101800	240750
a = 6, b = 600	3	4	9.7944	1.9334	18.9365	2108	100180	228820
a = 8, b = 600	3	3	9.6816	1.9353	18.7368	2601	91331	210130

TABLE 6 Different Results Based on Different Groups of Coefficients (k ^* = 5)

Coefficients of demand function	α	β	p m	k	p r	Q	∏ m	∏ r
a = 2, b = 600	3	4	9.3223	4.6885	43.7076	2221	95188	914110
a = 4, b = 600	5	2	9.8255	4.4534	43.7569	2210	90668	747830
a = 6, b = 600	5	3	9.5964	4.7733	45.8065	1127	62795	609250
a = 8, b = 600	2	2	9.8184	4.3010	42.2289	3408	54517	441050

FIGURE 2 Net profit curves of the manufacturer and the retailer under the condition of k* = 2. (Color figure available online.)

FIGURE 3 Net profit curves of the manufacturer and the retailer under the condition of k* = 5. (Color figure available online.)

FIGURE 4 Comparison of the manufacturer's net profits. (Color figure available online.)

FIGURE 5 Comparison of the retailer's net profits. (Color figure available online.)

APPENDIX 1 Linear Bilevel Programming Problems

Problem 1	Problem 2
Problem 3	Problem 4

APPENDIX 2 Nonlinear Bilevel Programming Problems

Problem 5	Problem 6
Problem 7	Problem 8