Performance evaluation of a multi-product system under CONWIP control

Figures & data

Fig. 1 Queuing model of multi-product CONWIP system.

Fig. 2 Characterization of fork/join synchronization stations J _r,i .

Fig. 3 Characterization of manufacturing station m.

Fig. 4 Equivalent single-class networks defined for determination of f _r,m .

Table 1 Results for Experiment 1: results are shown for the case where μ _r,m = 6, ∀ r,m; K _r,0 = K _r,2 = 4; λ _{P r,0} = 6

					Throughput			Mean queue length									Backorders
	(K r , K 2, K 3)	λ F r,2	c P r,0 ^2 = c F r,2 ^2 = c s r,m ^2		λ1	λ2	λ3	1,1	2,1	3,1	1,2	2,2	3,2	1,3	2,3	3,3	F 1,2	F 2,2	F 3,2
1	(4, 4, 4)	1.5	0.7	A	1.486	1.486	1.486	0.486	0.486	0.486	0.493	0.493	0.493	0.496	0.496	0.496	0.332	0.332	0.332
				S	1.490	1.491	1.495	0.678	0.678	0.680	0.687	0.686	0.688	0.696	0.696	0.697	0.185	0.180	0.185
				E	−0.271	−0.309	−0.613	−4.800	−4.800	−4.850	−4.850	−4.825	−4.875	−5.000	−5.000	−5.025	3.687	3.804	3.687
2	(4, 4, 4)	1.5	2	A	1.413	1.413	1.413	0.676	0.676	0.676	0.739	0.739	0.739	0.770	0.770	0.770	0.982	0.982	0.982
				S	1.360	1.370	1.371	0.678	0.678	0.680	0.687	0.686	0.688	0.696	0.696	0.697	0.916	0.915	0.917
				E	3.884	3.076	3.037	−0.050	−0.050	−0.100	1.300	1.325	1.275	1.850	1.850	1.825	1.652	1.683	1.616
3	(4, 4, 4)	1.75	1	A	1.633	1.633	1.633	0.785	0.785	0.785	0.847	0.847	0.847	0.868	0.868	0.868	1.118	1.118	1.118
				S	1.620	1.622	1.618	1.015	1.017	1.015	1.054	1.053	1.052	1.076	1.078	1.076	0.999	0.994	0.989
				E	0.837	0.683	0.970	−5.750	−5.800	−5.750	−5.175	−5.150	−5.125	−5.200	−5.250	−5.200	2.964	3.093	3.031
4	(4, 4, 4)	1.75	2	A	1.520	1.520	1.520	0.874	0.874	0.874	0.979	0.979	0.979	1.023	1.023	1.023	1.557	1.557	1.557
				S	1.458	1.455	1.458	1.028	1.028	1.029	1.071	1.070	1.072	1.085	1.085	1.086	1.437	1.435	1.452
				E	4.220	4.434	4.238	−3.850	−3.850	−3.875	−2.300	−2.275	−2.325	−1.550	−1.550	−1.575	2.998	3.049	2.633
5	(4, 4, 4)	2	0.7	A	1.761	1.761	1.761	0.935	0.935	0.935	1.025	1.025	1.025	1.045	1.045	1.045	1.519	1.519	1.519
				S	1.760	1.758	1.761	1.210	1.209	1.211	1.233	1.234	1.234	1.238	1.238	1.239	1.603	1.619	1.607
				E	0.064	0.169	0.001	−6.875	−6.850	−6.900	−5.200	−5.225	−5.225	−4.825	−4.825	−4.850	−2.097	−2.512	−2.209
6	(4, 6, 5)	2	0.7	A	1.669	1.882	1.808	1.015	1.144	1.100	1.115	1.257	1.208	1.136	1.280	1.230	1.826	0.917	1.302
				S	1.611	1.931	1.825	1.245	1.539	1.433	1.276	1.583	1.474	1.283	1.601	1.486	2.089	0.675	1.265
				E	3.621	−2.541	−0.926	−5.750	−6.583	−6.660	−4.025	−5.433	−5.320	−3.675	−5.350	−5.120	−6.587	6.040	0.915
7	(3, 3, 3)	6	2	A	1.563	1.563	1.563	0.915	0.915	0.915	1.020	1.020	1.020	1.061	1.061	1.061	3.692	3.692	3.692
				S	1.466	1.466	1.466	0.996	0.997	0.996	1.000	1.000	1.001	1.004	1.002	1.002	3.667	3.664	3.664
				E	6.585	6.585	6.585	−2.700	−2.733	−2.700	0.667	0.667	0.633	1.900	1.967	1.967	0.635	0.691	0.709
8	(4, 6, 5)	6	2	A	1.336	2.002	1.670	1.236	1.852	1.545	1.359	2.036	1.698	1.396	2.092	1.745	3.732	3.601	3.652
				S	1.296	1.944	1.620	1.335	2.001	1.667	1.329	1.994	1.663	1.336	2.005	1.670	3.708	3.466	3.596
				E	3.076	2.959	3.051	−2.475	−2.483	−2.440	0.750	0.700	0.700	1.500	1.450	1.500	0.608	3.366	1.392
A: Analytical; S: Simulation, E: % Error.

Fig. 5 Performance insights using the analytical method.

Fig. 6 Queuing network model for system in Experiment 2 with 12 machines and five products.

Table 2 Routing and service process parameters for network in Experiment 2

	Machine service rates and SCV at each machine: (μ r,m , c ^2 sr,m )
	1	2	3	4	5	6	7	8	9	10	11	12
Product 1	(1,1)	(0.5, 1)	(1,1)		(1, 0.5)	(1.33, 8.5)	(2, 1)	(1, 1)	(0.8, 1)		(4, 1)	(1.33, 1)
Product 2	(2,1)		(1,1)		(1, 5)	(2.5, 2.5)		(1, 1)	(0.8, 1)		(0.5, 1)	(2, 1)
Product 3	(4,1)		(1,1)	(2, 1)	(2, 2.2)	(1.11, 2.1)				(0.67, 1)		(1, 1)
Product 4	(10,1)	(0.5, 1)			(0.5, 5)		(2, 1)				(1, 1)
Product 5	(1.33,1)			(2, 1)		(0.8, 5.8)			(0.8, 1)			(4, 1)

Table 3 Performance results for Experiment 2

	λ1	λ2	λ3	λ4	λ5
Case 1
A	0.123	0.164	0.227	0.214	0.304
S	0.117	0.180	0.253	0.238	0.322
E	5.119	−8.791	−10.313	−10.319	−5.624
Case 2
A	0.150	0.195	0.192	0.228	0.302
S	0.151	0.210	0.221	0.248	0.314
E	−0.912	−7.160	−13.397	−8.300	−4.104
Case 3
A	0.153	0.190	0.231	0.227	0.365
S	0.151	0.203	0.256	0.253	0.384
E	1.463	−6.392	−9.897	−10.200	−4.760
A: Analytical, S: Simulation, E: %Error.

Table 4 Performance results for Experiment 3.1: comparison with Baynat and Dallery (Baynat and Dallery, 1966)

(K 1,K 2,K 3,K 4,K 5)		λ1	λ2	λ3	λ4	λ5
(6, 8, 10, 7, 9)	Baynat and Dallery	0.195	0.352	0.439	0.241	0.431
	This paper	0.182	0.329	0.411	0.211	0.422
	Simulation	0.182	0.327	0.433	0.237	0.410
(12, 15, 11, 13, 10)	Baynat and Dallery	0.237	0.460	0.364	0.240	0.403
	This paper	0.226	0.434	0.318	0.218	0.411
	Simulation	0.227	0.430	0.360	0.237	0.387
(16, 20, 18, 14, 19)	Baynat and Dallery	0.250	0.445	0.441	0.227	0.444
	This paper	0.243	0.425	0.391	0.205	0.467
	Simulation	0.243	0.423	0.434	0.226	0.431

Table 5 Performance results for Experiment 3.2: comparison with Ryan et al. (2000)

(K 1, K 2)		Network Throughput
(5, 10)	Ryan et al.	0.253
	This Paper	0.243
	Simulation	0.247
(7, 16)	Ryan et al.	0.261
	This Paper	0.253
	Simulation	0.255
(8,19)	Ryan et al.	0.263
	This Paper	0.256
	Simulation	0.258
(8,10)	Ryan et al.	0.277
	This Paper	0.269
	Simulation	0.271

Ryan , S. M. , Baynat , B. and Choobineh , F. 2000 . Determining inventory levels in a CONWIP controlled job shop . IIE Transactions , 32 : 105 – 114 .

 Web of Science ®Google Scholar