Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game

Narges Torabi Golsefid; Maziar Salahi

doi:10.3934/naco.2021032

Article Contents

2022, Volume 12, Issue 4: 719-743. Doi: 10.3934/naco.2021032

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Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game

Narges Torabi Golsefid^1, and
Maziar Salahi^2, ,

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2.
Department of Applied Mathematics, Faculty of Mathematical Sciences, and Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, Rasht, Iran

^* Corresponding author: Maziar Salahi
^* Corresponding author: Maziar Salahi

Received: February 29, 2020

Revised: May 31, 2021

Early access: August 2021

Published: December 2022

The authors would like to thank Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan for supporting this research

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Abstract

A vital issue in many organizations is the fair allocation of fixed cost among its subsets. In this paper, using data envelopment analysis, first we study fixed cost allocation based on both additive and multiplicative efficiency decompositions in the cooperative context for a two-stage structure in the presence of exogenous inputs and outputs. A conic relaxation formulation of multiplicative decomposition is given. Then, fixed cost allocation based on the leader-follower paradigm are presented. In the sequel, for allocating a fair fixed cost between the stages, using the results of the leader-follower model, we present the nonlinear Nash bargaining game model that independent of the efficiency score of each unit, allocates fixed cost to the stages. The nonlinear model is reformulated as a second order cone program which is an imporvement over the parametric linear models in the literature. Finally, two examples are used to illustrate the proposed models and compare their results with the existing models.

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Mathematics Subject Classification: Primary: 90B50; Secondary: 90C25.

Citation:

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Figure 1. Two-stage system with fixed costss

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Figure 2. The allocated cost to stages: (a) model (23) and Chu et al. [14] (b) model (23) and Li et al. [24]

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Table 1. Classification of the literature

Reference	exogenous input or output	efficiency decomposition	system	approach of fair cost allocation	allocation principles
Li et al. [25]	$ \times $	-	single-stage	Satisfaction degree	Efficiency maximization
Du et al. [18]	$ \times $	-	single-stage	Cross efficiency	Efficiency maximization
Yu et al. [37]	$ \times $	AED	two-stage	Cross efficiency	Effficiency maximization
Zhu et al. [39]	$ \checkmark $	AED	two-stage	Based on different objectives in reality	Effficiency maximization
Ding et al. [17]	$ \checkmark $	AED	two-stage	Maximal average satisfaction degree	-
An et al. [3]	$ \checkmark $	-	three-stage	The Shapley value	-
Li et al. [24]	$ \times $	AED	two-stage	By repeatedly minimizing the maximum deviation between the efficient and size allocation	Efficiency invariance & efficiency maximization
Li et al.[26]	$ \times $	-	single-stage	Cooperative game	-
Chu et al. [14]	$ \times $	AED	two-stage	Leader-follower & the Nash bargaining	-
The present study	$ \checkmark $	AED and MED	two-stage	Cooperative, leader-follower & the Nash bargaining	Independent from efficiency rank and size

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Table 2. Notations

Notation	Description
$ n $	number of DMUs
$ R $	allocated total cost to system
$ m_1 $	number of exogenous inputs of stage 1
$ m_2 $	number of exogenous inputs of stage 2
$ g_1 $	number of exogenous outputs of stage 1
$ g_2 $	number of exogenous outputs of stage 2
$ l $	number of intermediate products
$ x_{ij}^1 $	$ i $th input of DMU$ j $ of stage 1
$ x_{hj}^2 $	$ h $th input of DMU$ j $ of stage 2
$ r_{j}^1 $	allocated cost to DMU$ j $ of stage 1
$ r_{j}^2 $	allocated cost to DMU$ j $ of stage 2
$ y_{pj}^1 $	$ p $th output of DMU$ j $ of stage 1
$ y_{fj}^2 $	$ f $th output of DMU$ j $ of stage 2
$ z_{dj} $	$ d $th intermediate product of DMU$ j $
$ v_i $	weight assigned to the $ i $th input of stage 1
$ q_h $	weight assigned to the $ h $th input of stage 2
$ \mu_p $	weight assigned to the $ p $th output of stage 1
$ u_f $	weight assigned to the $ f $th output of stage 2
$ \gamma_d^1 $	weight assigned to the $ d $th intermediate product of stage 1
$ \gamma_d^2 $	weight assigned to the $ d $th intermediate product of stage 2

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Table 3. Data of the 27 bank branches

branches	Inputs			Exogenous input	Intermediate products		Exogenous output	Outputs
1	25	619	538	854	77237	34224	2101	2947	913	224
2	27	419	489	125	88031	56559	1023	3138	478	516
3	40	1670	1459	120	164053	62776	1440	5494	1242	877
4	42	2931	1497	86	145369	65226	2458	3144	870	1138
5	52	2587	797	133	166424	85886	2202	6705	854	618
6	45	2181	697	149	215695	30179	1653	8487	1023	2096
7	33	989	1217	144	114043	43447	1919	4996	767	713
8	107	6277	2189	735	727699	294126	2486	21265	6282	6287
9	88	3197	949	101	186642	53223	648	8574	1537	1739
10	146	6222	1824	399	614241	121784	2007	21937	8008	3261
11	57	1532	2248	83	241794	83634	626	8351	1530	2011
12	42	1194	1604	447	150707	57875	1538	5594	858	1203
13	132	5608	1731	141	416754	168798	1263	15271	4442	2743
14	77	2136	906	145	276379	38763	2686	10070	2445	1487
15	43	1534	438	750	133359	48239	538	4842	1172	1355
16	43	1711	1069	106	157275	27004	2419	6505	1469	1217
17	59	3686	820	119	150827	60244	2927	6552	1209	1082
18	33	1479	2347	41	215012	78253	2975	8624	894	2228
19	38	1822	1577	232	192746	76284	2472	9422	967	1367
20	162	5922	2330	148	533273	163816	1597	18700	4249	6545
21	60	2158	1153	180	252568	77887	1745	10573	1611	2210
22	56	2666	2683	469	269402	158835	1035	10678	1589	1834
23	71	2969	1521	65	197684	100321	2108	8563	905	1316
24	117	5527	2369	175	406475	106073	1300	15545	2359	2717
25	78	3219	2738	661	371847	125323	2900	14681	3477	3134
26	51	2431	741	164	190055	142422	2316	7964	1318	1158
27	48	2924	1561	183	332641	94933	1529	11756	2779	1398

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Table 4. Cost allocation results based on cooperative models

	Model (3)	Model (6)	Model (3)			Model (6)
branches	$ E $	$ E $	$ r_j^1 $	$ r_j^2 $	$ r_j^1 + r_j^2 $	$ r_j^1 $	$ r_j^2 $	$ r_j^1+r_j^2 $
1	1(1)	1(1)	246.9783	229.0669	476.0452(22)	249.9640	241.3908	491.3548(26)
2	0.8798(14)	0.8152(14)	247.3826	256.3008	503.6833(12)	250.6172	248.6579	499.2751(14)
3	0.7632(23)	0.5949(22)	245.3606	263.9743	509.3349(8)	249.0929	248.1273	497.2202(24)
4	0.7382(24)	0.5849(24)	244.5442	261.6055	506.1497(11)	249.7255	248.2464	497.97198(20)
5	0.8589(18)	0.7690(16)	244.4590	256.2162	500.6752(16)	251.6313	248.5930	500.2244(10)
6	1(1)	1(1)	248.9866	274.3444	523.3309(4)	250.5248	250.8240	501.3489(7)
7	0.8644(16)	0.7747(15)	245.7330	256.5588	502.2918(14)	249.5825	248.5230	498.1055(19)
8	0.9617(4)	0.9546(4)	226.5395	244.7763	471.3158(25)	255.3597	252.1110	507.4707(1)
9	0.7943(21)	0.5885(23)	245.7459	255.0998	500.8457(15)	250.7254	248.7602	499.4857(12)
10	0.9336(8)	0.8670(10)	235.6404	295.7199	531.3602(3)	255.4656	249.2222	504.6878(4)
11	0.8978(12)	0.8599(11)	244.5437	232.8622	477.4059(20)	248.6555	248.8462	497.5017(21)
12	0.8010(20)	0.69733(19)	245.1240	230.6952	475.8192(23)	249.2772	248.0615	497.3387(23)
13	0.8624(17)	0.7247(18)	237.5553	280.0552	517.6105(5)	252.4906	248.5284	501.0190(8)
14	1(1)	1(1)	249.5513	265.0496	514.6009(6)	256.0044	248.5349	504.5393(5)
15	0.9587(5)	0.9166(7)	246.8357	228.7804	475.6061(24)	248.9389	243.5603	492.4992(25)
16	0.9172(11)	0.8314(13)	249.2900	260.6817	509.9717(7)	251.1524	248.2845	499.4369(13)
17	0.9685(2)	0.9580(3)	244.1696	264.5992	508.7689(9)	250.3736	248.2632	498.6368(17)
18	1(1)	1(1)	245.3942	215.7705	461.1648(26)	249.4011	250.3788	499.7799(11)
19	0.9638(3)	0.9277(5)	244.6776	253.1032	497.7808(17)	249.8272	248.9734	498.8005(16)
20	0.8664(15)	0.7326(17)	236.1256	304.6363	540.7618(2)	251.2982	252.9057	504.2039(6)
21	0.8936(13)	0.8452(12)	244.6708	263.000	507.6708(10)	249.7751	248.5778	498.3529(18)
22	0.9210(10)	0.9234(6)	242.6463	245.2438	487.8901(18)	255.0194	251.9952	507.0145(2)
23	0.8016(19)	0.6033(21)	243.8905	259.6044	503.4949(13)	252.5173	248.3064	500.8237(9)
24	0.7912(22)	0.6329(20)	240.8019	300.7017	541.5036(1)	250.0492	248.9657	499.0150(15)
25	0.9220(9)	0.8732(8)	243.2701	232.7953	476.0654(21)	250.1477	247.1931	497.3408(22)
26	0.9511(6)	0.9983(2)	244.5997	233.8111	478.4109(19)	256.7994	249.7160	506.5153(3)
27	0.9353(7)	0.8705(9)	0.2413	0.2289	0.4702(27)	0.0355	0.0015	0.0370(27)

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Table 5. Cost allocation results arising from leader-follower and Nash game theory

	Stage 1 as the leader		Stage 1 as the follower		Model (23)
branches	$ r_j^1 $	$ r_j^2 $	$ r_j^1 $	$ r_j^2 $	$ r_j^1 $	$ r_j^2 $	$ r_j^1+r_j^2 $
1	0	135.7885	521.7512	0	260.8756	67.8943	328.7698 (21)
2	0	140.1356	260.8284	0	130.4142	70.0678	200.4820 (27)
3	0	214.8582	357.783	0	178.8915	107.4291	286.3206(24)
4	0	0.0000	602.2915	0	301.1457	0.0000	301.1457(23)
5	0	341.3245	562.1265	0	281.0632	170.6623	451.7255(17)
6	0	476.9271	441.9432	0	220.9715	238.4636	459.4352(15)
7	0	295.2004	469.1469	0	234.5734	147.6002	382.1736(19)
8	0	687.6763	735.7962	0	367.8980	344.8382	712.7362(4)
9	0	556.8296	181.5706	0	90.7853	278.4148	369.2001(20)
10	0	967.3349	599.2162	0	299.6080	483.6675	783.2756(1)
11	0	379.7576	157.2898	0	78.6449	189.8788	268.5237(25)
12	0	281.0507	374.9296	0	187.4647	140.5254	327.9901 (22)
13	0	726.8723	369.9592	0	184.9796	363.4362	548.4158(10)
14	0	460.0660	705.4757	0	352.7378	230.0330	582.7708(7)
15	0	248.9499	154.1661	0	77.0830	124.4750	201.5580(26)
16	0	371.5729	606.3652	0	303.1825	185.7865	488.9690(14)
17	0	390.0909	735.7962	0	367.8980	195.0455	562.9435(9)
18	0	501.3772	725.2381	0	362.6190	250.6887	613.3076(6)
19	0	623.6072	615.4395	0	307.7197	311.8037	619.5234(5)
20	0	967.3349	465.8196	0	232.9097	483.6675	716.5773(3)
21	0	623.5219	462.2460	0	231.1230	311.7610	542.8840(11)
22	0	569.3522	253.7010	0	126.8505	284.6762	411.5266(18)
23	0	503.6960	528.6428	0	264.3214	251.8480	516.1694(13)
24	0	788.8750	360.5408	0	180.2704	394.4376	574.7079(8)
25	0	806.3711	735.7962	0	367.8980	403.1856	771.0837 (2)
26	0	447.1736	597.3741	0	298.6870	223.5869	522.2738(12)
27	0	492.2555	418.7666	0	209.3832	246.1278	455.5110(16)

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Table 6. Evaluaition results with AED of [14] and models (3) and (6)

	Stage 1 of (3)	Stage 2 of (3)	AED of [14]	Model (3)	Model (6)	Model(3)	Model(6)
branches	$ E^1 $	$ E^2 $	$ E $	$ E $	$ E $	$ r^1_j+r^2_j $	$ r^1_j+r^2_j $
1	0.7597	1	0.8634(13)	0.8799(14)	0.7884(16)	306.9362	306.6713
2	1	0.7595	0.8798(11)	0.8798(13)	0.8375(12)	299.0361	307.4420
3	0.7204	0.7839	0.7470(21)	0.7521(24)	0.5810(23)	305.4121	306.6072
4	0.5595	0.7034	0.5726(27)	0.6314(27)	0.4292(27)	300.6593	306.9801
5	0.6884	0.8412	0.7102(24)	0.7848(22)	0.5998(21)	298.6402	307.7304
6	0.9994	0.9996	0.9304(5)	0.9995(2)	0.9979(4)	295.3327	308.3022
7	0.7285	0.9343	0.8152(17)	0.8314(19)	0.7136(19)	308.9644	306.5066
8	0.9983	0.9565	0.9762(2)	0.9774(4)	0.9544(5)	294.8752	310.4954
9	0.5872	1	0.7399(22)	0.7936(21)	0.5872(22)	306.4590	307.0616
10	1	0.8625	0.9225(8)	0.9313(8)	0.8624(9)	453.1254	309.8544
11	0.9380	0.8120	0.8770(12)	0.8750(15)	0.8297(13)	307.0382	306.8330
12	0.7732	0.8288	0.7975(19)	0.8010(20)	0.6945(20)	307.1054	306.7443
13	0.7248	1	0.8402(15)	0.8624(17)	0.7247(18)	299.7402	308.6713
14	1	1	0.9499(4)	1(1)	1(1)	289.1324	308.8396
15	0.9136	0.9942	0.9513(3)	0.9538(5)	0.9128(7)	304.4242	306.7419
16	0.6846	1	0.8023(18)	0.8423(18)	0.8124(14)	311.8755	306.8844
17	0.5531	0.9534	0.6934(25)	0.7533(23)	0.5380(25)	302.9726	306.6634
18	1	0.9752	0.9876(1)	0.9876(3)	0.9997(2)	307.5180	306.7759
19	0.8078	1	0.9036(10)	0.9039(11)	0.8090(15)	308.7589	306.9259
20	0.7328	1	0.8458(14)	0.8664(16)	0.7328(17)	309.7008	307.5808
21	0.8525	0.9232	0.8850(10)	0.8878(12)	0.8454(11)	306.4648	306.8614
22	1	0.8421	0.7935(20)	0.9210(10)	0.9253(6)	286.0109	309.9431
23	0.5549	0.8946	0.6407(26)	0.7247(26)	0.5231(26)	298.3907	308.0263
24	0.5918	0.9036	0.7177(23)	0.7477(25)	0.5791(24)	300.7911	307.3677
25	0.8710	0.9938	0.9239(7)	0.9324(7)	0.8731(7)	308.9031	307.0074
26	1	0.9022	0.8276(16)	0.9511(6)	0.9989(3)	281.1311	310.4377
27	1	0.8505	0.9241(6)	0.9252(9)	0.8504(10)	0.6013	0.0448

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Table 7. Comparison results

	Model (23)			Chu et al. [14]			Li et al. [24]
branches	$ r_j^1 $	$ r_j^2 $	$ r_j^1+r_j^2 $	$ r_j^1 $	$ r_j^2 $	$ r_j^1+r_j^2 $	$ r_j^1 $	$ r_j^2 $	$ r_j^1+r_j^2 $
1	44.2099	41.7809	86.2678	29.2523	54.8286	84.0809	34.8394	99.5911	134.4306
2	50.3884	43.1185	93.5068	33.3404	61.0391	94.3795	57.6112	4.1483	61.7595
3	93.9028	56.1099	160.0128	62.1326	102.6452	164.7778	66.9629	83.9139	150.8768
4	83.2082	0.0000	83.2082	55.0563	50.3562	105.4125	42.5563	5.3306	47.8869
5	95.2600	105.0225	200.2825	63.0305	130.1635	193.1941	105.6896	19.9776	125.6671
6	123.4624	146.7462	270.2086	81.6912	185.8070	267.4982	149.2000	46.4915	195.6915
7	65.2774	90.8305	156.1080	43.1920	105.1728	148.3648	27.0534	58.2580	85.3113
8	416.5300	212.2072	628.7372	275.6048	412.7297	688.3345	662.6315	455.6247	1118.2562
9	106.8326	171.3315	278.1641	70.6878	192.9383	263.626	55.8416	158.4308	214.2724
10	351.5874	297.6403	649.2277	232.6343	421.0993	653.7337	428.1954	448.9607	877.1561
11	138.4013	116.8480	255.2492	91.5758	170.7374	262.3132	91.0412	79.0747	170.1160
12	86.2637	86.4768	172.7404	57.0780	116.1171	173.1951	38.3474	35.4254	73.7728
13	238.5472	223.6521	462.1993	157.8392	305.5089	463.3481	257.0286	451.538	708.5666
14	158.1975	141.5582	299.7556	104.6743	195.2432	299.9174	164.9817	245.7357	410.7174
15	76.3338	76.5996	152.9334	50.5077	104.8872	155.3949	85.7256	103.1206	188.8462
16	90.0231	114.3296	204.3528	59.5655	138.6559	198.2214	62.9564	161.9881	224.9445
17	86.3324	120.0275	206.3598	57.1234	140.0173	197.1407	65.8529	108.8606	174.7134
18	123.0714	154.2693	277.3407	81.4325	192.0992	273.5317	84.4604	4.1483	88.6087
19	110.3265	191.8783	302.2048	72.9996	207.1934	280.193	100.2016	53.5750	153.7766
20	305.2419	297.6403	602.8821	201.9689	421.0993	623.0683	306.3559	340.4241	646.7800
21	144.5682	191.8521	336.4203	95.6562	230.5790	326.2353	164.1695	115.4121	279.5816
22	154.2039	175.1846	329.3884	102.0319	220.4889	322.5207	131.0596	56.5463	187.6059
23	113.1530	154.9827	268.1357	74.8698	181.0649	255.9346	76.9843	13.9235	90.9078
24	232.6635	242.7298	475.3933	153.9462	316.4011	470.3472	196.4414	133.2967	329.7382
25	212.8427	248.1131	460.9559	140.8313	313.3001	454.1314	195.7660	344.9362	540.7023
26	108.7862	137.5913	246.3775	71.9804	164.8987	236.8791	163.0222	66.0493	229.0715
27	190.4015	151.4626	341.8641	125.9827	218.2432	344.2259	250.7998	239.4424	490.2422

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Table 8. The characteristics of two different techniques

Model	Model(23)	Li et al. [24]
Technique	Bargaining game	Goal programming
Max-Cost(Efficiency rank)	DMU 10 (7)	DMU 8 (4)
Min-Cost(Efficiency rank)	DMU 4 (27)	DMU 4 (27)
Mean cost (stage1, stage2)	(148.1488,148.1475)	(150.5813,145.712)

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Table 9. CPU time of SOCP model (23) and Chu et al. [14] model

	Model(23)	Chu et al. [14]	Chu et al. [14]	Chu et al. [14]
Model	$ (\gamma_d^1=\gamma_d^2) $	(100 iterations)	(500 iterations)	(1000 iterations)
Time (s)	1.7	109.7	515.7	1000
$ s_1 $	0.52	0.52	0.52	0.52
$ s_2 $	0.51	0.54	0.53	0.53
Mean cost
(stage1, stage2)	(97.9866,198.3097)	(98.6419,197.6544)	(97.8199,198.4764)	(98.0254,198.2709)

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