A new efficiency evaluation approach with rough data: An application to Indian fertilizer

Alka Arya; Adel Hatami-Marbini

doi:10.3934/jimo.2022168

Article Contents

2023, Volume 19, Issue 7: 5183-5208. Doi: 10.3934/jimo.2022168

This issue Previous Article Abatement technology innovation modelling with spillover effects under different market structures Next Article Hybrid dynamic optimal tracking control of hydraulic cylinder speed in injection molding industry process

A new efficiency evaluation approach with rough data: An application to Indian fertilizer

Alka Arya^1, , and
Adel Hatami-Marbini^2,

1.
Department of Operations Management & Decision Sciences, Indian Institute, of Management Kashipur, Uttarakhand, India
2.
School of Leadership, Management and Marketing, De Montfort University, Hugh Aston Building, The Gateway, Leicester LE1 9BH, UK

^*Corresponding author: Alka Arya
^*Corresponding author: Alka Arya

The authors would like to thank the handling editor and two anonymous reviewers for their valuable and constructive comments

Received: January 2022

Revised: June 2022

Early access: September 2022

Published: July 2023

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Abstract

In the world of chaos, nothing is certain. In such an unpredictable world, measuring the efficiency of any individual is inevitable. In a conventional data envelopment analysis (DEA) model, exact input and output quantity data are needed to measure the relative efficiencies of homogeneous decision-making units (DMUs). However, in many real-world applications, the exact knowledge of data might not be available. The rough set theory allows for handling this type of situation. This paper tries to construct a rough DEA model by combining conventional DEA and rough set theory using optimistic and pessimistic $ \beta $ confidence values of rough variables, all of which help provide a way to quantify uncertainty. In the proposed method, the same set of constraints (production possibility sets) is employed to build a unified production frontier for all DMUs that can be used to properly assess each DMU's performance in the presence of rough input and output data. Besides, a ranking system is presented based on the approaches that have been proposed. In the presence of uncertain conditions, this article investigates the efficiency of the Indian fertilizer supply chain for over a decade. The results of the proposed models are compared to the existing DEA models, demonstrating how decision-makers can increase the supply chain performance of Indian fertilizer industries.

Keywords:

Mathematics Subject Classification: Primary: 90C08, 60L99; Secondary: 90B06.

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Figure 1. Foodgrains and fertilizers data

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Figure 2. Production frontier

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Figure 3. inputs-outputs of the fertilizer supply chain

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Table 1. DEA in SCM

Method	Data type	Application	Reference	Year
DEA	precise	Supply chain	[67]	1999
DEA network	precise	Swedish pharmacies	[42]	1999
DEA	precise	Supply chain	[46]	2001
DEA	precise	Petroleum industry	[19]	2002
DEA	precise	Taiwan hotels	[25]	2003
DEA	precise	Power plants	[34]	2004
DEA	precise	Automotive manufacturer	[12]	2008
Rough DEA	uncertain	Furniture industry	[73]	2009
DEA	precise	Enterprise Resource Planning	[10]	2009
Bilevel DEA	precise	Banking and manufacturing	[72]	2010
DEA	precise	Korean electronics industry	[37]	2012
DEA	precise	Green supply chain	[45]	2014
DEA	precise	Hyundai Steel Company and its suppliers	[43]	2015
DEA	uncertain	Indian fertilizer supply chain	[35]	2017
Hybrid DEA	precise	Biodiesel supply chain	[8]	2017
Network DEA	precise	Tourism supply chain	[24]	2018
Bootstrap DEA	precise	Healthcare supply chain	[33]	2019
DEA	precise	Sustainability supply chain	[16]	2020
DEA	precise	Fairness in supply chain	[77]	2021
DEA	uncertain	Three-level supply chain	[22]	2022
DEA	precise	fashion company	[55]	2022

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Table 2. Algorithm based on optimistic and pessimistic

Algorithm based on optimistic	Algorithm based on pessimistic
Step O.1. Run model (4.3) for each DMU using rough input-output data and determine $ E_{k}^{\sup(\beta)*} $, $ \beta\in [0.6,1] $.	Step P.1. Run model (4.4) for each DMU using rough input-output data and identify $ E_{k}^{\inf(\beta)*},\beta \in[0.6,1] $.
Step O.2. Let $ I=\{1,2,...,n\} $ be an index set. Let $ S^{\beta}=\{DMU_j:E_{k}^{\sup(\beta)}=1, \beta \in[0.6,1],j\in \bar{I}\} $ and $ S_1^{\beta}=\{DMU_j:E_{k}^{\sup(\beta)}<1, \beta \in[0.6,1],j\in I-\bar{I}\}. $	Step P.2. Let $ I=\{1,2,...,n\} $ be an index set. Let $ T^{\beta}=\{DMU_j:E_{k}^{\inf(\beta)}=1, \beta\in[0.6,1],j\in \bar{I}\} $ and $ T_1^{\beta}=\{DMU_j:E_{k}^{\inf(\beta)}<1, \beta\in[0.6,1],j\in I-\bar{I}\}. $
Step O.3. Run model (4.11) for $ DMU_j\in S^{\beta} $.	Step P.3. Run model (4.12) for $ DMU_j\in T^{\beta} $.

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Table 3. Rough input data of 17 fertilizer supply chains

Supply chains	Equity capital* (I1)	Expense* (I2)	Asset* (I3)
F1	([7509.2, 7509.2], [7509.2, 7509.2])	([240.5, 603.8], [211.5, 736.8])	([4373.5, 6321.3], [3862.9, 6382.1])
F2	([289.4,289.4], [289.4, 289.4])	([20206.1, 22745.8], [17372.7, 23111.5])	([16753.3, 23778.3], [14329.7, 23826.8])
F3	([1554.2, 1554.2], [1554.2, 1554.2])	([49408.6, 55172.2], [49380.9, 57958.6])	([79968.6, 87233.5], [76429.3, 106448.6])
F4	([797, 797], [797, 797])	([60184.5, 75291.4], [52649.7, 86734.3])	([94936, 107283.5], [86621.2, 109167.3])
F5	([6865.4, 6865.4], [6865.4, 6865.4])	([213.1, 3886], [55, 3887.4])	([2376.9, 2537.5], [1959.2, 2546.5])
F6	([0.5, 0.5], [0.5, 0.5])	([24.5, 52], [21.9, 75.5])	([103, 174.3], [94.4, 174.4])
F7	([1185.5, 1185.5], [1185.5, 1185.5])	([25125.8, 31324.5], [24489.5, 31515])	([26084.9, 29497.8], [23998.3, 30706.6])
F8	([598.1, 598.1], [598.1, 598.1])	([24220.7, 39241.2], [22108.4, 41624])	([42063.1, 48246.3], [38859.1, 55072.3])
F9	([4905.8, 4905.8], [4905.8, 4905.8])	([82817.4, 129528], [78476.1, 132068.2])	([105554.7, 143484.9], [100050.6, 149129.1])
F10	([5516.9, 5516.9], [5516.9, 5516.9])	([74661.9, 90973.8], [70526.9, 95530.9])	([66844.3, 91464.8], [65371.9, 104968.3])
F11	([4162.1, 4162.1], [4162.1, 4162.1])	([74774.8, 104192.5], [71595, 111565.3])	([81230.6, 161534.8], [79681.5, 168028])
F12	([291.7, 292.5], [291.3, 293])	([106353.4, 118939.7], [94563, 133142.9])	([91745.1, 104428.4], [87303.5, 108638])
F13	([4208.5, 4249.8], [4205.5, 6275.7])	([224320.6, 289446.7], [221938.8, 295581.9])	([229687.6, 305328.3], [208104.5, 353605.4])
F14	([5754.5, 5754.5], [5754.5, 5754.5])	([36494.8, 48006.1], [36223.8, 48545.5])	([41315.1, 49726.4], [35857.2, 56277.5])
F15	([10, 170.5], [10, 170.5])	([927.3, 30459.4], [442.9, 39144.7])	([1204, 53449.8], [755.5, 60297.2])
F16	([2036.4, 2036.4], [2036.4, 2036.4])	([18625.4, 21720.7], [15415.4, 26060.8])	([14791.5, 20521], [12659.8, 21011.4])
F17	([2548.2, 2548.2], [2548.2, 2548.2])	([51823, 79787.4], [27860.5, 96902.6])	([132773.1, 143680], [129836.8, 153486])
^* Million Indian rupees

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Table 4. Rough output data of 17 fertilizer supply chains

Supply chains	Profit after tax* (O1)	Sale* (O2)	Production** (O3)
F1	([301, 1647.8], [86.5, 1909.8])	([3.1, 11.675], [1.3, 25.1])	([7.78, 8.52], [6.08, 10.72])
F2	([96.4, 200.3], [18.4, 342])	([20148.8, 21829.4], [16663.3, 22826])	([5.16, 5.53], [4.02, 5.83])
F3	([4988.5, 7411.7], [1726.8, 7895.2])	([50137, 61345.1], [48489, 61863.7])	([8.66, 8.98], [8.42, 9.11])
F4	([4093.5, 4757.3], [987, 4936.8])	([61559.3, 76279.7], [54825.3, 85836.4])	([14.6, 16.47], [14.41, 16.9])
F5	([-3805, 221.7], [-3807.5, 93401.4])	([19.25, 21], [ 17, 21.6])	([2.3, 3.03], [2.05, 3.24])
F6	([-9.5, -2.4], [-11.3, -2.2])	([18.2, 43.9], [12, 68.7])	([11.37, 11.84], [11.15, 12.08])
F7	([194.1, 605.8], [-2401.3, 645.5])	([26937.4, 29820.8], [24948.1, 30743.9])	([6.42, 6.73], [5.85, 6.85])
F8	([-4723.5, -927.4], [-4907.3, -215.2])	([8188.1, 16560.4], [4611.5, 39776.5])	([3.65, 7.11], [1.99, 7.92])
F9	([1986.2, 2127.7], [-1710.1, 2984.5])	([78146.3, 122427.9], [76619.8, 128111.3])	([37.97, 38.1], [37.27, 39.61])
F10	([1391.7, 1792.6], [788, 2081.5])	([72511.5, 88707.2], [36887.3, 94173.6])	([29.37, 30.17], [29.18, 31.87])
F11	([4251, 5452.7], [-111.4, 12243.1])	([74678.2, 100961.4], [74323.4, 122088.8])	([20.94, 25.03], [20.02, 32.66])
F12	([4767.8, 7139.1], [3578.6, 10591.7])	([101767.1, 131165.2], [82741, 131922.4])	([22.71, 27.27], [22.21, 27.55])
F13	([6847, 8415.8], [ 6368.7, 9371.7])	([158206, 222900.2], [124865, 279049.9])	([64.84, 69.25], [61.39, 72.02])
F14	([650.9, 1505.9], [433.3, 1590.5])	([30863.3, 41876.1], [24684.2, 43579.1])	([12.02, 13.15], [10.63, 13.21])
F15	([22.8, 270.1], [-60.9, 545.9])	([950, 30707.2], [375.1, 37593.8])	([2.54, 5.04], [0.89, 5.65])
F16	([263.5, 533.4], [247.7, 569.4])	([18422.5, 21110.3]) ([15026.6, 26377.7])	([5.63, 6.52], [5.5, 6.59])
F17	([6927.1, 17669.6], [6662, 68402.2])	([31860.1, 38370.4], [30205.6, 84695])	([16.31, 18.62], [15.71, 18.7])
^* Million Indian rupees; ^** Lakh metric tonnes

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Table 5. CC between the optimistic rough data and CC between the optimistic rough data

	I1	I2	I3	O1	O2	O3
I1	1	0.9495	-0.1234	0.9767	0.1153	0.9498
I2	0.9587	1	-0.0671	0.9133	0.0966	0.8906
I3	-0.1583	-0.1199	1	-0.2131	-0.3323	-0.0909
O1	0.9792	0.9246	-0.2373	1	0.0946	0.9253
O2	0.1156	0.1015	-0.2509	0.0943	1	0.2584
O3	0.9553	0.9005	-0.093	0.934	0.2531	1

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Table 6. PRCC: CC between rough data

	I1	I2	I3	O1	O2	O3
I1	1
I2	0.9546	1
I3	-0.1457	-0.098	1
O1	0.9781	0.9195	-0.2295	1
O2	0.11547	0.0992	-0.2865	0.0945	1
O3	0.9527	0.896	-0.0924	0.9299	0.2558	1

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Table 7. Optimistic-pessimistic efficiencies of 17 fertilizer supply chains

Supply chains	$ [E^{\sup},E^{\inf}] $
	$ \beta=0.6 $	$ \beta=0.7 $	$ \beta=0.8 $	$ \beta=0.9 $	$ \beta=1 $
F1	[1, 1]	[1, 1]	[1, 1]	[1, 1]	[0.3425, 1]
F2	[0.9872, 1]	[0.8179, 1]	[0.6183, 1]	[0.3632, 0.9967]	[0.0328, 0.6352]
F3	[0.7056, 0.7569]	[0.4495, 0.5212]	[0.259, 0.3301]	[0.1211, 0.1728]	[0.0134, 0.0238]
F4	[0.6851, 0.7649]	[0.4494, 0.5482]	[0.2721, 0.3567]	[0.1283, 0.303]	[0.0134, 0.0548]
F5	[1, 1]	[1, 1]	[1, 1]	[1, 1]	[1, 1]
F6	[1, 1]	[1, 1]	[1, 1]	[1, 1]	[1, 1]
F7	[0.7648, 0.8211]	[0.5905, 0.6737]	[0.4265, 0.5737]	[0.2035, 0.4648]	[0.0282, 0.1084]
F8	[0.2804, 0.3701]	[0.1719, 0.3134]	[0.0974, 0.2717]	[0.0405, 0.1758]	[0.005, 0.0258]
F9	[0.662, 0.7939]	[0.456, 0.6442]	[0.2936, 0.5055]	[0.138, 0.3888]	[0.0234, 0.0489]
F10	[0.6951, 0.7816]	[0.5282, 0.669]	[0.3626, 0.5807]	[0.2322, 0.5154]	[0.0268, 0.0955]
F11	[0.6829, 0.8046]	[0.4391, 0.6369]	[0.2563, 0.4842]	[0.1207, 0.2977]	[0.0169, 0.0492]
F12	[0.7725, 0.8524]	[0.6113, 0.7439]	[0.4437, 0.5948]	[0.193, 0.3024]	[0.0181, 0.0333]
F13	[0.5416, 0.6307]	[0.3856, 0.5308]	[0.2498, 0.4122]	[0.1279, 0.2601]	[0.0132, 0.0358]
F14	[0.6408, 0.739]	[0.504, 0.6875]	[0.3826, 0.6373]	[0.2596, 0.5622]	[0.0397, 0.0955]
F15	[0.5755, 1]	[0.3489, 1]	[0.1943, 1]	[0.0794, 1]	[0.0046, 1]
F16	[0.7656, 0.8421]	[0.6133, 0.7544]	[0.4707, 0.6803]	[0.3119, 0.6086]	[0.0445, 0.0907]
F17	[0.3663, 0.468]	[0.2219, 0.4166]	[0.1242, 0.3285]	[0.0567, 0.2402]	[0.0101, 0.0398]

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Table 8. Optimistic-pessimistic super-efficiencies of 17 fertilizer supply chains

Supply chains	$ [SE^{\sup},SE^{\inf}] $
	$ \beta=0.6 $	$ \beta=0.7 $	$ \beta=0.8 $	$ \beta=0.9 $	$ \beta=1 $
F1	[7.3013, 9.9515]	[4.9053, 9.2883]	[3.0676, 8.2182]	[1.5116, 6.2238]	[0.3425, 1.0961]
F2	[0.9872, 1.2397]	[0.8179, 1.2952]	[0.6183, 1.0878]	[0.3632, 0.9967]	[0.0328, 0.6352]
F3	[0.7056, 0.7569]	[0.4495, 0.5212]	[0.259, 0.3301]	[0.1211, 0.1728]	[, 0.0238]
F4	[0.6851, 0.7649]	[0.4494, 0.5482]	[0.2721, 0.3567]	[0.1283, 0.303]	[0.0134, 0.0548]
F5	[1.8877, 1.9]	[1.7865, 1.9]	[1.6869, 1.9]	[1.5879, 1.9]	[1.386, 4.7603]
F6	[10.2122, 12.2894]	[9.0896, 13.1988]	[7.8615, 14.1662]	[6.124, 14.8885]	[4.3193, 15.2126]
F7	[0.7648, 0.8211]	[0.5905, 0.6737]	[0.4265, 0.5737]	[0.2035, 0.4648]	[0.0282, 0.1084]
F8	[0.2804, 0.3701]	[0.1719, 0.3134]	[0.0974, 0.2717]	[0.0405, 0.1758]	[0.005, 0.0258]
F9	[0.662, 0.7939]	[0.456, 0.6442]	[0.2936, 0.5055]	[0.138, 0.3888]	[0.0234, 0.0489]
F10	[0.6951, 0.7816]	[0.5282, 0.669]	[0.3626, 0.5807]	[0.2322, 0.5154]	[0.0268, 0.0955]
F11	[0.6829, 0.8046]	[0.4391, 0.6369]	[0.2563, 0.4842]	[0.1207, 0.2977]	[0.0169, 0.0492]
F12	[0.7725, 0.8524]	[0.6113, 0.7439]	[0.4437, 0.5948]	[0.193, 0.3024]	[0.0181, 0.0333]
F13	[0.5416, 0.6307]	[0.3856, 0.5308]	[0.2498, 0.4122]	[0.1279, 0.2601]	[0.0132, 0.0358]
F14	[0.6408, 0.739]	[0.504, 0.6875]	[0.3826, 0.6373]	[0.2596, 0.5622]	[0.0397, 0.0955]
F15	[0.5755, 1.2517]	[0.3489, 1.7588]	[0.1943, 2.6778]	[0.0794, 4.7892]	[0.0046, 10.3892]
F16	[0.7656, 0.8421]	[0.6133, 0.7544]	[0.4707, 0.6803]	[0.3119, 0.6086]	[0.0445, 0.0907]
F17	[0.3663, 0.468]	[0.2219, 0.4166]	[0.1242, 0.3285]	[0.0567, 0.2402]	[0.0101, 0.0398]

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Table 9. Comparision between the original DEA and proposed model

Supply chains	Original DEA model ([14])	Rank	Proposed model	Rank
F1	24.4374	2	0.3025	3
F2	3.6104	4	0.0477	5
F3	1.0227	6	0.0127	15
F4	1.0157	7	0.0149	13
F5	1	8	0.1184	4
F6	158.8404	1	0.5425	1
F7	1	8	0.0213	8
F8	1	8	0.0059	17
F9	1	8	0.0186	11
F10	1	8	0.0213	9
F11	1	8	0.0175	12
F12	1.5248	5	0.0202	10
F13	1	8	0.0133	14
F14	1	8	0.0221	7
F15	4.5376	3	0.3512	2
F16	1	8	0.0248	6
F17	1	8	0.0094	16

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