Mat-2.4142 Seminar on optimization Capacity utilization 12.12.2007
Contents Summary of chapter 14 Related DEA-solver models Illustrative examples Measure of technical capacity utilization Price-based measure of capacity utilization Home assignment
Summary of chapter 14 The chapter has covered two subjects Economies of scope Capacity utilization Technical Price-based
Related DEA-solver models Economies of scope [Cost] input-oriented BCC (BCC-I) [Cost-W] and [Cost-Merger] input-oriented super BCC (Super-BCC-I) Capacity utilization [SBM 0 -Restricted], [SBM 0 -Relaxed] and [New-Tech 0 ] output-oriented SBM codes under VRS (SBM-O-V) [Profit 0 -Restricted] and [Profit 0 -Relaxed] new-profit model under VRS (New-Profit-V)
Illustrative examples 1 / 2: Measure of technical capacity utilization Hospital data set as an example, Table 14.4. Two inputs, Doctors (fixed) and Nurses (variable) Model: VRS; [SBM 0 -Restricted] and [SBM 0 - Relaxed] Results in Table 14.5.
Illustrative examples 1 / 2: Measure of technical capacity utilization Inputs Outputs Doctor Nurse Outpatients Inpatient DMU Number Number Cost Number Price Number Price A 20 151 100 100 550 90 2010 B 19 131 80 150 400 50 1800 C 25 160 90 160 480 55 2200 D 27 168 120 180 600 72 3500 E 22 158 70 94 400 66 3050 F 55 255 80 230 430 90 3900 G 33 235 100 220 540 88 3300 H 31 206 85 152 420 80 3500 I 30 244 76 190 350 100 2900 J 50 268 75 250 410 100 2600 K 53 306 80 260 540 147 2450 L 38 284 70 250 295 120 3000 Table 14.4. Data for 12 hospitals. Source: [1].
Illustrative examples 1 / 2: Measure of technical capacity utilization DMU Restricted Relaxed Capacity utilization Nurse utilization * φ * * * * φ F κ ( = φ / φ F ) rate ( δ * ) A 1.000 1.000 1.000 1.000 B 1.000 1.000 1.000 1.000 C 1.163 1.284 0.906 1.150 D 1.000 1.107 0.903 1.179 E 1.342 1.342 1.000 0.992 F 1.164 1.382 0.842 1.200 G 1.000 1.091 0.917 1.041 H 1.219 1.308 0.932 1.128 I 1.011 1.011 1.000 0.916 J 1.000 1.224 0.817 1.125 K 1.000 1.000 1.000 1.000 L 1.000 1.000 1.000 1.000 Table 14.5. Technical capacity utilization. Source: [1].
Illustrative examples 2 / 2: Price-based measure of capacity utilization Price-based data set: Table 14.6. Current profit G y F V V { x, x = c x, y = p y, y = p y )} ( j j j j 1 j 1 j 1 j 2 j 2 j 2 j = 1 + y2 Three LPs: [New-Tech], [Profit-Restricted] and [Profit-Relaxed] with profits G *, G ** and G ***, and losses L, L * and L **, respectively Table 14.7. V x
Illustrative examples 2 / 2: Price-based measure of capacity utilization DMU Doctor Nurse Outpat Inpat F x V y x y1 2 A 20 15,100 55,000 180,900 B 19 10,480 60,000 90,000 C 25 14,400 76,800 121,000 D 27 20,160 108,000 252,000 E 22 11,060 37,600 201,300 F 55 20,400 98,900 351,000 G 33 23,500 118,800 290,400 H 31 17,510 63,840 280,000 I 30 18,544 66,500 290,000 J 50 20,100 102,500 260,000 K 53 24,480 140,400 360,150 L 38 19,880 73,750 360,000 Table 14.6. Price based data set. Source: [1].
Illustrative examples 2 / 2: Price-based measure of capacity utilization Profit Loss DMU Current Tech Restrict Relaxed Tech Alloc Capacity Total G * G ** G *** G L * L ** L A 220,800 220,800 220,800 220,800 0 0 0 0 B 139,520 139,520 139,520 139,520 0 0 0 0 C 183,400 237,098 237,098 305,830 53,698 0 68,732 122,430 D 339,840 339,840 339,840 339,840 0 0 0 0 E 227,840 227,840 227,840 254,212 0 0 26,372 26,372 F 429,500 429,500 429,500 476,070 0 0 46,570 46,570 G 385,700 385,700 385,700 385,700 0 0 0 0 H 326,330 339,193 345,123 369,772 12,863 5,931 24,649 43,442 I 337,956 337,956 337,956 337,956 0 0 0 0 J 342,400 407,318 407,318 463,630 64,918 0 56,312 121,230 K 476,070 476,070 476,070 476,070 0 0 0 0 L 413,870 413,870 413,870 413,870 0 0 0 0 Sum 131,478 5,931 222,635 360,044 Table 14.7. Profits and losses. Source: [1].
Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H The current and maximum costs, revenues and profits for each DMU, Table 14.8. Current profit G H = 152 420 + 80 3500 85 206 = 326330 = 63840 + 280000 17150
Illustrative examples 2 / 2: Price-based measure of capacity utilization Current Maximum Cost Revenue Cost Revenue Loss DMU Nurse Outpat Inpat Nurse Outpat Inpat A 15,100 55,000 180,900 15,100 55,000 180,900 0 B 10,480 60,000 90,000 10,480 60,000 90,000 0 C 14,400 76,800 121,000 18,714 92,858 231,686 122,430 D 20,160 108,000 252,000 20,160 108,000 252,000 0 E 11,060 37,600 201,300 16,503 69,415 201,300 26,372 F 20,400 98,900 351,000 24,480 140,400 360,150 46,570 G 23,500 118,800 290,400 23,500 118,800 290,400 0 H 17,510 63,840 280,000 21,978 111,750 280,000 43,442 I 18,544 66,500 290,000 16,544 66,500 290,000 0 J 20,100 102,500 260,000 23,560 127,070 360,120 121,230 K 24,480 140,400 360,150 24,480 140,400 360,150 0 L 19,880 73,750 360,000 19,880 73,750 360,000 0 Table 14.8. Comparison of current and maximum profits. Source: [1].
Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H Optimal solution of [New-Tech H ] is φ * H For technically efficient H * the optimal cost of nurses Similarly, optimal number of doctors = 31 However, optimal revenue from outpatients = 76 703 Optimal revenue from inpatients = 280 000 * * * * = 1.101, λd = 0.277, λe = 0.316, λk = 0.073, λl G H* =339 193 = 0.334 = 0.277 20160 + 0.316 11060 + 0.073 24480 + 0.334 19880 = 17510
Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H For [Profit H -Restricted] G ** H = 345 123 Revenue from outpatients = 71 804 Revenue from inpatients = 290 829 Cost of nurses = 17 510 Relaxation of nurse capacity may give more profit
Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H For [Profit H -Relaxed] G *** H =369 772 Revenue from outpatients = 111 750 Revenue from inpatients = 280 000 Cost of nurses = 21 978 Maximum profit by relaxing the cost-based nurse capacity from 17 510 to 21 978
Questions?
Home assignment Why the relaxation of capacity constraint for nurses can improve profit for DMU H, eventhough this allows greater costs(the salary of nurses)? - Some answer 3p, one with an idea 6p Why is it justified to consider doctors without cost, but nurses with one? (The example given in this presentation) - Some answer 1p, one with an idea 3p Compact answers 1p For any clarification considering the home assignment, contact mikko.loimula@tkk.fi
References [1] William W. Cooper, Lawrence M. Seiford, Kaoru Tone: Data Envelopment Analysis, 2005