Capacity utilization

Transkriptio

1 Mat Seminar on optimization Capacity utilization

2 Contents Summary of chapter 14 Related DEA-solver models Illustrative examples Measure of technical capacity utilization Price-based measure of capacity utilization Home assignment

3 Summary of chapter 14 The chapter has covered two subjects Economies of scope Capacity utilization Technical Price-based

4 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)

5 Illustrative examples 1 / 2: Measure of technical capacity utilization Hospital data set as an example, Table Two inputs, Doctors (fixed) and Nurses (variable) Model: VRS; [SBM 0 -Restricted] and [SBM 0 - Relaxed] Results in Table 14.5.

6 Illustrative examples 1 / 2: Measure of technical capacity utilization Inputs Outputs Doctor Nurse Outpatients Inpatient DMU Number Number Cost Number Price Number Price A B C D E F G H I J K L Table Data for 12 hospitals. Source: [1].

7 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 B C D E F G H I J K L Table Technical capacity utilization. Source: [1].

8 Illustrative examples 2 / 2: Price-based measure of capacity utilization Price-based data set: Table 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 V x

9 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, ,900 B 19 10,480 60,000 90,000 C 25 14,400 76, ,000 D 27 20, , ,000 E 22 11,060 37, ,300 F 55 20,400 98, ,000 G 33 23, , ,400 H 31 17,510 63, ,000 I 30 18,544 66, ,000 J 50 20, , ,000 K 53 24, , ,150 L 38 19,880 73, ,000 Table Price based data set. Source: [1].

10 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, , , , B 139, , , , C 183, , , ,830 53, , ,430 D 339, , , , E 227, , , , ,372 26,372 F 429, , , , ,570 46,570 G 385, , , , H 326, , , ,772 12,863 5,931 24,649 43,442 I 337, , , , J 342, , , ,630 64, , ,230 K 476, , , , L 413, , , , Sum 131,478 5, , ,044 Table Profits and losses. Source: [1].

11 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 Current profit G H = = =

12 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, ,900 15,100 55, ,900 0 B 10,480 60,000 90,000 10,480 60,000 90,000 0 C 14,400 76, ,000 18,714 92, , ,430 D 20, , ,000 20, , ,000 0 E 11,060 37, ,300 16,503 69, ,300 26,372 F 20,400 98, ,000 24, , ,150 46,570 G 23, , ,400 23, , ,400 0 H 17,510 63, ,000 21, , ,000 43,442 I 18,544 66, ,000 16,544 66, ,000 0 J 20, , ,000 23, , , ,230 K 24, , ,150 24, , ,150 0 L 19,880 73, ,000 19,880 73, ,000 0 Table Comparison of current and maximum profits. Source: [1].

13 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 = Optimal revenue from inpatients = * * * * = 1.101, λd = 0.277, λe = 0.316, λk = 0.073, λl G H* = = = = 17510

14 Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H For [Profit H -Restricted] G ** H = Revenue from outpatients = Revenue from inpatients = Cost of nurses = Relaxation of nurse capacity may give more profit

15 Illustrative examples 2 / 2: Price-based measure of capacity utilization Case: DMU H For [Profit H -Relaxed] G *** H = Revenue from outpatients = Revenue from inpatients = Cost of nurses = Maximum profit by relaxing the cost-based nurse capacity from to

16 Questions?

17 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

18 References [1] William W. Cooper, Lawrence M. Seiford, Kaoru Tone: Data Envelopment Analysis, 2005