The Northside Bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank, including 9:00 A.M. to 7:00 PM. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here. Time No. of Tellers Time No. of Tellers 9:00 a.m. 10:00 a.m. 6 2:00 p.m.-3:00 p.m. 6 10:00 a.m.-11:00 a.m. 4 3:00 p.m.-4:00 p.m. 4 11:00 a.m.-Noon 8 4:00 p.m.-5:00p.m. 7 Noon-1:00p.m. 10 5:00 p.m.-6:00 p.m. 6 1:00p.m. -2:00p.m. 9 6:00p.m.-7:00p.m. 6 Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning hour ($105 a day), and part-time employees cost the bank 58 per hour ($32 per day). (a) formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. (Hint: Let xi nu time employees coming on duty at the beginning of hour J.) If your answer is zero enter "0" and if the constant is "1" it must be entered in the box. Min + 5 5 4 4 4 4 x10 x11 y9 s.t. + 1 1 1 x9 x9 x9 + + 0 1 1 x10 x10 x10 + + + 0 0 1 x11 X11 *11 + + + 1 1 1 y9 y9 y9 + + 7 1 1 y10 + y10 M + Y10 10 + 0 0 1 Y11 y11 + 3+ y11 y11 + 0 0 0 y12 y12 y12 y12 + + + 7 0 0 + ** E Y1 + y1 + 0 * y2 se se s 3 S.E. 1 1 1 1 1 S 6x 1 1 x9 6x > 6x * 6x X + x9 1 & 2 & 9 6x + + 6x + +6 + 0 + 1 1 1 1 0 1 1 1 9 0 x10 *10 +( x10 x10 + (3) + x10 x10 x10 (+ x10 x10 + 0 + 0 1 1 1 1 0 1 1 S 1 *11 *11 x11 x11 X11 x11 X11 x11 + x11 + + + + + + + 1 1 1 + 1 0 0 0 x9 0 x9 x10 *11 y9 Xi, yj 20 and integer for 9, 10, 11 and j 9, 10, 11, 12, 1, 2, 3 0 0 S y9 S y9 y9 +6 > y9 y9 e s y9 y9 + y9 + + + + + + + + 7 1 1 1 1 0 0 0 0 * Y10 y10 y10 Y10 y10 Y10 y10 Y10 Y10 + y10 + + + + + + + + +90 0 0 1 1 1 0 0 0 0 > y11 y11 y11 FOFO F y11 y11 Y11 y11 > > y11 y11 Y11 + + + + + + + + + + 0 0 1 1 1 Y12 (+ 0 y12 1 0 0 > 0 Y12 Y12 y12 y12 y12 + y12 + Y12 + + + (+ Y12 + + + 7 0 0 0 1 1 1 1 0 0 X + NON = = = = IA y1 y1 IA IA +6 y1 + YI + y1 + +6 y1 + 66 6 e se se s C +6 O > & S & S & S Y2 y2 V2 + + + + + + + e 0 0 1 1 1 1 0 0 0 0 Y11 y11 y11 y11 Y11 y11 y11 y11 y11 y11 y11 + + + + + + + + + + + 4 0 0 0 1 1 1 0 0 0 y12 (+ y12 Y12 y12 (+ y12 y12 + y12 1+ y12 y12 + y12 Y12 + + + + + + 4 7 0 0 0 1 1 1 1 0 0 Y1 y1 y1 y1 y1 y1 > + y1 MON y1 y1 + C + y1 + y1 S+ + + + + + 4 0 0 0 0 0 1 1 1 1 0 Y2 y2 ON ON O N Y2 Y2 SON ON S Y2 Y2 y2 + + + + + + + + + + + 7 0 0 0 0 1 1 y3 SO SO SO y3 y3 Y3 Y3 y3 y3 y3 Y3 y3 2 > > > > > > > > S (> 6 4 8 10 9 6 7 6 6 > > > Time (9:00 a.m.-10:00 a.m.) Time (10:00 a.m.-11:00 a.m.) Time (11:00 a.m.-Noon.) Time (Noon.-1:00 p.m.) Time (1:00 p.m.-2:00p.m.) Time (2:00 p.m.-3:00 p.m.) Time (3:00 p.m.-4:00p.m.) Time (4:00 p.m.-5:00p.m.) Time (5:00 p.m.-6:00 p.m.) Time (6:00 p.m.-7:00p.m.) (b) Solve the LP Relaxation of your model in part (a). Enter the total cost value under the optimal solution. Total Cost: $ 950 (c) Solve for the optimal schedule of tellers. Enter the total number of full-time and part-time employees required. If your answer is zero enter "0"; Total Number of Full-Time Employees: 630 Total Number of Part-Time Employees: 320 (d) After reviewing the solution to part (c), the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full-tim Revise your model to incorporate these additional requirements, and solve for the optimal solution. Enter the total number of full-time and part-time employees required If required, round your answers to the nearest whole number. If your answer is zero enter "0" Total Number of Full-Time Employees: 735 Total Number of Part-Time Employees: 256 Total Cost: $ 991 The Northside Bank is working to develop an efficient work schedule for full-time and part-time tellers. The schedule must provide for efficient operation of the bank, including 9:00 A.M. to 7:00 PM. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized here. Time No. of Tellers Time No. of Tellers 9:00 a.m. 10:00 a.m. 6 2:00 p.m.-3:00 p.m. 6 10:00 a.m.-11:00 a.m. 4 3:00 p.m.-4:00 p.m. 4 11:00 a.m.-Noon 8 4:00 p.m.-5:00p.m. 7 Noon-1:00p.m. 10 5:00 p.m.-6:00 p.m. 6 1:00p.m. -2:00p.m. 9 6:00p.m.-7:00p.m. 6 Each full-time employee starts on the hour and works a 4-hour shift, followed by a 1-hour break and then a 3-hour shift. Part-time employees work one 4-hour shift beginning hour ($105 a day), and part-time employees cost the bank 58 per hour ($32 per day). (a) formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. (Hint: Let xi nu time employees coming on duty at the beginning of hour J.) If your answer is zero enter "0" and if the constant is "1" it must be entered in the box. Min + 5 5 4 4 4 4 x10 x11 y9 s.t. + 1 1 1 x9 x9 x9 + + 0 1 1 x10 x10 x10 + + + 0 0 1 x11 X11 *11 + + + 1 1 1 y9 y9 y9 + + 7 1 1 y10 + y10 M + Y10 10 + 0 0 1 Y11 y11 + 3+ y11 y11 + 0 0 0 y12 y12 y12 y12 + + + 7 0 0 + ** E Y1 + y1 + 0 * y2 se se s 3 S.E. 1 1 1 1 1 S 6x 1 1 x9 6x > 6x * 6x X + x9 1 & 2 & 9 6x + + 6x + +6 + 0 + 1 1 1 1 0 1 1 1 9 0 x10 *10 +( x10 x10 + (3) + x10 x10 x10 (+ x10 x10 + 0 + 0 1 1 1 1 0 1 1 S 1 *11 *11 x11 x11 X11 x11 X11 x11 + x11 + + + + + + + 1 1 1 + 1 0 0 0 x9 0 x9 x10 *11 y9 Xi, yj 20 and integer for 9, 10, 11 and j 9, 10, 11, 12, 1, 2, 3 0 0 S y9 S y9 y9 +6 > y9 y9 e s y9 y9 + y9 + + + + + + + + 7 1 1 1 1 0 0 0 0 * Y10 y10 y10 Y10 y10 Y10 y10 Y10 Y10 + y10 + + + + + + + + +90 0 0 1 1 1 0 0 0 0 > y11 y11 y11 FOFO F y11 y11 Y11 y11 > > y11 y11 Y11 + + + + + + + + + + 0 0 1 1 1 Y12 (+ 0 y12 1 0 0 > 0 Y12 Y12 y12 y12 y12 + y12 + Y12 + + + (+ Y12 + + + 7 0 0 0 1 1 1 1 0 0 X + NON = = = = IA y1 y1 IA IA +6 y1 + YI + y1 + +6 y1 + 66 6 e se se s C +6 O > & S & S & S Y2 y2 V2 + + + + + + + e 0 0 1 1 1 1 0 0 0 0 Y11 y11 y11 y11 Y11 y11 y11 y11 y11 y11 y11 + + + + + + + + + + + 4 0 0 0 1 1 1 0 0 0 y12 (+ y12 Y12 y12 (+ y12 y12 + y12 1+ y12 y12 + y12 Y12 + + + + + + 4 7 0 0 0 1 1 1 1 0 0 Y1 y1 y1 y1 y1 y1 > + y1 MON y1 y1 + C + y1 + y1 S+ + + + + + 4 0 0 0 0 0 1 1 1 1 0 Y2 y2 ON ON O N Y2 Y2 SON ON S Y2 Y2 y2 + + + + + + + + + + + 7 0 0 0 0 1 1 y3 SO SO SO y3 y3 Y3 Y3 y3 y3 y3 Y3 y3 2 > > > > > > > > S (> 6 4 8 10 9 6 7 6 6 > > > Time (9:00 a.m.-10:00 a.m.) Time (10:00 a.m.-11:00 a.m.) Time (11:00 a.m.-Noon.) Time (Noon.-1:00 p.m.) Time (1:00 p.m.-2:00p.m.) Time (2:00 p.m.-3:00 p.m.) Time (3:00 p.m.-4:00p.m.) Time (4:00 p.m.-5:00p.m.) Time (5:00 p.m.-6:00 p.m.) Time (6:00 p.m.-7:00p.m.) (b) Solve the LP Relaxation of your model in part (a). Enter the total cost value under the optimal solution. Total Cost: $ 950 (c) Solve for the optimal schedule of tellers. Enter the total number of full-time and part-time employees required. If your answer is zero enter "0"; Total Number of Full-Time Employees: 630 Total Number of Part-Time Employees: 320 (d) After reviewing the solution to part (c), the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full-tim Revise your model to incorporate these additional requirements, and solve for the optimal solution. Enter the total number of full-time and part-time employees required If required, round your answers to the nearest whole number. If your answer is zero enter "0" Total Number of Full-Time Employees: 735 Total Number of Part-Time Employees: 256 Total Cost: $ 991