Question 4 (20 points): A production plant has 13 work-in-progress items that needs to be processed by 5 machines. To process each item, only one machine is required. All these machines do the same process, but they work based on different manufacturing technologies. Therefore, as indicated in Table 2, the duration of processing each item depends on the machine that is used to process the item. Moreover, the electricity cost of using machines are different for processing each item. The electricity cost of each machine for processing each item is given in Table 3. 4/9 Table 2- Duration of performing items by different machines in terms of minutes. Item 1 19tem tem tem tem tem tem Item 12 tem 13 Item 2 SISI 76 Machine 1 Machine 2 Machine Machine 4 Machine 5 52 55 51 90 72 75 92 100 62 61 66 | Fa UN 8188 18 - I 100 99 86 94 68 DIS MR818 . 65 84 64 55 64 58 93 85 53 94 kem 108 Rally 100 8H2819 57 54 50 50 67 6 66 79 98 52 66 Rem 11 Sails NOD 96 94 71 60 N 50 60 64 81 70 67 100 66 62 69 76 93 60 91 52 T 11 DIE 55 al 93 ci Table 3. The electricity cost of machine for performing different items in dollars) rem N Machine 1 280 Machine 2 172 Machine 265 Machine 4 Machines 149 126 188 145 179 288 400 137 156 349 113 387 348 327 315 140 261 227 150 366 21 117 356 350 270 246 351 355 200 240 395 203 196 267 387 110 333 383 360 374 383 289 121 148 316 311 231 215 125 109 176 329 2:10 372 114 299 221 301 302 291 Part a) Write a mathematical programming model to help the production manager decide how to use items to machine such that all items are processed, each machine does not work more than 500 minutes, and the total electricity cost is minimised. (6 points) Part b) How would you modify the model to ensure that at most 4 items are assigned to each machine (2 points) Part c) How would you modify the model to ensure that the total electricity cost of each machine with respect to assigned jobs will be less than 570072 points) Part d) How would you modify the model to ensure that it item is assigned to machine 2 then items must be assigned to machine 1 (2 points) Parte) How would you moday the model to ensure that, among items 3,5,6,10 and 11, at least of them can be assigned to machine 22 points Part f) How would you modity the model to ensure that it items 2 and 6 are assigned to machine, then