XYZ company produces metal shelves that are made of metal bars. There are three different sizes of metal bars that XYZ purchases from some suppliers (small-size bars, medium-size bars, large-size bar, extremely-large-size bar). In fact, XYZ buys "batches" of metal bars from 15 different suppliers. Table 1 provides information on the cost of a batch bought from different suppliers. This table also provides information on the number of bars with different sizes that a batch from each supplier includes. As an example, as Table 1 shows, one batch from Supplier 1 costs $4400 and includes 16 extremely large bars, 28 large bars, 45 medium bars, and 37 small bars. Similar information is available for other suppliers in Table 1. XYZ must buy at least 600 extremely large, 540 large, 650 medium, and 790 small bars each week. Because of some operational reasons, XYZ can buy at most 1000 bars per week from each supplier.

 

Table 1

                                                                                                                             

Supplier	Cost Per batch ($)	The number of extremely large bars in each batch	The number of large bars in each batch	The number of medium bar in each batch	The number of small bars in each batch
1	4400	16	28	45	37
2	3700	34	35	21	40
3	4400	30	19	26	34
4	3500	43	36	36	17
5	5300	37	24	23	18
6	4700	36	32	21	18
7	5100	26	28	26	41
8	5200	44	20	44	32
9	4700	40	27	14	33
10	3100	23	14	10	24
11	5600	23	20	34	35
12	4000	42	13	28	39
13	3300	30	32	44	48
14	3200	34	27	44	20
15	4100	36	27	44	46
16	4400	44	39	33	11
17	3500	24	16	43	33
18	5600	30	38	24	23
19	4200	28	26	35	12
20	4000	28	10	32	50

Part a)Formulate a mathematical programming model to minimize the total cost.

Part b)How would you modify the model to ensure that at least 60 percent of purchased small bars are purchased from Suppliers 4 and 5.