Calculating the extra payment: Let x 1 be the number of pairs of shoes that were produced during the month “t” with non-overtime labor, o t be the number of pairs of shoes that were produced during month “t” with the overtime labor, i t be the inventory of shoes at the end of month “t”, h t is the number of workers that were hired at the beginning of month “t”, f t be the number of workers who were fired at the beginning of month “t” and w t be the number of workers who were available for month “t”. For the objective function, the following cost must be considered as worker’s salaries, hiring costs, firing costs, holding costs, overtime costs and raw-material costs. Hence, the derived objective function is: min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 25 × 4 o t + 5 ( x t + o t ) min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. The constraints are described where the first constraint for inventory is, i 1 = 50 + x 1 + o 1 − 300 x 1 + o 1 − i 1 = 250 For the second month, i 2 = i 1 + x 2 + o 2 − 500 i 1 + x 2 + o 2 − i 2 = 500 In the same way, for next two months, x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 The second constraint for relation available workers to hiring and firing the workers, for the first month, w 1 = 3 + h 1 − f 1 w 1 + f − h 1 = 3 For the second month, w 2 = w 1 + h 2 − f 2 w 2 + f 2 − w 1 − h 2 = 0 In the same way, for next two months, w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 The third constraint for each month, the amount of shoes made with non-overtime labors is limited by the number of workers, here for the first month, 4 x 1 ≤ 160 w 1 4 x 1 − 160 w 1 ≤ 0 In the same way, for next three months, 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 The fourth constraint for each month, the amount of shoes made with overtime labor is limited by the number of workers, for the first month, 4 o 1 ≤ 20 w 1 4 o 1 − 20 w 1 ≤ 0 In the same way, for next three months, 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 Hence, the obtained Linear Programming min z= 1500 w t + 1600 h t + 2000 f t + 30 i t + 105 o t + 5 x t for t=1,2,3,4. Subjected to, x 1 + o 1 − i 1 = 250 i 1 + x 2 + o 2 − i 2 = 500 x 3 + o 3 + i 2 − i 3 = 100 x 4 + o 4 + i 3 − i 4 = 100 w 1 + f − h 1 = 3 w 2 + f 2 − w 1 − h 2 = 0 w 3 + f 3 − w 2 − h 3 = 0 w 4 + f 4 − w 3 − h 4 = 0 4 x 1 − 160 w 1 ≤ 0 4 x 2 − 160 w 2 ≤ 0 4 x 3 − 160 w 3 ≤ 0 4 x 4 − 160 w 4 ≤ 0 4 o 1 − 20 w 1 ≤ 0 4 o 2 − 20 w 2 ≤ 0 4 o 3 − 20 w 3 ≤ 0 4 o 4 − 20 w 4 ≤ 0 The following is the LINDO output for this Linear Programming: min z=