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=

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Chapter 5.3, Problem 9P

Explanation of Solution

Calculating the extra payment:

Let x₁ 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= 1500wt+1600ht+2000ft+30it+25×4ot+5(xt+ot)

min z= 1500wt+1600ht+2000ft+30it+105ot+5xt

for t=1,2,3,4.

The constraints are described where the first constraint for inventory is,

i1=50+x1+o1−300

x1+o1−i1=250

For the second month,

i2=i1+x2+o2−500

i1+x2+o2−i2=500

In the same way, for next two months,

x3+o3+i2−i3=100

x4+o4+i3−i4=100

The second constraint for relation available workers to hiring and firing the workers, for the first month,

w1=3+h1−f1

w1+f−h1=3

For the second month,

w2=w1+h2−f2

w2+f2−w1−h2=0

In the same way, for next two months,

w3+f3−w2−h3=0

w4+f4−w3−h4=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,

4x1≤160w1

4x1−160w1≤0

In the same way, for next three months,

4x2−160w2≤0

4x3−160w3≤0

4x4−160w4≤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,

4o1≤20w1

4o1−20w1≤0

In the same way, for next three months,

4o2−20w2≤0

4o3−20w3≤0

4o4−20w4≤0

Hence, the obtained Linear Programming

min z= 1500wt+1600ht+2000ft+30it+105ot+5xt

for t=1,2,3,4.

Subjected to,

x1+o1−i1=250

i1+x2+o2−i2=500

x3+o3+i2−i3=100

x4+o4+i3−i4=100

w1+f−h1=3

w2+f2−w1−h2=0

w3+f3−w2−h3=0

w4+f4−w3−h4=0

4x1−160w1≤0

4x2−160w2≤0

4x3−160w3≤0

4x4−160w4≤0

4o1−20w1≤0

4o2−20w2≤0

4o3−20w3≤0

4o4−20w4≤0

The following is the LINDO output for this Linear Programming:

min z=

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Chapter 5.3, Problem 7P

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