The parameter table given below shows the transportation problem formulation of Option 1 for the Better Products Co.problem presented in Sec.9.3 of the textbook.As stated in the textbook,the optimal solution for this transportation problem has the following basic variables (allocations): x₁₂ = 30,x₁₃ = 30,x₁₅ = 15,x₂₄ = 15,x₂₅ = 60,x₃₁ = 20,x₃₄ = 25 Verify that this optimal solution actually is optimal by applying just the optimality test portion of the transportation simplex method to this solution.

Question

Quizplus · Accepted Answer

The Answer is The optimality test requires solving for the u_i and v_j such that c_ij = u_i + v_j for each (i,j)such that x_ij is basic.Since the number of unknowns (the u_i and v_j )is one larger than the number of equations,and since Source 1 has the largest number of basic variables,we can begin by setting u_i = 0.We then solve for each of the remaining unknowns as outlined below.x₁₂: 27 = u₁ + v₂.Set u₁ = 0,so v₂ = 27,x₁₃: 28 = u₁ + v₃. v₃ = 28,x₁₅: 0 = u₁ + v₅. v₅ = 0.x₂₅: 0 = u₂ + v₅.Know v₅ = 0,so u₂ = 0.x₂₄: 23 = u₂ + v₄.Know u₂ = 0,so v₄ = 23.x₃₄: 21 = u₃ + v₄.Know v₄ = 23,so u₃ = -2.x₃₁: 37 = u₃ + v₁.Know u₃ = -2,so v₁ = 39.We next calculate c_ij - u_i - v_j for each of the nonbasic variables.x₁₁: c₁₁ - u₁ - v₁ = 41 - 0 - 39 = 2 x₁₄: c₁₄ - u₁ - v₄ = 24 - 0 - 23 = 1 x₂₁: c₂₁ - u₂ - v₁ = 40 - 0 - 39 = 1 x₂₂: c₂₂ - u₂ - v₂ = 29 - 0 - 27 = 2 x₂₃: c₂₃ - u₂ - v₃ = M - 0 - 28 = M - 28 x₃₂: c₃₂ - u₃ - v₂ = 30 + 2 - 27 = 5 x₃₃: c₃₃ - u₃ - v₃ = 27 + 2 - 28 = 1 x₃₅: c₃₅ - u₃ - v₅ = 0 + 2 - 0 = 2 The optimality test states that a BF solution is optimal if and only if

The optimality test requires solving for the u<sub>i</sub> and v<sub>j</sub> such that c<sub>ij</sub> = u<sub>i</sub> + v<sub>j</sub> for each (i,j)such that x<sub>ij</sub> is basic.Since the number of unknowns (the u<sub>i</sub> and v<sub>j</sub> )is one larger than the number of equations,and since Source 1 has the largest number of basic variables,we can begin by setting u<sub>i</sub> = 0.We then solve for each of the remaining unknowns as outlined below.x<sub>12</sub>: 27 = u<sub>1</sub> + v<sub>2</sub>.Set u<sub>1</sub> = 0,so v<sub>2</sub> = 27,x<sub>13</sub>: 28 = u<sub>1</sub> + v<sub>3</sub>. v<sub>3</sub> = 28,x<sub>15</sub>: 0 = u<sub>1</sub> + v<sub>5</sub>. v<sub>5</sub> = 0.x<sub>25</sub>: 0 = u<sub>2</sub> + v<sub>5</sub>.Know v<sub>5</sub> = 0,so u<sub>2</sub> = 0.x<sub>24</sub>: 23 = u<sub>2</sub> + v<sub>4</sub>.Know u<sub>2</sub> = 0,so v<sub>4</sub> = 23.x<sub>34</sub>: 21 = u<sub>3</sub> + v<sub>4</sub>.Know v<sub>4</sub> = 23,so u<sub>3</sub> = -2.x<sub>31</sub>: 37 = u<sub>3</sub> + v<sub>1</sub>.Know u<sub>3</sub> = -2,so v<sub>1</sub> = 39.We next calculate c<sub>ij</sub> - u<sub>i</sub> - v<sub>j</sub> for each of the nonbasic variables.x<sub>11</sub>: c<sub>11</sub> - u<sub>1</sub> - v<sub>1</sub> = 41 - 0 - 39 = 2 x<sub>14</sub>: c<sub>14</sub> - u<sub>1</sub> - v<sub>4</sub> = 24 - 0 - 23 = 1 x<sub>21</sub>: c<sub>21</sub> - u<sub>2</sub> - v<sub>1</sub> = 40 - 0 - 39 = 1 x<sub>22</sub>: c<sub>22</sub> - u<sub>2</sub> - v<sub>2</sub> = 29 - 0 - 27 = 2 x<sub>23</sub>: c<sub>23</sub> - u<sub>2 </sub> - v<sub>3</sub> = M - 0 - 28 = M - 28 x<sub>32</sub>: c<sub>32</sub> - u<sub>3</sub> - v<sub>2</sub> = 30 + 2 - 27 = 5 x<sub>33</sub>: c<sub>33</sub> - u<sub>3</sub> - v<sub>3</sub> = 27 + 2 - 28 = 1 x<sub>35</sub>: c<sub>35</sub> - u<sub>3</sub> - v<sub>5</sub> = 0 + 2 - 0 = 2 The optimality test states that a BF solution is optimal if and only if for every (i,j)such that x<sub>ij</sub> is nonbasic.Since the above calculations verify that this test passes,the solution is optimal.

for every (i,j)such that x_ij is nonbasic.Since the above calculations verify that this test passes,the solution is optimal.

The Parameter Table Given Below Shows the Transportation Problem Formulation