1. **Problem Statement:** Sunshine House has 5 production lines and 5 types of cookies. Each cookie must be assigned to exactly one production line to minimize the total sum of completion times. 2. **Understanding the Problem:** We want to assign each cookie to a unique production line such that the sum of the processing times is minimized. 3. **Method:** This is an assignment problem, which can be solved using the Hungarian algorithm or by checking permutations since the problem size is small. 4. **Given Processing Times (hours):** \begin{array}{c|ccccc} \text{Cookie} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Chocolate Mint} & 30 & 18 & 26 & 17 & 15 \\ \text{Peanut Butter} & 23 & 22 & 32 & 25 & 30 \\ \text{Shortbread} & 17 & 31 & 24 & 22 & 29 \\ \text{Fudge Delight} & 28 & 19 & 13 & 18 & 23 \\ \text{Macaroons} & 23 & 14 & 16 & 20 & 27 \end{array} 5. **Step-by-step solution using Hungarian algorithm:** - Step 1: Construct cost matrix $C$ with rows as cookies and columns as lines. - Step 2: Subtract the row minimum from each row. - Step 3: Subtract the column minimum from each column. - Step 4: Cover zeros with minimum number of lines and adjust matrix until an optimal assignment is found. 6. **Row minima subtraction:** Row minima: Chocolate Mint = 15, Peanut Butter = 22, Shortbread = 17, Fudge Delight = 13, Macaroons = 14 Subtracting row minima: \begin{array}{c|ccccc} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Chocolate Mint} & 30-15=15 & 18-15=3 & 26-15=11 & 17-15=2 & 15-15=0 \\ \text{Peanut Butter} & 23-22=1 & 22-22=0 & 32-22=10 & 25-22=3 & 30-22=8 \\ \text{Shortbread} & 17-17=0 & 31-17=14 & 24-17=7 & 22-17=5 & 29-17=12 \\ \text{Fudge Delight} & 28-13=15 & 19-13=6 & 13-13=0 & 18-13=5 & 23-13=10 \\ \text{Macaroons} & 23-14=9 & 14-14=0 & 16-14=2 & 20-14=6 & 27-14=13 \end{array} 7. **Column minima subtraction:** Column minima: Col1=0, Col2=0, Col3=0, Col4=2, Col5=0 Subtracting column minima: \begin{array}{c|ccccc} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Chocolate Mint} & 15 & 3 & 11 & 0 & 0 \\ \text{Peanut Butter} & 1 & 0 & 10 & 1 & 8 \\ \text{Shortbread} & 0 & 14 & 7 & 3 & 12 \\ \text{Fudge Delight} & 15 & 6 & 0 & 3 & 10 \\ \text{Macaroons} & 9 & 0 & 2 & 4 & 13 \end{array} 8. **Finding optimal assignment:** - Assign Chocolate Mint to line 5 (cost 0) - Assign Peanut Butter to line 2 (cost 0) - Assign Shortbread to line 1 (cost 0) - Assign Fudge Delight to line 3 (cost 0) - Assign Macaroons to line 4 (cost 4) 9. **Calculate total minimum sum of completion times:** $$30 + 22 + 17 + 13 + 20 = 102$$ 10. **Answer:** Assign cookies as follows to minimize total completion time: - Chocolate Mint: Line 5 - Peanut Butter: Line 2 - Shortbread: Line 1 - Fudge Delight: Line 3 - Macaroons: Line 4 Total minimum sum of completion times is **102 hours**.