Inventory Problems as Transportation Problems Sailco Corporation: Formulation (Costs) Cost Coefficients: ($/Sailboat) Supply Demand Dummy I R OT R 4 M OT 5 M R 6 M M OT 7 M M R 8 M M M OT 9 M M M M is a very large, arbitrary, positive number.
Assignment Problems Personnel Assignment: Time (hours) Job 1Job 2Job 3Job 4 Person Person Person Person
Assignment Problem Formulation Define Variables: Let Xij = 1 if ith person is assigned to jth job Xij = 0 if ith person is not assigned to jth job Objective Function: Min z = 14*X11 + 5*X12 + … + 10*X44 Personnel Constraints: X11 + X12 + X13 + X14 = 1 X21 + X22 + X23 + X24 = 1 X31 + X32 + X33 + X34 = 1 X41 + X42 + X43 + X44 = 1 Demand Constraints: X11 + X21 + X31 + X41 = 1 X12 + X22 + X32 + X42 = 1 X13 + X23 + X33 + X43 = 1 X14 + X24 + X34 + X44 = 1 Binary Constraints: Xij = 0 or Xij = 1
Assignment Problem Algorithm Row Minimum Column Minimum Subtract Row Minimum from Each Row:
Assignment Problem Algorithm Subtract Column Minimum from Each Column: Subtract Minimum uncrossed value from uncrossed values and add to twice-crossed values: Draw lines to cross out zeros and read solution from zeros 0 Solution:
Hungarian Method Step 1: Find the minimum element in each row of the m x m cost matrix. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix (called the reduced cost matrix) by subtracting from each cost the minimum cost in its column. Step 2: Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all the zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are needed, proceed to Step 3. Step 3: Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in Step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to Step 2.