The production scheduling problem of the J. Mehta Company can be formulated as a transportation model to minimize cost as follows: **Decision variables:** * $x_{ij}$: the number of sinks produced in month $i$ and shipped to month $j$ * $y_i$: the number of sinks produced in month $i$ in overtime * $z_i$: the number of sinks purchased from the outside supplier in month $i$ * $i = 1, 2, 3, 4$ * $j = 1, 2, 3, 4$ **Objective function:** ``` Minimize: 100x_{ij} 130y_i 150z_i ``` **Subject to:** * Demand constraints: ``` x_{11} x_{12} x_{13} x_{14} = 120 x_{21} x_{22} x_{23} x_{24} = 160 x_{31} x_{32} x_{33} x_{34} = 240 x_{41} x_{42} x_{43} x_{44} = 100 ``` * Production constraints: ``` x_{11} x_{12} x_{13} x_{14} y_1 <= 150 x_{21} x_{22} x_{23} x_{24} y_2 <= 150 x_{31} x_{32} x_{33} x_{34} y_3 <= 150 x_{41} x_{42} x_{43} x_{44} y_4 <= 150 ``` * Inventory constraints: ``` x_{11} x_{21} x_{31} x_{41} - z_1 = 40 x_{12} x_{22} x_{32} x_{42} - z_2 = x_{11} - 10y_1 x_{13} x_{23} x_{33} x_{43} - z_3 = x_{12} - 10y_2 x_{14} x_{24} x_{34} x_{44} - z_4 = x_{13} - 10y_3 ``` * Non-negativity constraints: ``` x_{ij}, y_i, z_i