wild lumber company ships construction materials from three wood - processing plants to three retail stores…

wild lumber company ships construction materials from three wood - processing plants to three retail stores. the shipping cost, monthly production capacities, and monthly demand for framing lumber are given below.\n\nplant store a store b store c capacity\n1 $7.30 $6.80 $6.10 610\n2 $4.60 $9.00 $8.30 580\n3 $8.10 $5.30 $5.30 230\ndemand 430 310 620\n\nformulate a linear - optimization model for this problem, implement your model on a spreadsheet, and use excel solver to find a solution that minimizes total distribution costs. if the constant is equal to one, enter 1.\n\nlet:\n$x_{11}$ = number of components produced in plant 1 and supplied to store a\n$x_{12}$ = number of components produced in plant 1 and supplied to store b\n$x_{13}$ = number of components produced in plant 1 and supplied to store c\n$x_{21}$ = number of components produced in plant 2 and supplied to store a\n$x_{22}$ = number of components produced in plant 2 and supplied to store b\n$x_{23}$ = number of components produced in plant 2 and supplied to store c\n$x_{31}$ = number of components produced in plant 3 and supplied to store a\n$x_{32}$ = number of components produced in plant 3 and supplied to store b\n$x_{33}$ = number of components produced in plant 3 and supplied to store c\n\nmin $x_{11}$+ $x_{12}$+ $x_{13}$+ $x_{21}$+ $x_{22}$+ $x_{23}$+ $x_{31}$+ $x_{32}$+ $x_{33}$\n\nsubject to the constraints:\n $x_{11}$+ $x_{12}$+ $x_{13}$ select \n $x_{21}$+ $x_{22}$+ $x_{23}$ select \n $x_{31}$+ $x_{32}$+ $x_{33}$ select \n $x_{11}$+ $x_{21}$+ $x_{31}$ select \n $x_{12}$+ $x_{22}$+ $x_{32}$ select \n $x_{13}$+ $x_{23}$+ $x_{33}$ select \n$x_{11},x_{12},x_{13},x_{21},x_{22},x_{23},x_{31},x_{32},x_{33}geq0$\n\nwhat is the value of total distribution costs at the optimal solution? use the excel solver to find the optimal solution. round your answer to the nearest dollar.\n

wild lumber company ships construction materials from three wood - processing plants to three retail stores. the shipping cost, monthly production capacities, and monthly demand for framing lumber are given below.\n\nplant store a store b store c capacity\n1 $7.30 $6.80 $6.10 610\n2 $4.60 $9.00 $8.30 580\n3 $8.10 $5.30 $5.30 230\ndemand 430 310 620\n\nformulate a linear - optimization model for this problem, implement your model on a spreadsheet, and use excel solver to find a solution that minimizes total distribution costs. if the constant is equal to one, enter 1.\n\nlet:\n$x_{11}$ = number of components produced in plant 1 and supplied to store a\n$x_{12}$ = number of components produced in plant 1 and supplied to store b\n$x_{13}$ = number of components produced in plant 1 and supplied to store c\n$x_{21}$ = number of components produced in plant 2 and supplied to store a\n$x_{22}$ = number of components produced in plant 2 and supplied to store b\n$x_{23}$ = number of components produced in plant 2 and supplied to store c\n$x_{31}$ = number of components produced in plant 3 and supplied to store a\n$x_{32}$ = number of components produced in plant 3 and supplied to store b\n$x_{33}$ = number of components produced in plant 3 and supplied to store c\n\nmin $x_{11}$+ $x_{12}$+ $x_{13}$+ $x_{21}$+ $x_{22}$+ $x_{23}$+ $x_{31}$+ $x_{32}$+ $x_{33}$\n\nsubject to the constraints:\n $x_{11}$+ $x_{12}$+ $x_{13}$ select \n $x_{21}$+ $x_{22}$+ $x_{23}$ select \n $x_{31}$+ $x_{32}$+ $x_{33}$ select \n $x_{11}$+ $x_{21}$+ $x_{31}$ select \n $x_{12}$+ $x_{22}$+ $x_{32}$ select \n $x_{13}$+ $x_{23}$+ $x_{33}$ select \n$x_{11},x_{12},x_{13},x_{21},x_{22},x_{23},x_{31},x_{32},x_{33}geq0$\n\nwhat is the value of total distribution costs at the optimal solution? use the excel solver to find the optimal solution. round your answer to the nearest dollar.\n

Answer

Explanation:

Step1: Define the objective function

The objective is to minimize the total distribution cost. The cost - per - unit from each plant to each store is given. The objective function $Z$ is: $$Z = 7.30x_{11}+6.80x_{12}+6.10x_{13}+4.60x_{21}+9.00x_{22}+8.30x_{23}+8.10x_{31}+5.30x_{32}+5.80x_{33}$$

Step2: Define production capacity constraints

Plant 1 has a capacity of 610 units. So, $x_{11}+x_{12}+x_{13}\leq610$. Plant 2 has a capacity of 580 units. So, $x_{21}+x_{22}+x_{23}\leq580$. Plant 3 has a capacity of 230 units. So, $x_{31}+x_{32}+x_{33}\leq230$.

Step3: Define demand constraints

Store A has a demand of 430 units. So, $x_{11}+x_{21}+x_{31}=430$. Store B has a demand of 310 units. So, $x_{12}+x_{22}+x_{32}=310$. Store C has a demand of 620 units. So, $x_{13}+x_{23}+x_{33}=620$.

Step4: Non - negativity constraints

$x_{ij}\geq0$ for $i = 1,2,3$ and $j=1,2,3$.

To find the optimal solution, you would input the objective function and constraints into Excel Solver.

Answer:

The linear optimization model is: Minimize $Z = 7.30x_{11}+6.80x_{12}+6.10x_{13}+4.60x_{21}+9.00x_{22}+8.30x_{23}+8.10x_{31}+5.30x_{32}+5.80x_{33}$ Subject to: $x_{11}+x_{12}+x_{13}\leq610$ $x_{21}+x_{22}+x_{23}\leq580$ $x_{31}+x_{32}+x_{33}\leq230$ $x_{11}+x_{21}+x_{31}=430$ $x_{12}+x_{22}+x_{32}=310$ $x_{13}+x_{23}+x_{33}=620$ $x_{ij}\geq0$ for $i = 1,2,3$ and $j = 1,2,3$