Distributing Goods through a Distribution Network

The Problem. The DISTRIBUTION UNLIMITED CO$.$ will be producing the same

new product at two different factories, and then the product must be shipped to two warehouses, where either factory can supply either warehouse. The distribution network available for shipping this product is shown in Fig. $3\mathrm{.}13,$ where F$1$ and F$2$ are the two factories, W$1$ and W$2$ are the two warehouses, and DC is a distribution center. The amounts to be shipped from F$1$ and F$2$ are shown to their left, and the amounts to be received at W$1$ and W$2$ are shown to their right. Each arrow represents a feasible shipping lane. Thus, F$1$ can ship directly to W$1$ and has three possible routes $($F$1$ DC $W2,F1F2DCW2,$ and $F1W1W2)$ for shipping to W $2.$ Factory F $2$ has just one route to $W2(F2DCW2)$ and one to $W1(F2DCW2W1).$ The cost per unit shipped through each shipping lane is shown next to the arrow. Also shown next to $F1F2$ and $DCW2$ are the maximum amounts that can be shipped through these lanes. The other lanes have sufficient shipping capacity to handle everything these factories can send.

FIGURE $3\mathrm{.}13$

The distribution network for Distribution Unlimited Co$.$ supplied.

c$.$ Modify the formulated mathematical model for the example in section $6\mathrm{.}1,$ page $804$ to that of section $3\mathrm{.}4,$ page $89($Hint: enter the data from the formulated Minimum Cost flow problem on page $91$ into the appropriate cell in the spreadsheet and make modifications where necessary$).$

d$.$ Obtain the optimal solution for the minimum cost flow problem using Solver as

explained in the Watch item, Minimum Cost Flow Problem $-$ Excel Solver

Distribution Unlimited Co$.$ Minimum Cost Flow Problem excel provided