$6\mathrm{.}13$ Case study $4$: Supplier Selection and Order Allocation

In this case study, you will solve a supplier order allocation problem with two products, two buyers, and two suppliers. The problem has three conflicting criteria, namely, total cost, lead$-$time, and quality $($measured by rejects$).$ All three objectives have to be minimized.

Minimize total cost, which consists of fixed cost and the variable cost. Fixed cost is a one$-$time cost that is incurred if a supplier is used irrespective of the number of units bought from that supplier.

Minimize the weighted average lead$-$time.

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Supply Chain Engineering: Models and Applications

$3.$ Minimize the weighted average quality.

For objectives $2$ and $3,$ you can use the total demand as the denominator for the weighted average expressions. The constraints in the model include capacity constraints of the supplier and the demand constraint of the buyers.

The data regarding the various model parameters are given in Tables $6\mathrm{.}43$ through $6\mathrm{.}46.$ The fixed costs are $$1200$ and $$1325$ for suppliers $1$ and $2,$ respectively.

Questions

$($a$)$ Formulate the supplier order allocation problem as a three criteria mixed integer linear programming $($MILP$)$ problem. You must define your variables clearly, write out the

TABLE $6\mathrm{.}43$

Product Demand $($Exercise $6\mathrm{.}13)$

$\backslash$table$[[$Product$,$Buyer,Demand$],[1,1,150],[1,2,175],[2,1,200],[2,2,180]]$

TABLE $6\mathrm{.}44$

Supplier Capacities $($Exercise $6\mathrm{.}13)$

$\backslash$table$[[$Product$,$Supplier,Capacity$],[1,1,300],[1,2,350],[2,1,280],[2,2,360]]$

TABLE $6\mathrm{.}45$

Product Prices $($Including Shipping Cost$)$ and the Lead$-$Time of Products in Days $($Exercise $6\mathrm{.}13)$

$\backslash$table$[[$Lead$-$Time of Products in Days $($Exercise $6\mathrm{.}13)],[$Product$,$Buyer,Supplier,Price Nnit,Lead$-$Time$],[1,1,1,$$$190,15],[1,1,2,175,17],[1,2,1,194,19],[1,2,2,200,18],[2,1,1,335,24],[2,1,2,360,21],[2,2,1,370,11],[2,2,2,365,12]]$