EM$222$ Deterministic Operations Research Case Study

EMU$27$ is a publisher that publishes and distributes kids' books. The books are published in three plants located in Ankara, $$stanbul$,$ and Eski$$ehir$.$ They are then sent to five wholesalers in Bursa, Antalya, Edirne, $$zmir$,$ and Zonguldak.

The online orders have increased in the last year, which in turn forced the company to open new distribution centers $($DCs$)$ to lower the transportation costs. The DCs will act as intermediate points between the plants and the wholesalers. The orders coming from the plants will be first collected there and then sent to the wholesalers. The company considers five possible locations for the DCs: A$,$ B$,$ C$,$ D$,$ and E$.$ The company is planning to open at most two of those DCs due to budgetary limitations$.$ Each location has a different fixed cost $($given in Table $1)$ that accounts for the monthly amortized cost of that distribution center. The distances between the plants and the DCs$,$ and the distances between the DCs and the wholesalers are given in Tables $2$ and $3,$ respectively.

Table $1.$ Fixed cost of operating distribution centers $($in TL$)$

$\backslash$table$[[$DC$,$Fixed Cost$],[$A$,140,000],[$B$,150,000],[$C$,220,000],[$D$,145,000],[$E$,160,000]]$

Table $2.$ Distance between the plants and the distribution centers $($in $km)$

$\backslash$table$[[$Plant$/$DC$,$A$,$B$,$C$,$D$,$E$],[$Ankara$,207,235,114,168,175],[$Istanbul$,206,344,281,391,312],[$Eskischir$,361,313,260,149,267]]$

Table $3.$ Distance between the distribution centers and the wholesalers $($in $km)$

$\backslash$table$[[$DC$/$Wholesaler$,$Bursa,Antalya,Edirne,$$mir$,$Zonguldak$],[$A$,158,162,164,175,380],[$B$,168,212,224,364,145],[$C$,145,180,200,251,215],[$D$,190,410,320,157,216],[$E$,175,198,148,287,120]]$

The books they publish can be classified in two groups: Early childhood books $($ECB$),$ and middle$/$late childhood books $($MLCB$).$ The books in each group are sold to the wholesalers in sets. The monthly demand of the wholesalers for each group $($in terms of sets$)$ are given in

Table $4.$ The plants have different capacities, and their maximum monthly supply amounts are given in Table $5.$

Table $4.$ Monthly demand of wholesalers $($in sets$)$

$\backslash$table$[[,$ECB,MLCB$],[$Bursa$,900,700],[$Antalya$,700,500],[$Edirne$,500,650],[$Izmir$,1200,1000],[$Zonguldak$,800,450]]$

Table $5.$ Monthly supply of the plants $($in sets$)$

$\backslash$table$[[,$ECB,MLCB$],[$Ankara$,1500,1300],[$stanbul$,1400,1200],[$Eskisehir$,1400,1000]]$

EMU$23$ is planning to determine a distribution plan that minimizes the total monthly cost of operating the facilities they own, while fulfilling the demand of the wholesalers. The transportation cost is proportional to the distance and the amount shipped. It is $2\mathrm{.}5TL$ per each ECB set per $km,$ and $1\mathrm{.}5TL$ per each MLCB set per $km$ between the plants and the DCs$,$ and between the DCs and the wholesalers. The DCs can store at most $2200$ sets of ECB and $2000$ sets of MLCB monthly.

Find a transportation plan as well as a recommendation on the DCs to open. Summarize your transportation plan in Tables.

Consider the following changes in the environment. Evaluate each change independently. How would you account for the change? $($You should write the changes in the mathematical model explicitly to get full grade from these parts.$)$ What are the new optimal solutions?

$($a$)$ The trucks used in transportation of ECBs from distribution center D to any wholesaler can carry at most $500$ sets. If more than $500$ sets are carried to any wholesaler, a larger vehicle needs to be rented for a cost of $3,000$ TL between distribution center D and that wholesaler. Note that for each wholesaler, a new vehicle should be rented. Include this change to your model and report the new optimal solution $($the DCs that are opened, and the transportation plan$).$

$($b$)$ The company is planning to make a contract with a carrier that makes discounts for sets transported from distribution center A to the wholesaler in Antalya. If the number of ECB sets transported from A to Antalya is more than $200$ units, the unit transportation cost becomes $0\mathrm{.}5TL$ instead of $2\mathrm{.}5TL$ for the units above $200.$ Include this change to your model and report the new optimal solution $($the DCs that are opened, and the transportation plan$).$