5. (10 pts) the cost of production and inventory. The production cost, demandand

ID: 389302 • Letter: 5

Question

5. (10 pts) the cost of production and inventory. The production cost, demandand production capacity vary fromquarterto quarter. The maximumamount ofinventory which can be held is 120 unitsand managementwantstokeep at least 60units on hand. Quarterly inventory holding cost is 4% of the cost of production. There are currently 50 units in inventory. The companywantsto produce at no less than one half of its maximum capacity in any quarter. Formulate an LP model. Swagelok is planning production forthenext 4 quarters. Theywantto minimize Quarter Units Demanded Maximum Production 100 150 Holding cost $2.2 150 150 $2 180 160 $2 120 130 $1.8 6. (15 pts) Fedexproviding shipping for a major U.S.based company fortheirnew product launch. Theircontract calls for a total of 250,000 tons of materialto be delivered overa three-week period. Fedexcan use multi modes of transportation:trucks, rairoads, and air cargo. Ther contract requires l 10,000 tons delivered by the end of week one, 70% of the totaldelivered by theend ofweek two, and the entire amount delivered bythe end of w eek three. Contra cts in pla ce with th e tra nsportation comp a nies call for at least 45% of the total delivered be delivered by trucking, at least 40% ofthe total delivered be delivered by railway, and up to 15% ofthe total delivered be delivered by ar cargo. Unfortunately,competingdemands limit the availability of each mode of transportation Week Trucking Limits Raiway Limits Air Cargo Limits 50 45 s per tons of the three weeks to the following levels (all in thousands oftons); Formulate an LP model.

Explanation / Answer

Q. 5:

Decision Variable:

Let,

Xi = units of item to be produced in ith quarter

Ei = units of item to be stored at the end of ith quarter

i = 0, 1, 2, 3, 4 quarters

Objective Function:

Let, Ci, and Hi be the cost of production and inventory holding cost for item in ith quarter respectively.

Objective is to minimize the production and holding cost:

Min Z = (Ci)(Xi) + (Hi)(Ei)

Min Z = $50X1 + $55X2 + $50X3 + $45X4 + $2E1 + $2.2E2 + $2E3 + $1.8E4

Subject To:

Production capacity of quarters i

X1 <= 150

X2 <= 150

X3 <= 160

X4 <= 130

Minimum utilization of capacity (at least 50% capacity should be used)

X1 >= 0.5*150 or X1 >= 75

X2 >= 75

X3 >= 80

X4 >= 65

Demand for item in quarter I (beginning inventory for quarter 1 is 50 units)

Xi + Ei-1 >= Di + Ei

Xi + Ei-1 – Ei >= Di

X1 + 50 – E1 >= 100

X2 + E1 – E2 >= 150

X3 + E2 – E3 >= 180

X4 + E3 – E4 >= 120

Storing limit for a given quarter

E1 <= 120

E2 <= 120

E3 <= 120

E4 <= 120

Minimum requirement of ending inventory

E1 >= 60

E2 >= 60

E3 >= 60

E4 >= 60

Non-negative constraint

All, Xi, Ei, >= 0