You are the general manager of a Soda Company (SC). There are three soda factori

Question

You are the general manager of a Soda Company (SC). There are three soda factories A, B, and C that require to be supplied at least 1750, 1620, and 2250 m3 of purified water in order to meet the market demand. There are 4 water treatment plants (TP) as potential suppliers of the purified water. The price of water purification in TP1 is 2.3 CAD/m3, in TP2 is 1.9 CAD/m3, in TP3 is 2.45 CAD/m3, and in TP4 is 2.13 CAD/m3. The cost of delivering each m3 of water in from water treatment plant to factories is listed in CAD in the following table. SF-A SC-B SC-C TP1 $0.54 $0.72 $0.61 TP2 $0.42 $0.62$0.51 TP3 $0.65 $0.47 $0.59 TP4 $0.77 $0.58 $0.66 Also, TP1, TP2, TP3 and TP4 have a daily limitation of treating 1680, 1920, 1340, and 1570 m3 of water, respectively. Formulate an LP that results in the least-cost strategy for your company to supply the factories demand with purified water. Clearly define decision variables (10 pts), objective function (5 pts), and constraints (10 pts)

Explanation / Answer

Let,

Pi = production cost (CAD/m3) of water purification at each TP i

Tij = Transportation cost for cubic meter of water transported from treatment plants i to Soda Company j   

Decision Variable:

Xij be the cubic meter of water to be transported from treatment plants i to Soda Company j,

Where,

i = 1,2,3,4 for TP1, TP2, TP3, and TP4 respectively

j = a, b, and c for SF-A, SF-B, and SF-C respectively

Objective Function:

Objective is to minimize the total production and transportation cost:

Min. Z = Pi(Tij x Xij)

Min. Z = $2.3($0.54X1a + $0.72X1b + $0.61X1c) + $1.9($0.42X2a + $0.62X2b + $0.51X2c) + $2.45($0.65X3a + $0.47X3b

+ $0.59X3c + $2.13($0.77X4a + $0.58X4b + $0.66X4c)

Subject to:

Supply Constraint:

Treatment Plant 1: X1a + X1b + X1c = 1680

Treatment Plant 2: X2a + X2b + X2c = 1920

Treatment Plant 3: X3a + X3b + X3c = 1340

Treatment Plant 4: X3a + X3b + X3c = 1570

Demand constraint:

Soda Company A: X1a + X2a + X3a + X4a <= 1750

Soda Company B: X1b + X2b + X3b + X4b <= 1620

Soda Company C: X1c + X2c + X3c + X4c <= 2250

Nonnegative Constraint: Xij>= 0

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