Comprehensive Exam Figure I illustrates the nctwork reprcsenting the distributio

Question

Comprehensive Exam Figure I illustrates the nctwork reprcsenting the distribution sysiem o f a lir using intermediate warclouses to distribute a product. Figure 1 Product Distribulion Systcm Planls Warehouses Cusloimers -3 -5 3 -4 4 -5 The firm has 2 plants, denotcd by A and B, 3 warehouses, denoted by X. Y, and Z. and-4 customer areas denoted by I, 2, 3, 4. The numbers adjacent to cach node denote the availability of material at that node. Plant A, for example, has 9 units available to be shipped. Customer 3, on the other hand, has -4 units. which means it requires 4 units to be shipped in. The number above each arc is the cost per unit shipped along that arc. For example, if 5 of plant A's 9 units are shipped to warehouse Y. then a cost of 5 x 2 10 will be incurred as a direct result. The problem is to determine the uimoutii stipped along cach arc so that total costs arc minimizcd and every customer has his requirements satisfied. Set up the problem as a linear program. Do not solve it. Define all variables used.

Explanation / Answer

Linear Program is following:

Let AX, AY, BX, BY, BZ, X1, X2, Y1, Y2, Y3, Z2, Z3, Z4 be the quantity shipped from respective plants to warehouse and warehouse to customers.

Objective: Min 1AX+2AY+3BX+1BY+2BZ+5X1+7X2+9Y1+6Y2+7Y3+8Z2+7Z3+4Z4

s.t.

AX+AY = 9

BX+BY+BZ = 8

AX+BX = X1+X2

AY+BY = Y1+Y2+Y3

BZ = Z2+Z3+Z4

X1+Y1 = 3

X2+Y2+Z2 = 5

Y3+Z3 = 4

Z4 = 5

All variables >= 0

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