Question

A corporation has decided to produce three new products. Five branch plants now have excess production capacity. The unit manufacturing cost of the first product would be $31, $29, $32, $28, and $29 in plants 1, 2, 3, 4, and 5 respectively. The unit manufacturing cost of the second product would be $45, $41, $46, $42, and $43 in plants 1, 2, 3, 4, and 5 respectively. The unit manufacturing cost of the third product would be $38, $35, and $40 in plants 1, 2, and 3 respectively. Plants 4 and 5 do not have the capability for producing this product. Sales forecasts indicate that 1500, 2500, and 2000 units of products 1, 2, and 3 respectively should be produced daily. Plants 1, 2, 3, 4, and 5 have the capacity to produce 2000, 1000, 2000, 1500, 2500 units daily respectively. Assume that any plant has the capability and capacity to produce any combination of the products in any quantity Management wishes to know how to allocate the new products to the plants to minimize total manufacturing cost.(a) Formulate this problem as a transportation problem by constructing the appropriate cost and requirements table.(b) Use the transportation simplex method to solve the problem as formulated in (a).

Do a) above only!!!                      LINEAR PROGRAMMING   - DO    a) ABOVE   ONLY

Answer 1

Total demand of three products = 1500+2500+2000 = 6000 units

Total capacity of five plants = 2000+1000+2000+1500+2500 = 9000 units

Total capacity exceeds the total demand, therefore, a dummy product with demand of 3000 units is added to make it a balanced transportation problem. The manufacturing cost for dummy product is $ 0

Plants 4 and 5 do not have the capacity to product 3, therefore, hypothetically large cost (say $ 999) is assigned to corresponding cells

Resulting Parameter table, showing costs and requirements, for transportation model is following:

	Product 1	Product 2	Product 3	Dummy	Capacity
Plant 1	31	45	38	0	2000
Plant 2	29	41	35	0	1000
Plant 3	32	46	40	0	2000
Plant 4	28	42	999	0	1500
Plant 5	29	43	999	0	2500
Requirement	1500	2500	2000	3000

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Linear Programming model is following:

Let Xij be the number of product i to be produced in plant j

Minimize 31X11+45X12+38X13+29X21+41X22+35X23+32X31+46X32+40X33+28X41+42X42+999X43+29X51+43X52+999X53

s.t.

X11+X12+X13+X14 = 2000

X21+x22+X23+X24 = 1000

X31+x32+X33+X34 = 2000

X41+X42+X43+X44 = 1500

X51+X52+X53+X54 = 2500

X11+X21+X31+X41+X51 = 1500

X12+X22+X32+X42+X52 = 2500

X13+X23+X33+X43+X53 = 2000

X14+X24+X34+X44+X54 = 3000

Xij >= 0