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ABC Corporation has three plants with production capacity and plans to begin a production of a product that can be made in three sizes – large, medium and small – that yield a net unit profit of $420, $360 and $300 respectively. Plants 1, 2 and 3 have capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved.

The amount of available storage space also imposes a limitation on the production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000 and 5,000 square feet respectively, of in-process storage space available for a day’s production of this product. Each unit of the large, medium and small sizes produced per day requires 20, 15 and 12 square feet, respectively.

Finally, management has decided that all three plants must use the same percentage of their capacity. For example, suppose Plant 1 uses half of its capacity (produces 375 units), Plant 2 and Plant 3 must also utilizes half of its capacity as well.

Managements wishes to know how much of each size should be produced by each plant to maximize profit.

Hint: You may define decision variables as x_ij = the amount produced of size j at factory i

Part A. Formulate (mathematically) an LP model that will maximize profit for ABC Corporation.

Part B. Use Excel Solver to find the optimal solution.

Answer 1

The decision variables can be denoted and defined as follows:

xP1L   = number of large units produced per day at Plant 1,

xP1M = number of medium units produced per day at Plant 1,

xP1S   = number of small units produced per day at Plant 1,

xP2L   = number of large units produced per day at Plant 2,

xP2M = number of medium units produced per day at Plant 2,

xP2S   = number of small units produced per day at Plant 2,

xP3L   = number of large units produced per day at Plant 3,

xP3M = number of medium units produced per day at Plant 3,

xP3S   = number of small units produced per day at Plant 3.

Also letting P (or Z) denote the total net profit per day, the linear programming model for this problem is

Maximize   P =    420 xP1L + 360 xP1M + 300 xP1S + 420 xP2L + 360 xP2M + 300 xP2S

+ 420 xP3L + 360 xP3M + 300 xP3S,

subject to

xP1L + xP1M + xP1S       750

xP2L + xP2M + xP2S       900

xP3L + xP3M + xP3S       450

20 xP1L + 15 xP1M + 12 xP1S       13000

20 xP2L + 15 xP2M + 12 xP2S       12000

20 xP3L + 15 xP3M + 12 xP3S         5000

xP1L + xP2L + xP3L           900

xP1M + xP2M + xP3M       1200

xP1S + xP2S + xP3S            750

( xP1L + xP1M + xP1S ) – ( xP2L + xP2M + xP2S ) = 0

( xP1L + xP1M + xP1S ) – ( xP3L + xP3M + xP3S ) = 0

and

xP1L  0, xP1M  0, xP1S  0, xP2L  0, xP2M  0, xP2S  0,

xP3L  0, xP3M  0, xP3S  0.

The above set of equality constraints also can include the following constraint:

= 0.

However, any one of the three equality constraints is redundant, so any one (say, this one) can be deleted.