Performance management and control - Essay Example

Performance management and control Introduction Linear programming isa controlled maximization, whereby the objective and constraints functions are all linear. They are helpful in choosing the optimum course of action (Henderson 253). A practical scenario will be used to describe leaner programming in this paper. A producer makes two types of soaps; Yellow soaps and blue soaps. He can sell up to 280 of the Yellow soaps and up to 400 of the blue soaps per day. Each of the yellow soaps occupies s space of 0.

01 m3 in the depot while each of the small soaps occupies 0.008m3 of the depot space. The capacity of the depot is 4m3. Every day, 8 hours are spent producing the soaps, and the producer can produce Yellow soaps at the rate of 40 per hour, and the Blue soaps at the rate of 80 per hour. Each of the Yellow soaps attracts a profit of $5 and $3 for the Blue soap. We are going to use linear programming technique to find out the number of Yellow and Blue soaps that should be produced in order to maximise the profits.

SolutionThe first step is to formulate the problemWe shall take Y to represent the number of Yellow soaps and B to represent the number of the Blue soaps. Therefore, the objective function is:X = 5Y+3B ProfitThe following are the constraints:Y≤ 280; which is the maximum number of Yellow soaps Line 1B ≤ 400; which is the maximum number of Blue soaps Line 210Y + 8B ≤ 4000; which is the depot space Line 325Y + 12.5B ≤ 8000; which is the hours Line 4 Y, B ≥ 0 Non-negativity function Using the graphical method, the functions are plotted as follows:The area ABCDEF is the viable region.

The optimum viable solutions are at the corner points. The maximum problem is to be taken as the upper points.At corner A, Y and Line 2 = 0 intersectAt corner point A, Line 2 and Y=0 overlap; therefore, Y= 400 The following is the result of putting these figures in the objective function:X = 5Y + 3B=5*0 + 3*400=1200; Profit AAt corner B, Line 2 and Line 3 overlap; therefore: Substituting these figures in the objective function produces the following equation:X = 5Y + 3B=5*80 + 3*400=1600 Profit BAt corner C, Line 3 and Line 4 overlapThe following functions can be used to represent these lines:Line 3 10Y + 8B = 4000 (1)Line 4 25Y + 12.

5B = 8000 (2)Multiplying function (1) by 25 and function (2) by 10 produces the following(10Y + 8B = 4000) *25(25Y + 12.5B = 8000) * 10250Y + 200 B= 100,000 (3) 250Y + 125B = 80,000 (4)Deducting function (4) from (3) gives:75Y =20,000⇒B =266.7⇒Y = 186.7 Combining these values in the objective equation leads to the following:X= 5Y + 3B=5 *186.7 + 3 * 266.7=1733 Profit CAt corner D, Line 1 and Line 4 overlap, Y= 280 The following equation represents the profit at this point:X = 5Y + 3B=5*280 + 3*80=1640 Profit DAt E, Line 1 and B = 0 overlapB = 0 ⇒ Y = 280z = 5Y + 3B=5*280 + 3*0=1400 Profit EDecisionOf all these profits, it is C where profit is maximized; hence 186 is the number of Yellow soaps produced and 266 is the number of Blue soap produced and 266.

7 is the maximum profit. Three ways in which the overall contribution to sales ratio could be improved1. The selling price can be increased; however, care should be taken so this strategy does not lead reduction in sales volume. If the selling price is increased without affecting the sales volume significantly, then the contribution sales ratio is increased directly.2. Reduction of variable costs per unit can lead to a better contribution sales ratio. This can be achieved by acquiring inputs at cheaper prices; for example by buying in large qualities in order to enjoy quantity discounts.3. An entity can concentrate on production of the product mix that promises the highest contribution in order to augment the contribution sales ratio.

Works CitedCui, Mark et al “Linear Programming Detection and Decoding for MIMO System”, IEEE Commun. Lett 9.5 (2006): 423-425. Print.Henderson, Charles. An Introduction to Linear Programming. New York: John Willy, 1993. Print.