Question

A manufacturer of clothing makes pants, shirts, and jackets. The profit on a pair of pants is $12.00, on a shirt is $8.00, and on a jacket is $18.00. Both pants, shirts and jackets require the work of sewing operators and cutters. There are 60 hours of sewing operator time and 48 hours of cutter time available. It takes 10 minutes to sew one pair of pants, 6 minutes to sew a shirt, and 14 minutes to sew one jacket. Cutters take 6 minutes on pants, 8 minutes on shirts, and 10 minutes on a jacket. Furthermore, each item should undergo a final quality control inspection. An inspection takes 1 minute on pants, 2 minutes on shirts, and 3 minutes on a jacket. There are 12 hours of quality control operator time available.
The problem is to find the maximum profit and the amount of pants, shirts and jackets to maximize the profit.
1) Write the linear programming model.
2) Find the optimal solution using Excel Solver.
Answer to the following questions without solving again the model:
3) What will be the impact of increasing the number of producing one additional shirt?
4) Assume that the new profit on a jacket is $30. Find the new optimal solution and profit.
5) Management is considering adding an extra 50 minutes labor time. Would you recommend adding this time to sewing, cutting, or quality control? (Provide a detailed justification of your recommendation).

Answer 1

First, we design the model:

Decision variables :

P = no. of pants to produce

S = no.of shirts to produce

J = no.of jackets to produce

Objective function :

Z (MAX) = 12P + 8S + 18J

Constraints :

Sewing operator 10P + 6S + 14J <= 3600

Cutter 6P + 8S + 10J <= 2880

Quality inspection 1P + 2S + 3J <= 720

Non negavtivity P,S,J>=0

Now, we put the model in Excel

Here, sumproduct is between the constraints/objective and the solution statement.

G11 (minutes) AVAILABILITY PANT SHIRT JACKET SUMPRODUCT SIGN 10 3 SEWING 4 CUTTER 5 QUALITY 3600 10 1 7 OBJECTIVE 12 18 8 SOLUTION

Now we go to DATA -- SOLVER :

Solver Parameters Home nsert Page Layout Formulas Dt Review View Connections Set Objectve: From From From From Other Existing Refresh Access Web Text Sources Sort Filter e, Max ConnectionsAll Edit Links Get External Data Connections Sort & Filter By Changing Variable Cels: PANT SHIRT JACKET Subject to the Constraints: (minutes) AVAILABILITY 3 SEWING 4 CUTTER 5 QUALITY SUMPRODUCT SIGN 14 10 10 0 < 0<= Delete 7 OBJECTIVE 8 SOLUTION 12 Reset A 10 Make Unconstrained Variables Non-Negative 12 Select a Solving Method: GRG Noninear Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for Inear Solver Problems, and select the Evolutonary engine for Solver problems that are non-smooth. 15 16 17 18 19 21

Click OK

Here is the result :

Boo Data Review View File Insert Page Layout Formulas Home Clea 的! Connections Al Reap Properties FrAdva Sort Filter From From From From Other Existing Access Web Text SourcesConnections All Refresh Edit Links Sort & Filter Connections Get External Data G11 (minutes) AVAILABILITY PANT SHIRT JACKET SUMPRODUCT SIGN 2 3 SEWING 4 CUTTER 5 QUALITY 3600 <= 2520仁 720仁 3600 10 10 1 4590 12 7 OBJECTIVE 8 SOLUTION 10 12

Sensitivity report is as follows:

File Home Insert Page Layout Formulas Data Review View 최FR 的! Connections Al Properties From From From From Other Existing Access Web Text SourcesConnections All Refresh Sort Filter Edit Links Get External Data Connections Sort & Filte A1 f Microsoft Excel 14.0 Sensitivity Report A B 1 Microsoft Excel 14.0 Sensitivity Report 2 Worksheet: [Book1]Sheet3 3 Report Created: 25-03-2019 12:49:50 4 6 Variable Cells Final Reduced Value Gradient 0 0.25 Cell 9 B$8 SOLUTION P Name 45 0 225 10 SC$8 SOLUTION S 11 ŞD$8 SOLUTION J 12 13 Constraints 14 15 Cell 16 ŞES3 SEWING SUMPRODUCT 3600 17 $ES4 CUTTER SUMPRODUCT 2520 18 ŞES5 QUALITY SUMPRODUCT 720 19 20 21 Final Lagrange Value Multiplier 1.125 0 0.75 Name

Now, we answer the questions from the information in the sensitivity report:

3) Producing an additional shirt will cost us $0.25. So, the profit will reduce.

4) If profit on Jacket is now $30. the optimal solution will remain the same. New profit will be: $7290

Old profit + additional profit on jacket i.e.(30-18)*225

= 4590 + 2700 = $7290

5) Extra minutes can be added to the sewing operator. It has the highest Lagrange Multiplier value. As we add 50minutes to sewing operator, the profit increases by 50minutes*1.125 = $56.25

New profit will be $4646.25