Range of Feasibility For phase 2 of manufacturing, the allowable increase is 40 (LINDO) Thus the dual price of $125 is valid for any increase in the allowable hours for phase 2, up to 960 + 40 = 1000 i.e. any increase n (<= 40) added to 960 will increase the objective value by $125n. Any increase x (> 40) added to 960 may or may not increase the objective value by $125x.

Example The range of feasibility for the phase 2 constraint is [800, 1000]. What is the value of objective function if the right-hand side is changed to: 1000: 125*40=5000  220000 + 5000 = 225000 860: 125*-100 = -12500 220000-12500 = 207500 700: don’t know

The 100% rule for constraint right-hand sides To determine whether simultaneous changes will not change dual prices For each constraint: Compute change as a percentage of the allowable change Sum all percentage changes If the sum is less than or equal to 100%, the dual prices will not change If the sum exceeds 100%, the dual prices may change

Consolidated Electronics As part of a quality improvement initiative, Consolidated Electronics employees complete a three-day training program on teaming and a two-day training program on problem solving. The manager of quality improvement has requested that at least 8 training programs on teaming and at least 10 training programs on problem solving be offered during the next six months. In addition, senior-level management has specified that at least 25 training programs must be offered during this period. Consolidated Electronics uses a consultant to teach the training programs. During the next six months, the consultant has up to 84 days of training time available. Each training program on teaming costs $10,000 and each training program on problem solving costs $8,000. Determine the number of training programs on each teaming and problem solving that minimizes cost.

Post-Optimality Analysis What will happen to the optimal solution if the cost of each training program on teaming is decreased to $9000? What will happen to the optimal solution if the cost of each training program on problem solving is increased to $9500? What if both changes occur simultaneously? What will be the new overall minimum cost if the minimum total number of courses is increased from 25 to 27? What will be the new overall minimum cost if the total number of days available for training is decreased from 84 to 64? 54? What will be the new overall minimum cost if the minimum total number of training courses on teaming is decreased from 8 to 5? What will be the new overall minimum cost if the minimum total number of training courses on teaming is decreased from 8 to 2 but the minimum total number of courses is increased from 25 to 26?

Integer Programming Each year CrossChek decides which lines of golf clubs and clothing it will market. Consider that each line of golf clubs is expected to generate $400K profit for the year, while each new line of golf clothing generates $300K profit for the year. Each new offering requires time for marketing (clubs 50 hrs, clothing 60 hrs), which is limited to 300 hrs, as well as storage space (clubs 900 sq ft, clothes 400 sq ft), which is limited to 3600 sq ft. CrossChek will not consider marketing any more than 4 lines of clothing. How many new lines of golf clubs and clothing should be offered?

Integer Programming Solution: 2.82 lines of golf clubs 2.65 lines of golf clothing

Integer Programming Solution: Could round down: Feasible. But optimal? 2.82 lines of golf clubs 2.65 lines of golf clothing Could round down: 2 lines of golf clubs 2 lines of golf clothing Profit = $1.4 million Feasible. But optimal?