1. **Problem Statement:** (a) Define Operations Research (OR) and explain its growing importance. (b) Use the Northwest Corner Rule to find an initial feasible solution for the given transportation problem. (c) Determine the best alternative using the Maximin principle from the given payoff matrix. (d) Calculate earliest start, latest start, and total float for each task in the project. --- 2. **(a) Define Operations Research:** Operations Research is a discipline that deals with the application of advanced analytical methods to help make better decisions. It uses mathematical models, statistics, and algorithms to optimize complex systems. **Importance:** - Helps in resource allocation and scheduling. - Improves efficiency and reduces costs. - Supports decision-making in business, engineering, and other fields. --- 3. **(b) Northwest Corner Rule:** This method finds an initial feasible solution for transportation problems by starting allocation from the top-left (northwest) cell and moving right or down. Given: Supply: $P=4$, $Q=8$, $R=9$ Demand: $A=3$, $B=3$, $C=4$, $D=5$, $E=6$ Step-by-step allocation: - Allocate to $P-A$: min(4,3) = 3; update supply $P=4-3=1$, demand $A=3-3=0$ - Move right to $P-B$: min(1,3) = 1; update supply $P=1-1=0$, demand $B=3-1=2$ - Move down to $Q-B$: min(8,2) = 2; update supply $Q=8-2=6$, demand $B=2-2=0$ - Move right to $Q-C$: min(6,4) = 4; update supply $Q=6-4=2$, demand $C=4-4=0$ - Move right to $Q-D$: min(2,5) = 2; update supply $Q=2-2=0$, demand $D=5-2=3$ - Move down to $R-D$: min(9,3) = 3; update supply $R=9-3=6$, demand $D=3-3=0$ - Move right to $R-E$: min(6,6) = 6; update supply $R=6-6=0$, demand $E=6-6=0$ Initial feasible solution matrix: $$\begin{bmatrix} P & 3 & 1 & 0 & 0 & 0 \\ Q & 0 & 2 & 4 & 2 & 0 \\ R & 0 & 0 & 0 & 3 & 6 \end{bmatrix}$$ --- 4. **(c) Maximin Principle:** Choose the alternative whose minimum payoff is the maximum among all minimum payoffs. Calculate minimum payoffs: - $X$: min(8,0,-10,6) = -10 - $Y$: min(-4,12,18,-3) = -4 - $Z$: min(14,6,0,8) = 0 Maximum of these minimums is $0$ for alternative $Z$. **Decision:** Choose alternative $Z$. --- 5. **(d) Project Scheduling:** Tasks: A(15), B(17), C(21), D(19), E(22), F(18), G(15) Predecessors: B and C depend on A; D and E depend on B; F depends on C and D; G depends on E and F. Calculate Earliest Start (ES) and Earliest Finish (EF): - $A$: ES=0, EF=0+15=15 - $B$: ES=15, EF=15+17=32 - $C$: ES=15, EF=15+21=36 - $D$: ES=32, EF=32+19=51 - $E$: ES=32, EF=32+22=54 - $F$: ES=max(EF of C,D)=max(36,51)=51, EF=51+18=69 - $G$: ES=max(EF of E,F)=max(54,69)=69, EF=69+15=84 Calculate Latest Finish (LF) and Latest Start (LS) starting from project completion at 84: - $G$: LF=84, LS=84-15=69 - $F$: LF=LS of G=69, LS=69-18=51 - $E$: LF=LS of G=69, LS=69-22=47 - $D$: LF=LS of F=51, LS=51-19=32 - $C$: LF=LS of F=51, LS=51-21=30 - $B$: LF=min(LS of D,E)=min(32,47)=32, LS=32-17=15 - $A$: LF=min(LS of B,C)=min(15,30)=15, LS=15-15=0 Calculate Total Float (TF) = LS - ES: - $A$: 0-0=0 - $B$: 15-15=0 - $C$: 30-15=15 - $D$: 32-32=0 - $E$: 47-32=15 - $F$: 51-51=0 - $G$: 69-69=0 --- **Final answers:** - (a) OR defined and importance explained. - (b) Initial feasible solution by Northwest Corner Rule: $$\begin{bmatrix}3 & 1 & 0 & 0 & 0 \\ 0 & 2 & 4 & 2 & 0 \\ 0 & 0 & 0 & 3 & 6 \end{bmatrix}$$ - (c) Choose alternative $Z$ by Maximin principle. - (d) Earliest and latest times and total float calculated as above.