To develop a linear programming model for Johnson Industry’s bulldozer transportation problem, we need to minimize the total cost of moving bulldozers from Simei and Yishun to other locations.

Step 1: Define Variables

Let the decision variables represent the number of bulldozers moved from Simei and Yishun to each of the five other locations. Denote these as:

• $x_1$: Number of bulldozers moved from Simei to Bishan
• $x_2$: Number of bulldozers moved from Simei to Kallang
• $x_3$: Number of bulldozers moved from Simei to Jurong
• $x_4$: Number of bulldozers moved from Simei to Bedok
• $x_5$: Number of bulldozers moved from Simei to Tampines
• $y_1$: Number of bulldozers moved from Yishun to Bishan
• $y_2$: Number of bulldozers moved from Yishun to Kallang
• $y_3$: Number of bulldozers moved from Yishun to Jurong
• $y_4$: Number of bulldozers moved from Yishun to Bedok
• $y_5$: Number of bulldozers moved from Yishun to Tampines

Step 2: Objective Function

The objective is to minimize the total cost of transporting the bulldozers. The cost function is given by the sum of the costs per bulldozer for each route: $\text{Minimize} \ Z = 360x_1 + 400x_2 + 350x_3 + 460x_4 + 240x_5 + 330y_1 + 320y_2 + 560y_3 + 600y_4 + 380y_5$

Step 3: Constraints

We need to satisfy the following constraints:

• Each client (other than Simei and Yishun) must receive at least 10 bulldozers but ideally 18. This means that the total number of bulldozers sent to each location must satisfy:

  • $x_1 + y_1 \geq 10$ (Bishan)
  • $x_2 + y_2 \geq 10$ (Kallang)
  • $x_3 + y_3 \geq 10$ (Jurong)
  • $x_4 + y_4 \geq 10$ (Bedok)
  • $x_5 + y_5 \geq 10$ (Tampines)

• Additionally, the total number of bulldozers sent from Simei and Yishun cannot exceed their respective available excess bulldozers:

  • $x_1 + x_2 + x_3 + x_4 + x_5 \leq 32$ (Simei’s excess bulldozers)
  • $y_1 + y_2 + y_3 + y_4 + y_5 \leq 45$ (Yishun’s excess bulldozers)

• The decision variables must be non-negative integers, as fractional bulldozers cannot be moved.

Step 4: Linear Program

The linear programming model for this problem is as follows:

$\text{Minimize} \ Z = 360x_1 + 400x_2 + 350x_3 + 460x_4 + 240x_5 + 330y_1 + 320y_2 + 560y_3 + 600y_4 + 380y_5$

Subject to:

• $x_1 + y_1 \geq 10$
• $x_2 + y_2 \geq 10$
• $x_3 + y_3 \geq 10$
• $x_4 + y_4 \geq 10$
• $x_5 + y_5 \geq 10$
• $x_1 + x_2 + x_3 + x_4 + x_5 \leq 32$
• $y_1 + y_2 + y_3 + y_4 + y_5 \leq 45$
• $x_1, x_2, x_3, x_4, x_5, y_1, y_2, y_3, y_4, y_5 \geq 0$

This linear programming model can then be solved using Excel Solver or other optimization tools to minimize the transportation cost while satisfying all the constraints.

Would you like assistance in solving this model using Excel Solver, or do you need further details on any specific part of the problem?